THESEdeDOCTORATenCOTUTELLE
de I'UNIVERSITE PARIS VII et de L'INSTITUT de PHYSIQUE
de I'ACADEMIE des SCIENCES de LETTONIE
speciaiite: Physique des Liquides
presentee par
Sandris LAcISI.-
pour obtenir le grade DOCTEUR de l'UNIVERSITE PARIS VII et de L'INSTITUT de
PHYSIQUE de l'ACADEMIE des SCIENCES de LETTONIE
COMPORTEMENT D'UNE GOUTTE DE LIQUIDE MAGNETIQUE
DA.~SUN CHAMP MAGNETIQUE VARIABLE:
THEORIE ET SIMULATION
MAGNETISKA SKIDRUMA PILIENA DINAMlKA MAINIGOS,
MAGNETISKOS LAUKOS:
TEORIJA UN SKAITLISKA MODELESANA
BEHAVIOUR OF A MAGNETIC FLUID DROP
IN A TIME DEPENDENT MAGNETIC FIELD:
THEORY AND SIMULATION
English version
soutenue Ie 15 fevrier 1996 devant Ie jury compose de :
G. BOSSIS
A GAILITIS
J.-C. BACRI
A CEBERS
R GOLDSTEIN
Rapporteur
Rapporteur
S38. 95S -itCJ't (043)
6~ 1."3 a.(- "oil (IJ 1t!J
To my grandmother Marta Plikause
Acknowledgements
I am very much obliged to prof. Jaques Duran and prof. Alain Mauger for the
warm hospitality and the stimulating conditions for scientific work in Laboratoir
d'Acoustique et Optique de fa Matiere Condensee, University Paris 6. I am very
grateful to prof. Olgerts Lielausis for his support to my scientific work in Institute of
Physics, Latvia.
I am very much obliged to both my thesis-directors, prof. Jean-Claude Bacri
and prof. Andrejs Cebers. I have had an opportunity to profit from the doubled
experience of two directors as well as 1 have used non-ordinary chance to look on
physical phenomena from two viewpoints, joining together the theoretical and
experimental knowledge
The thesis will never be possible without the assistance of prof. Regine
Perzynski during both helpful discussions and everyday situations.
I would like to express my deepest gratitude to prof. George Bossis and prof. Agris
Gailitis, who have agreed to be "rapporteurs" of my thesis and thus have made the
enormous work in rather limited time
I am also very thankful to prof Raymond Goldstein who have found possible
to take care of my work being a member of the thesis jury
I feel it my pleasant duty to express my thankfulness to all the people of
AOMC. laboratory, who have stimulated my research I would like especially to
mention Urnberto dOrtona for his help during my first steps in UNIX enviroment.
The French version of the theses will never see the day-light without the
excellent work of Jean-Christophe Dabadie translating it from English.
And last but not least, I would like to thank my Latvian colleges, who have helped me
to keep connection to Latvia and to manage formalities in Latvia during my absence,
especially I wish to thank assocprof. Leonids Buligins and Ivars Drikis.
Contents
Introduction 9
Chapter 1
Physical properties and models of magnetic liquids 13
1.1. Magnetic fluids and their properties 13
1.1.1. General properties 13
1.1.2. Concentrated phase 18
1.2. Ferrohydrodynamical surface instabilities 20
1.2.1. Free ferro fluid droplet in a static field 20
1.2.2. Sessile ferrofluid droplet in a static field 21
1.2.3. Free ferrofluid surface instability in vertical field 21
1.2.4. Instabilities of ferrofluid in a narrow gap between two parallel plates 22
1.2.5. Instabilities of ferro fluid droplet in rotating field 23
1.3. Choice of the characteristical units 24
1.4. Boundary integral equations 25
1.4.1. Stokes flow 25
1.4.2. Boundary conditions for a velocity field 26
1.4.3. Boundary integral formulation for a creeping flow 27
1.4.4. Partial differential equations for a magnetic field 32
1.4.5. Boundary conditions for a magnetic field 33
5
1.4.6. Derivation of boundary integral equations 33
Direct formulation of boundary integral equations 33
Indirect formulation of boundary integral equations 35
Cone! usions 38
CHAPTER 2
Approximation of boundary integral equations 39
2.1. Interpolation of a planar boundary 39
2.2. General remarks about discretization of boundary integral equations 41
2.3. Equations for a magnetic field 43
Separation of the external field components .45
The shape periodicity of the droplet 45
The accuracy of the magnetic field calculations 46
2.4. Equations for the motion of a free boundary 48
The accuracy of the boundary velocity calculations 50
CHAPTER 3
High-frequency rotating magnetic field: the energy approach 57
3.1. Shape generation for energy calculations 57
Included paper. 63
Conclusions 85
6
CHAPTER 4
High-frequency rotating magnetic field: dynamic simulation 87
4.1. Theoretical predictions '" 88
4.2. n-lobe perturbation decrements in the case of creeping flow 90
4.3. Dynamical simulation of the 2D droplet in the high-frequency
rotating magnetic field 97
Included paper. 117
Conclusions 127
CHAPTERS
Magnetic fluid droplet in low-frequency rotating magnetic fields .. 129
Included papers 141
Conclusions 155
Chapter 6
Droplet in low-frequency elliptically polarised field: mode
locking and Devil' s staircase '" 157
Included paper 167
Conclusions 195
General Conclusions 197
References 203
Table of figures 213
7
Introduction
The behaviour of a ferrofluid droplet in a rotating field is quite intricate
including even transition to chaotic dynamics. If at present the behaviour of droplet in
static field is quite well understood both theoretically and experimentally then in
rotating field or even more in the field of more complicate character there are great
variety of rather complex phenomena for which at present moment adequate
theoretical understanding does not still exists. Existing theoretical approaches are
based on simple assumptions of ellipsoidal shapes of droplets as in 3D and 2D. More
complex shapes is possible to study enough thoroughly only by numerical means.
The goal of the present thesis work was to perform analytical and numerical
studies of a 2D ferrofluid droplet in a rotating magnetic field including elaboration of
the corresponding numerical tools. For that small perturbation theory is used in the
case of simple geometry to find the growth rate of perturbations and boundary element
method is used to solve the moving free boundary problem of droplet behaviour. In
the last case the satellite problem of the creeping flow and the magnetic field have to
be solved in the every time step.
In the first chapter the short overview about most important properties of
magnetic fluids and especially about the concentrated phase of a magnetic fluid is
given. The concentrated phase due to its abnormally high magnetic susceptibility and
low surface tension on the interface between two phases, leading to high values of a
magnetic Bond number, enables to observe many intrinsic phenomena, which are not
to observe in experiments with conventional ferrofluids. In the rest of this chapter the
low Reynolds number flow (called creeping flow) problem and the satellite problem
of a magnetic field solution are defined for a 2D droplet using boundary integral
equation formulation.
The second chapter contains approximation of boundary integral equations,
given in the first chapter. The boundary element method (BEM) accuracy is tested
using exact analytical solutions. The results of test display that BEM accuracy allows
9
Introduction
to use it to simulate numerically unsteady motion of a 2D droplet in an external
magnetic field. On the basis of given discrete algorithms the various computer codes
are elaborated to perform BEM simulation in the case of the magnetic field problem at
given geometry (chapter 3) and in the cases of coupled magnetic field and creeping
flow problems (chapters 4 and 5).
In the case of a high-frequency rotating magnetic field time averaging of surface
forces could be used if the rotation period is much smaller than characteristic
relaxation time for a droplet. In the third chapter the equilibrium figures of a 2D
droplet are studied from minimal energy viewpoint in high-frequency field. The
favourite shape with a minimal energy is found from an a priori given set of shapes at
a given magnetic field strength amplitude, using time-averaging for a magnetic
energy. Such an approach allows to detect the magnetic field threshold at which the
perturbation from the circular shape of a droplet appears, if a magnetic field strength
is increased from the zero value. The decrease of a magnetic field strength displays
the hysteresis effect, if magnetic permeability of the ferro fluid is high enough. The
magnetic field is calculated by BEM, families of shapes are generated by the special
technique described in this chapter.
The constraints of the droplet shape in the third chapter interfere in the obtaining
transitions between shapes with different numbers of spikes. In the fourth chapter the
solution of the whole free moving boundary problem of a 2D ferro fluid droplet by
BEM in the case of time-averaged surface forces provides information about the
evolution of shapes. It has been illustrated by direct numerical simulation that starting
from random initial state transitions to configurations with two and three spikes are
observed. The four spike shape appears to be unstable in 2D.
The fifth chapter contains results of a 2D droplet simulation in a low-frequency
rotating magnetic field. It has been shown that two different scenarios for a behaviour
of a droplet exist in dependence on the value of the magnetic Bond number. In the
case of large magnetic Bond number values there is the critical frequency for a
magnetic field rotation up to which droplet is able to follow the field rotation. Beyond
this critical frequency droplet is no more phase locked to the magnetic field and
exhibits rather intrinsic behaviour, peculiarities of which is analysed in more detail in
10
Introduction
the case of the more general elliptic field polarisation in the next, namely sixth
chapter. In the fifth chapter under the assumption of an elliptic shape of a droplet two
sets of the equations of motion are derived and used to obtain the characteristics of a
droplet behaviour in the fifth chapter. The comparison with the BEM shows fairly
well agreement. The BEM simulations display that the role of the bending instability
of a droplet increases, if the viscosity of a droplet is increased with respect to the one
of a surrounding fluid.
The last, namely sixth chapter is devoted the behaviour of a droplet in an
elliptical polarised rotating magnetic field. Elliptical polarisation of a field is the
source of the second modulation frequency, interaction of which with the frequency of
the droplet's rotation causes such effects as the mode-locking represented by the
devil's staircase, overlapping of mode-locking intervals, period doubling and the
transition to a chaos.
Throughout the present work the cgs system, and particularly the
electromagnetic cgs system of units, centred on gauss unit for a magnetic field is
used.
11
Chapter 1
Physical properties and models of
magnetic liquids
1.1. Magnetic fluids and their properties
1.1.1. General properties
The studies of magnetic fluids started in the early 1960s. Most of applications of
colloidal magnetic fluids (ferro fluids) were recognised slightly latter, when real
ferro fluids become available, because they are pure artificial product: they are not
found in nature. The name "ferrofluids" was proposed by R.E.Rosensweig [77], the
name magnetic fluids was used in [99] by M.I.Shliomis. Perhaps the name
"ferrohydrodynamics" (FHD) matches the topic of hydrodynamics of magnetic fluids
at best, since the name "magnetohydrodynamics" (MHD) is mostly used for
phenomena where the interaction between magnetic fields and electric currents in
fluids takes place. Nevertheless, it should be noted that the term MHD is still used as
more general, covering both topics, "classical" MHD and ferrohydrodynamics.
Exhausting overviews about magnetic fluids and their most important physical and
chemical properties are given in [22,25,91].
Ferrohydrodynamics studies the flow of magnetic fluid under the action of
strong forces of magnetic polarisation. The main difference between MHD and FHD
is that in MHD the body force acting on the fluid is the Lorentz force that arise when
electric current flows at an angle to the direction of a magnetic field. In FHD usually
there is no electric current flowing in the fluid. The body force in FHD is due to
13
Chapter 1
polarisation force, which in turn requires material magnetisation in the presence of
magnetic field gradients or discontinuities.
The uniqueness of ferrofluids is in giant magnetic response to magnetic field. As
a result many surprising phenomena are exhibited by the magnetic fluids in response
to magnetic fields:
• normal field instability (formation of a pattern of spikes on the fluid surface);
• the labyrinthine pattern formation in thin layers;
• the generation of body couple in rotary fields, which IS manifested as
antisymmetric stress;
• self levitation of an immersed magnet;
• enhanced convective cooling in ferrofluids having a temperature-dependent
magnetic moment;
• and many others.
During the last thirty years since ferrofluids become available, great variety of
applications of them in different branches are found. Actual commercial usage
presently includes [25,87,91]:
• novel zero leakage rotary shaft seals, used in computer disk drives [15];
• vacuum feedthroughs for semiconductor manufacturing [76];
• pressure seals for compressors and blowers [90];
• liquid-cooled loudspeakers that employ a ferro fluid to conduct heat away from the
speakers coils [23,56];
• piloting the path of a drop of ferrofluid in the body, to bring drugs to a target site
[75];
• non-invasive circulatory measurements of the blood flow [78,84];
• artificial high specific gravity effect applied to separate mixtures of industrial scrap
metals such as titanium, aluminium, and zinc [47,98];
• high-speed, inexpensive, silent printers. using magnetic fluid ink [72];
14
Chapter 1
Here just more known applications are mentioned, in fact this list IS still under
growing.
Exist several types of magnetic fluids, but the principal type of them is
"colloidal ferrofluid". A colloid is a suspension of finely divided particles in a
continuous medium, including suspensions that settle out slowly. However, a true
ferrofluid have not to settle out, even long exposure of a force field causes a slight
concentration gradient. A good ferro fluid [95] have to be a concentrated, stable
suspension of very small magnetic particles in a liquid. A surfactant normally
provides the colloid stability. Therefore the traditional ferrofluids are composed of
small (3-15 run), solid, single-domain magnetic particles coated with a molecular
layer of a dispersant and suspended in a liquid carrier. The difference of the ionic-
ferrofluids [73] from the surfactant-stabilised ones is that they are stabilised due to the
strongly screened electrostatic repulsion of ionic particles. Brownian motion keeps
particles suspended, and the coatings prevent the particles from sticking to each other.
Vander Waals attraction is one of the main problems to obtain colloidal particles that
are stable with respect to mutual agglomeration. To prevent particles from
approaching so close to one another that van der Waals attraction prevail, the steric
repulsion mechanism is used. For that particle surface is coated by long chain
molecules in such a way that the polar "heads" of chains are adsorbed by the particle
but the "tail" creates "elastic bumper" layer. According to the simplest model of a
stable magnetic fluid [22], a particle inside it is characterised by three dimensions,
namely, magnetic core diameter dm, solid diameter d., and coating diameter d.,
Quantities d, and d., are not the same because due to chemical interactions between
stabiliser and the particles, the surface layer of the particles may lose magnetic
properties.
A low evaporation rate, a low viscosity and chemical inertness are desirable
properties fore the base liquid. A ferrofluid remains flowable in the presence of
magnetic field even when magnetised to saturation. Nonetheless, the reology is
affected by the presence of a field [25,100].
The stability of the magnetic liquid IS ensured by the balance between the
various interparticle interactions:
15
Chapter 1
• magnetic dipole-dipole interaction,
• van der Waals interaction,
• dipole-magnetic field interaction,
• and
• either steric repulsion via the solvent particles coated with a surfactant in the
case of surfactant-stabilised ferrofluids when particles are coated with surfactant
chains,
• or strongly screened electrostatic repulsion for ionic particles in the case of ionic
colloids when magnetic particles are made also macro-ions in order to create
electrostatic repulsion.
Stability in a magnetic field gradient is the stability against settling of particles
in a field gradient due to an external magnetic source. Thermal motion of particles
should prevent the attraction of them to the higher intensity regions of a magnetic
source. Stability against segregation is favoured by a high ratio of the thermal energy
to the magnetic energy: ksT / A1HV::::: 1, here kB: Boltzmann constant, T: temperature,
M: magnetisation, H: magnetic field strength, V: volume of the particle. This relation
forms an upper limit for particle size. The related topic is the stability against
magnetic agglomeration. Due to collisions between particles, they could agglomerate
rapidly if particles adhere together. Here again the thermal energy is responsible to
separate the particles, i.e. to overcome dipole-dipole interaction, characterised by ratio
of thermal and dipole-dipole contact energy. The requirement for stability in a
gravitational field is usually lower than the requirements for stability in magnetic
field.
There are two magnetic relaxation mechanisms in ferrofluids: relaxation by
rotation if particle in the liquid and the relaxation due to rotation of the magnetic
vector within the particle. Solidified ferro fluid has only the second mechanism. The
particle rotation mechanism is characterised by a Brownian rotation diffusion time 1'B
having hydrodynamical origin [25,50,91] and given by 1's = 3Vllo / kT where V is the
particle volume and 110 the viscosity of the carrier liquid. The second relaxation has
so-called Neel relaxation time [25,50,91]
16
Chapter 1
r = _1 exp( KV)
B fo kT
where KV is the height of energy barrier which separates the two possible
magnetisation orientations inside the particle in absence of field for single-domain
uniaxial ferromagnetic particle.
The two main magnetic fluid preparation kinds are sketched below.
Preparation by size reduction.
Size reduction by grinding can succeed in reducing bulk (micron-size) material
to the order of lOoA in size. The method discovered by S.Papell [81], further
developed by many researchers (see [91D. Preparation of ferrofluid consists from
following steps 91:
• urn size magnetic powder is mixed with solvent and surfactant dispersing agent;
• size reduction in ball grind;
• centrifugation to separate oversize solids;
• correction of particle concentration.
Typical preparation time is about 1000h.
Preparation of ferrofluids by chemical precipitation
There are many chemical methods to prepare ferrofluids. Here just three more
representative of them are mentioned according to [91]:
• magnetite precipitation with steric stabilisation;
• cobalt particles in an organic carrier;
• charge-stabilised magnetite.
The chemical methods are fast typical preparation time are few hours, they can
be cheap, but they are restricted to a few particular compounds. Special fluids are
produced mainly by size reduction. Additional stabilisation effect is achieved by steric
repulsion mechanism due to the presence of long chain molecules absorbed onto the
particle surface.
17
Chapter 1
1.1.2. Concentrated phase
In the general purpose of technical applications require industrial products stable
in time, proof against temperature variations and magnetic field action. Nevertheless
in some particular cases it is possible to take advantage of the phase separation in
magnetic liquids. The concentrated phase could be obtained due to failure of stability
leading to a phase separation in two liquids of different particle concentrations:
droplets of concentrated phase grow among the more dilute phase [31]. A phase
separation could be induced by many factors, for example; temperature lowering, for
sterically stabilised particles, some variations of free surfactant concentration; for
electrostatically stabilised particles, an increase of ionic strength.
The concentrated phase causes interest due to its outstanding physical
properties: it was found in [11] that susceptibility XSI=J.l-l~40 (magnetic permeability
fl~40) and surface tension cr~5·10-7J·m-2=5_IQ-4erg-ern" for an ionic ferrofluid and in
[86] for a surfactant stabilised ferrofluid XSI=fl-I-80 (fl~80) and cr~3_10-7J.m-2=3-1O-
4 -2erg-cm .
These susceptibilities are about an order of magnitude higher than for usual
magnetic fluids, what allows to observe some phenomena which appears only in the
case of high susceptibility. Such extraordinary values of susceptibility are possible
due to a high packing of particles in a concentrated phase, additional effect caused by
a packing is large values of viscosity. In contrary, surface tension at the boundary of
the two phases is very low. Magnetisation curve for concentrated phase exhibits [31]:
• a linear regime with very high initial susceptibility in low field, what means that
the particles keep their rotational degrees of freedom,
• a very large saturation value (~ 90 kAlm=1130 Oe), what means a volume fraction
of magnetic particles close to the maximum reasonable value (-24%) for a
ferro suspension retaining its fluidity [94].
Hence in order to obtain concentrated phase, repulsive interaction should be
decreased. For ionic ferrofluids it is much simpler to realise since they are directly
synthesised as micro-ions [73], but for by surfactant stabilised magnetic liquids it is
very complicate to decrease repulsion.
18
Chapter 1
In the concentrated phase, distance between particles are determined only by the
solvate shells. The concentration of the concentrated phase is always the same and
about equal to 25 % in volume. Every particle is surrounded by a shell of structural
water of thickness 1 or 2 run, so all the water shells mix into a single one in the whole
phase. A liquid-gas like phase separation always causes that a concentrated phase
appears as droplets inside a more diluted phase.
19
Chapter I
1.2. Ferrohydrodynamical surface instabilities
1.2.1. Free ferrofluid droplet in a static field
A deformation of a ferrofluid droplet in a static field is studied experimentally in
[2,46] and theoretically under assumption of an ellipsoidal shape of a droplet [24,32]
in zero gravity. Very interesting results are obtained in experiments with a
concentrated phase [11,12,14]. In [11,14] under assumption of the prolate ellipsoidal
shape of the ferrofluid droplet the equilibrium shapes are described from the balance
between magnetic energy and interfacial tension energy. For magnetic permeability of
the droplet ~=40 it is found that the droplet becomes unstable for a certain magnetic
field threshold: it jumps from a slightly elongated shape to a much more elongated
one. The ratio of semi -axes of the ellipsoidal shape changes from a1b~2 to a1b~13. At
decrease of the field the droplet from the elongated shape a1b~7 reduces to a less
elongated one a/b- 1.5 at a smaller magnetic field threshold value. Such a transition
takes place only if magnetic permeability is high enough, namely if ~>20.8 [89]. In
[25,32] it is shown that the virial method and the energy approach gives equivalent
results. The advantage of the virial method is that it could be used to study
perturbations from the ellipsoidal shape. The virial method is already successfully
used to calculate equilibrium shapes for selfgravitating rotating mass [42,41] and for
rotating charged drops [40,88,89]. The similar results concerning the hysteresis
phenomenon for the elongation of a droplet could be obtained both from virial and
energy approach methods in 2D case. The critical value of a magnetic permeability for
the hysteresis of the 2D droplet deformation in a static magnetic field is ~cr=27. In
some theoretical studies [68,97] it is reported about a conical tip formation for a
ferrofluid droplet and the eventual breakup of it, if a magnetic permeability exceeds
critical value ~=17.59 [68]. Nevertheless, such phenomena was never observed
experimentally for a ferro fluid droplet even for the concentrated phase with ~=40
[11,12,14].
20
Chapter 1
1.2.2. Sessile ferrofluid droplet in a static field
Equilibrium axisymmetric shapes of sessile drops of ferro fluid are studied in
[19] experimentally. In [21] on the basis of the energy balance the elongation of the
droplet is calculated as function of a magnetic Bond number under an assumption that
the shape of a sessile droplet could be described as a half of an axisymmetric
ellipsoid. In [96] the equilibrium state of partially and totally free ferrofluid droplet is
studied by energy minimisation accounting for the gravity. It is found that the
hysteresis phenomena take place if gravitational Bond number Bg = pg(VY/3 /cr IS
larger than critical value BgcR~8.5, P being the density, g : gravity acceleration, V:
volume of a drop, cr : surface tension. It means that for larger gravitational Bond
number values there is some threshold value of magnetic Bond number Bm, beyond
which drop elongates with jump. Decreasing a magnetic field, the jump back to the
less elongated state takes place at smaller value of Bm. The influence of nonlinear
magnetisation of the sessile drop is studied in [17,27,29]. The comparison with the
linear magnetisation shows [27,29] that at low field values both results are very close,
but increasing field above 140 gauss the difference increases and in particular case in
[27] even at a magnetic field 160 gauss the difference in the drop height is about 1.6
times. It should be mentioned that agreement with experimental observations is within
2%, thus it is proved that the nonlinear behaviour of ferrofluids can significantly
affect a drop shape at field strength as low as 140 gauss.
1.2.3. Free ferrofluid surface instability in vertical field
The ferrofluid surface instability in vertical field is one of the most studied
problems of FHD. The first observation and predictions in the frame of the linear
perturbation theory are given in [27]. By theoretical studies many phenomena were
found out: subcritical character of hexagonal pattern structure [51,52,62], transition
from hexagonal pattern to square one [52,62]. Most of predictions are proved
experimentally [13,28]. In [13] the large hysteresis effect due to the high magnetic
permeability /1-40 was observed. Interpretation of the hysteresis on the basis of a
simple model is in reasonably good agreement with experimental data. Periodic
pattern formation in normal field is observed in [19] for thin ferrofluid film. In [83]
21
Chapter I
for hexagonal ferrofluid bubble lattice at some field value abrupt increase of the
magnetic field leads to the following: large initial ferro fluid bubbles droplets shrink
and smaller bubbles appear, together forming inhomogeneous patterns.
1.2.4. Instabilities of ferrofluid in a narrow gap between two parallel
plates
The thin gap between two parallel plates forms the quasi-twa-dimensional
domain. The fluid flow between plates could be described by Darcy equation.
Presence of two fluid phases in the gap one of which is ferrofluid makes a system
where many rather complicate phenomena are observed. In many cases instabilities
lead to non-axisymmetric shapes. Presence of different instabilities in orthogonal with
respect to a layer magnetic field:
• elliptic deformation instability for ferrofluid drops and bubbles III ferrofluid
[36,38];
• bending instability [36,38];
• comblike instability in a vertical gap [37];
leads to the labyrinthine instability [38,92]. Instability of the vertex splitting has been
numerically studied in [34]. The character of these instabilities depends on the
magnetic field magnitude and on the conditions, how a field is applied. Dependence of
the wave number of growing perturbations on ramp rate has been considered in [58].
Here also equivalence of approaches based on fictitious magnetic charges and current
loops around magnetised body has been shown.
Instabilities of the flat layer are quite well studied by numerical simulation. The
magnetic field problem is mostly treated accounting for demagnetising field in the
first nonvanishing approximation (constant magnetisation approximation approach).
Along the same lines the behaviour of the droplet in the field along the boundaries of
the layer has been considered in [20,49]. Fluid flow is approximated by a Hele-Shaw
flow and treated by the boundary integral technique for a flow potential [39]. In [67]
the surface evolution is traced by the dissipative motion of closed 2D curves,
conserving area. The formalism is derived from a general energy functional. The
conformal mapping method for the simulation of the labyrinthine instabilities of
22
Chapter 1
magnetic fluids has been considered in [58]. Computational limitations in most of
cases restrict the studies to the simple patterns of labyrinth structure.
1.2.5. Instabilities of ferrofluid droplet in rotating field
For the first time the response of magnetic fluid micro drops to time dependent
magnetic fields is tested in [10,4]. The concentrated phase of ferrofluid with relatively
high magnetic permeability fl~25 is used. The first stage of shape deformation is an
oblate ellipsoid, the plane of symmetry is one of field rotation. Increasing a magnetic
field, oblate ellipsoid becomes unstable, leading to rather complicate shapes. In a high
frequency rotating field a "starfish shape" for a droplets is observed. The droplet
slowly rotates in the direction of a field, number of arms is proportional to the square
of applied field strength. For lower field frequencies shapes could be more irregular,
including loop-like and worm-like ones. Generally droplet tries to follow a field
rotation, but additional bending effects take place due to friction forces acting from an
external fluid. The ratio of the droplet viscosity versus the external fluid viscosity is
about 100.
Similar results are observed In crossed field: Hx = Hoi J2 ,
Hy = H, sin rot (static field crossed with linearly oscillating one) [10]. As in the case
of a rotating field, the ultimate shape is a disk crowned with peaks. Difference is that
there is no co-rotation of a "cog-wheel". Number of lateral peaks is still proportional
to the square of the field.
By virial method [10,4,35] the symmetry destroying perturbation analysis is
performed for oblate ellipsoid. The number of lateral peaks is explained by the most
unstable perturbation mode, that is found by dispersion equation for surface waves of
flat magnetic fluid surface and by lateral instability of a infinite ferrofluid cylinder.
Still unexplained is the behaviour of the microdroplet at finite surface perturbations,
including slowly rotating distorted worms and reptating snakes, which behaviour
strongly depends on frequency of the magnetic field.
23
Chapter J
1.3. Choice of the characteristical units
In order to obtain non-dimensional form for all equations, used to perform
numerical calculations, some set of characteristical units should be chosen. As
characteristic values are chosen: external field strength Ho, surface tension a, external
fluid viscosity TJexand unperturbed droplet (i.e. circle) radius R. Thus all the processes
are ruled by some non-dimensional numbers, characterising the balance between these
forces. The acting forces in the present work are magnetic forces, surface tension and
viscous stresses. Since surface tension is always present, it is chosen as reference
force. The magnetic forces are characterised by the ratio between the magnetic forces
and surface tension ones, called the magnetic Bond number is defined as
Bm = H~R/O". It is sometimes convenient to incorporate in Bm some multiplier,
containing magnetic permeability f.l mostly because it better corresponds to the nature
of magnetic forces. Here, in the present work it is avoided because usually it helps not
completely to eliminate /l from all equations, governing the certain phenomenon. The
other goal is that the same defmition is kept for all the processes described in the
present work.
In time dependent processes, the characteristic decay time for free surface
perturbations T=TJexR/cris introduced, in fact, this dimensional value characterises the
ratio of viscous forces and surface tension ones. The viscosity of the surrounding fluid
is chosen as the reference one, because the friction forces acting to the rotating droplet
are governed mostly by it. The ratio of viscosities A=TJin/TJexis used to characterise the
system droplet-surrounding media.
These definitions lead to the following definitions of non-dimensional
magnitudes:
• non-dimensional coordinates X=xdimlR,y=y dim/R,
• a non-dimensional velocity v=vdim'TIR,
• a non-dimensional magnetic field H=Hdim/HO'
• a non-dimensional time t=tdimlt,
• a non-dimensional angular frequency Q=Qdim'T,
• a non-dimensional surface force density fs=fs,dim'R!cr.
24
Chapter 1
1.4.Boundary integral equations
1.4.1. Stokes flow
In general case incompressible fluid motion IS described by the momentum
conservation law
d v , ~ acrijp-- = L.,-- + F;, i= 1,2,3,
dt )=\ ax)
and by the continuity equation
(1.1)
div v = o. ( 1. 2 )
Here p denotes the density of the fluid, F, are the components of the volume forces,
acting in the fluid, and dldt refers to the complete derivative, which is defined as
aj 0 t + v .V . The viscous stress tensor ()ij has property of symmetry
cr ij = cit > i= 1,2,3. ( 1.3 )
The law for material properties of a media in the present case is Stokes law
[54,55,66], which for a viscous fluid establishes a connection between stress tensor on
the one hand and the pressure p and the rate-of-deformation tensor
E .. = ~[OVi + av)]
u 2 Ox) Ox;
on the other. Stokes law is of the form
()ij=_P'6ij+T][ov;+OVi], i,j=1,2,3, (1.4)Ox) oXj
T] being the coefficient of dynamic viscosity, connected with the coefficient of
kinematic viscosity v by relation T]=PV. The substitution of (1.4) into the momentum
conservation law (1.1) accounting for the continuity equation (1.2) gives the Navier-
Stokes equation
(1.5)
Dealing with slow viscous flows (so-called creepmg flow [55], Reynolds
number Re = vL pjT] « 1, L: the characteristic dimension of a flow, v: the
characteristic velocity) Navier-Stokes equation could be linearized, that is, non-linear
25
Chapter J
term could be neglected since it is small compared with remaining terms. Other
important simplification could be applied to Navier-Stokes equation when dealing
Ovwith motions of a droplet of small size. In such a case inertial term p - couldat
although be neglected. What results is the Stokes equations for creeping flow,
describing the slow steady flow of a viscous fluid:
-Vp + ll~v + pF = 0,
divv = 0.
What have to be understood by
( 1 .6)
the term "steady flow" in the dynamics of
creeping flow for moving boundaries? Small shift of boundary causes velocity field in
all the domain, but in the same instant this flow is slowed down due to the relatively
large viscosity as a shift velocity is small. If volume forces are present, they are
compensated by pressure and viscous stresses, influence of inertial terms is negligible
as viscosity stresses dominate over inertia effects. The action of potential volume
forces appears as additional pressure on surface. Every instant viscous flow stresses
are compensated only by a pressure and volume forces. The known velocities of the
fluid at boundaries of domains completely determine the flow inside the domains at
given volume forces. Thus inside one fixed domain the problem of time dependent
creeping flow at fixed time instant is equivalent to the steady flow inside the domain
with given volume forces and given velocities at the boundary of this domain.
Since all the flow is driven by surface forces, the boundary integral equation
technique turns out to be very powerful tool to solve the creeping flow problems with
free moving boundaries.
1.4.2. Boundary conditions for a velocity field
The boundary conditions usually are split in kinematic ones and dynamic ones.
The kinematic boundary conditions describes the velocity behaviour at the boundary,
usually it is the continuity of a velocity field [66]:
ill 1 ex IV = V •r r
In particular cases velocity could have prescribed values at the boundary.
( 1 .7)
26
Chapter 1
The dynamic boundary condition expresses the influence of surface forces llf
to the motion of the fluid [66]:
cr~njlr -cr~njlr = ll/;.
H ~ ~ mod .ere stress tensor cr ij = -u ij p + cr ij consists
(1.8)
from the pressure term and from
viscous stresses. In general the discontinuity in the surface force llf is dependent upon
physical properties of fluids as well as upon the structure and thermodynamic
properties of the interface. These thermodynamic properties could involve a number
of physical constants [54,85], including the densities of the fluids, surface tension,
surface elasticity, surface viscosity etc. An interface is called active one if llf "* O. In
the present work the flow of fluids is caused by the surface tension and the magnetic
forces acting to the interface between magnetic liquid droplet and the surrounding
non-magnetic fluid. Thus the boundary condition (1.8) in absence of tangential surface
forces could be written separately for tangent component of viscous stresses:
i/ll exl 0crtn r -cr'/I r = . (1.9)
and for normal ones:
in in ex ex 2 (M)2 / R-p +cr,11l = -p +cr,11l + 1[ n -cr c. (1.10)
Here Rc stays for local curvature radius, which in 2D could be expressed in terms of a
normal vector in the following way: c] Rc = o div n [48].
1.4.3. Boundary integral formulation for a creeping flow
Stokes flow, governed by equations (1.6) has the following properties:
1. The reversibility of Stokes flow: reversing signs for velocity v and pressure p,
reversed flow is mathematically acceptable and physically viable solution [55, 85].
It should be noted that the direction of the force and torque acting on any surface
are also reversed.
2. Uniqueness of solution.
3. Stokes flow accumulates no kinetic energy since the fluid possesses no momentum
and hence, no inertial mass.
27
Chapter 1
4. Rate of viscous dissipation in Stokes flow is lower than that is in any other
incompressible flow that has same boundary values ofthe velocity [55]
Beside that the "Stokes paradox" [55] should be avoided for 2D external flows. The
essence of the Stokes paradox [64] is that a velocity solution of the homogeneous
system (1.6) which is equal to a on boundary S and to given v'" at infinity generally
does not exist. Thus, for 2D streaming motion perpendicular to the axis of a circular
cylinder there exist no solution of the creeping motion equations vanishing on the
cylinder that tends to infinity.
In order to write the boundary integral equations for some vector field, the
corresponding fundamental solutions should be found at first. The fundamental
solutions of a creeping flow were introduced in fact independently by Odqvist [79]
and Lichtenstein [69]. They have constructed the corresponding hydrodynamical
potentials and investigated their properties, and used them to solve the problems of a
creeping flow. The general overviews of the fundamental solution of a creeping flow
could be found in [64,85].
In the present work the boundary integral equations are used to describe the
motion of the 2D magnetic fluid droplet under the action of the external magnetic
field. The boundary integral equation formulation of a Stokes flow is based on 2D free
space Green's functions. In 2D the free space Green's functions of the Stokes flow
represents solutions of continuity equation Vv = a and the singularity forced equation
of Stokes flow [64,85]
-Vp + l1V2V + g<5(r - ro)= a ( 1. 11)
where g is an arbitrary constant vector, ro is an arbitrary point, and <5is the 2D delta-
function, V2 is 2D Laplacian operator. Introducing the free space Green's function G
we wrote the solution for the velocity in the form
1vj(r)= -Gij(r,ro)gJ. (1.12)41t11
Physically, formula (1.8) expresses the 2D flow due to a 2D point force g<5(r -ro)'
Let us introduce the fundamental solutions for the vorticity, pressure, and stress fields:
(1.13)
28
Chapter 1
1p(r) = -Pj(r,ro)gj'4n
1cr ik(r) = 4n Tijk(r, ro )gj"
Stress tensor T is defined by Stokes law [54,66]
(1.14)
(1.15)
(,)_ (,) 8Gij(r,ro) 8Gkj(r,rO)Tijk\.r,rO - -8;kPj\.r,ro +----+----.Oxk Ox;
Pressure vector P and stress tensor T associated
(1.16)
with a free space Green's
function for infinite unbounded or bounded flow, constitute two fundamental
solutions of Stokes flow.
The free-space Green's function for velocity or 2D Stokeslet [54,85] is
x.x .
Gij = -8 ij In r + ~ ' r = lxi, x = r - To • ( 1 . 1 7)r
The associated vorticity, pressure and stress fields are given by (1.13), (1.14), and
(1.15) with
(1.18)
x~=2--f,
r
(1.19)
X.XXkTijk = -4 I .~ • ( 1 . 2 0 )r
For convenience potential volume forces could be incorporated into a modified
pressure. For example, the magnetic volume force ~ V(MH), incorporated in a2
modified pressure under assumption of a constant magnetic permeability gives
mod _ fl-1 nliH2P -P---v\: .8n
Thus the body-foree-free Stokes equation written in terms of pmodis to be considered:
(1.21)
-v»:: + llV2V = O.
The use of (1.21) changes the boundary condition (1.10) to the following form [25]:
_pmOdJII + o".= -rr: + o " + 2rr(Mn)2 + (MH)(2 - rr div n .
1IJ1 nn (1.22)
Further in the text the superscript "mod" for a pressure is dropped, additional
pressure due to a magnetic field appears in boundary conditions.
29
Chapter 1
To describe such a motion, singularities (singular forces) of Green's ftmctions
should be placed on boundaries. So the motion inside the droplet at the point ro is
described by the boundary integral equation, which accounts for forces to acting to
the droplet from the interface of it:
v~'(ro)= ~1j/"(r)Gij(r,ro)1i(r)-_1 1v;(r)Tijk(r,rO)nk(r)di(r). (1.23)
4n11 L 4n L
Here v; (r) stays for the i-th velocity component on the boundary, 11in: the viscosity of
the droplet. Due to the continuity of velocities on the boundary this value is the same
from inside and outside of the droplet. It is convenient to derive another boundary
integral equation from the Lorentz reciprocal identity [55,70], what arrives [85] at
ft;ex (r)G ij(r, <}:il(r )-l1"X 1v, (r )T,ik(r, rO)nk (r )ii(r) = o. ( 1 . 2 4 )
L L
Here A=l1in/11ex, 11ex: the viscosity ofa fluid outside the droplet, fX are the forces acting
to the fluid outside the droplet from the interface of it. Now by combining (1.23) and
(1.24) the boundary integral equation in terms of the surface force ~f = f " - fin
could be written:
v';'(ro) = - 4n~;11 1~.t:(r)Gii(r,ro)1l(r)+
t.
I - A 1 v , (rv; (r, ro)n k (r 'yil (r)4nA L
According to the (1.8) and (1.22) the surface force discontinuity for a linearly
(1.25)
magnetizable ferrofluid droplet is
6.f = c di v n - ~ - 1 ~Il- IXHn y + H 2 ( 1. 2 6 )8n r
In order to obtain the boundary integral equation for velocity on the boundary,
the principal value (PV) of the second right hand side integral in (1.25) is derived as
follows: if L is a Lyapunov line, i.e. it has a continuously varying normal vector, and
the velocity v varies over L in a continuous manner,
PV
lim 1 v;(r)Tljk(r,rO)nk(r'yii(r)=
rn -40 L t.
PV
+2n v j (r) + 1v; (rv; (r, r,)nk (r 'yii(r)
L
(1.27)
30
Chapter 1
where minus sign applies when the point ro approaches L from the internal side, and
the plus sign otherwise. The normal vector is pointed outside from the droplet (see
Fig.l.l). The point rL lays on the boundary contour L and corresponds to the limit
position of the point ro when it tends to L. The superscript PV for a boundary integral
here and further indicates the principal value of the double-layer potential, defined as
the value of the improper double-layer integral corresponding to the case where the
point ro is located on L. Clearly, the double-layer potential undergoes a discontinuity
4nv across L.
Thus use of (1.27) in (1.25) gives the necessary boundary integral equation to
calculate the velocity v(r) on the boundary:
vj(rJ=- ,1 f4f: (r)G'i (r, ro)d/(r) +2ml'x(1+A) I. .
1- A PV
-- J v,(r~k(r,rJnk(x)dI(r)21t(1+A) 1 j
In (1.28) the only unknown is velocity values along the boundary (surface of a 2D
(1.28)
droplet). It will be noted that when the viscosities of the two fluids are equal, i.e. ).~=1,
the coefficients of the double-layer integral on the right-hand side of (1.28) vanish,
and the flow is expressed merely in terms of a single-layer potential with known
density ~f. Two other particular cases, namely A=O and A=OO, requires special
treatment, called the regularization of the double-layer potential. This treatment IS
discussed in details in [64,85], in the present work these cases are not to meet.
The analogue of the equation (1.25) for the velocities outside the droplet is
v:X(ro)= - 41t~eX 1~f(rY:;u(r,ro)d/(r)+
1- A J- ':f Vi (r )TUk (r, rO)nk (r)dl(r)
41t L
The limit ro~L arrives at the same equation (1.28).
(1.29)
So the equation (1.28) is used to obtain the velocity distribution along the
droplet's boundary, once it is calculated, (1.25) and (1.29) could be used to obtain the
velocity distribution in every point of internal and external domains.
31
Chapter I
1.4.4. Partial differential equations for a magnetic field
The equations for a magnetic field follow from the Maxwell's equations. A
magnetic field satisfies the magnetic flux continuity equation
divB = 0 ( 1 .30 )
and the second equation in the case when electric and displacement currents are absent
IS
rotH = 0 ( 1 . 3 1 )
Total magnetic field H is formed by given external magnetic field H, which causes
secondary field H M due to magnetisation of magnetic fluid domain: H = H, + H M .
Throughout the present work a linear magnetisation of magnetic fluid is
assumed:
M = XH (1.32)
where magnetic susceptibility X and therefore magnetic permeability fl = 47tX + 1 are
constant. Thus from relation B = flH it follows that in domains of a continuously
constant magnetic permeability fl magnetic field intensity H satisfies equation
divH = 0,
what leads to Laplace equation for magnetostatic potential:
(1.33)
~qJ = 0 ( 1. 3 4 )
Thus magnetic field intensity has both solenoidal (1.33) and potential (1.31)
character because it is created "from outside" by an external field, magnetisation of
local ferrofluid domains is taken into account by appropriate boundary conditions. It
is convenient according to Fig.l.l denote a magnetostatic potential inside the droplet
by \(11, but a potential outside the droplet denoted by \(1'2 is split in a potential Hor of
an external field and a potential \(12 due to a magnetisation of the droplet:
(1.35)
32
Chapter 1
1.4.5. Boundary conditions for a magnetic field
The boundary between a magnetic fluid (subscript "1") and a surrounding non-
magnetic media (subscript "2") has not special magnetic surface properties, hence the
boundary conditions are conventional, straight-forward following from (1.30), (1.31):
Bl/1 = Bl/2 or ~Hl/l = Hl/2
H,l = H'2·
Boundary conditions for potentials on droplet surface are
(1.36a)
(1.36b)
(1.36c)
(1.36d)
1.4.6. Derivation of boundary integral equations
Directformulation of boundary integral equations
Since surface force formula for a viscous flow (1.22) requires to know tangential
and normal components of a magnetic field, it is convenient to write boundary
integral equations for Hn, HI on the surface inside the ferrofluid droplet. The full
derivation of equations is given in [33].
hird Green's identity in 2D [61]
(1.37)
R(rL,r)= IrL - rl
is a base on which expressions for the potentials \II I' \112, Hor are obtained, taking into
account direction of a normal and identity (1.34):
(1.38)
33
Chapter I
1 PV[ 1 0\v2(r) a ( 1 J]\jJ2 (rJ = - - f In (, ) -- - \jJ2 (r)- In (, ) dl ,
7t L R\rL>r an an R\rL,r
1 jV[ 1 a(Hor) a ( 1 Jl(HOrL)=- J In (,) - (HorJ- In (, ) dl.rt L R\rL>r an an R\rL,r
Subscript "L" is used to denote "observation point" on boundary contour L, for which
(1.39)
(1.40)
equations are written, r without any subscript usually denotes integration coordinate.
An application of the boundary conditions for magneto static potentials (1.36c),
(1.36d) to the following combination of equations (1.38), (l.40):
1l\jJ1 (rJ+ \jJ2(rJ- (HOrL)=
~~Vln 1 [ll a\jJl(r)_a\lf2(rJ_ a(HorL)]dl_
7t L R(rL,r) an an an
1 PV a [ I J- f [ll\lfI(r)-\lf2(r)-(HorJ]- In ~ ) dl
7t L an R rL,r
allows to obtain after simplifications the boundary integral equation in terms of only
one potential \If I:
2 1 11 -1 PV a [ 1 J\jJJ(rJ=-(HorJ--- f \lfl(r)- In ( ) dl.11+ 1 7t fl + 1 L an R rL , r
Derivative along the boundary contour a/ aIL gives a boundary integral equation
(1.41)
for tangential field component H.:
2 1 11-1 PV a [ 1 JH,(rL)= -(Hot)+-- f H,(r)- In ( ) dl.II + 1 rt II + 1 I. an, R rL, r (1.42)
Here identity
o~,[:n [InR(r:J] ~ : [o~,[InR(r:J]
and integration by parts are used. IL denotes the observation point rL coordinates, I :
integration point r, t is a tangent unit vector.
The boundary integral equation for normal component [33]
2 1 11-1 PV a [ 1 JHII(r,)= -(Hoo)- -- f HII(r)- In ~ ) dl
II + 1 7t fl + 1 t. an I. R rL ' r (1.43)
is derived in analogous way.
34
Chapter 1
Indirect formulation of boundary integral equations
There are two indirect boundary integral equation formulations which use two
different physical analogies: surface charge distribution and equivalent surface
currents.
Surface charges
In this case imaginary magnetic surface charges are distributed by density K(rL)
over boundary in such a way that the external field together with the field of charges
satisfies the boundary conditions (1.36). Contour element dl(r) in 2D plane point ro
creates the field
dH(ro) = K(r)dl(r) ro -r2 •Iro - rl
Hence total field for all points ro which belong no to the boundary contour L is
PV
H(ro) = u, + 1 ro - r2 K(r)dl(r). ( 1 .44)
L Iro- rl
Applying boundary conditions in the cases when ro tends from inside and
outside to the boundary point rL the following boundary integral equation for surface
charge density K(rL) is obtained:
7t /l + 1 K(rL)= (HOn(rL))+ 1VK(r) n(rLXrL ~ r) dl. ( 1 .45)
/l - 1 L IrL - rl
Corresponding equations for magnetic field normal and tangent components are
HII(rL)=~K(rJ,/l-1
(,) () jV ()t(rLXrL - r)Ht\rL =HotrL +'j xtr 2 dl.
L IrL -rl
The equation (l.45) turns out to be equivalent
calculated from known values of H,
(1.46)
(1.47)
to (1.43), so Ht could be
35
Chapter 1
Equivalent currents
According to Biot and Savart's law in 2D [65], magnetic induction, created in
point roby surface currents with density j(r) is
B(ro)= n, + ~~j(r)x (ro ;r) dl(r) (1.48)c L Iro - rl
Since j(r) has only z-component, it could be substituted by a modified scalar density
v(r) in such a way that j(r) = ~ v(r) e z : The corresponding boundary integral
equation, which includes boundary conditions (1.36), is
(1.49)
(1.50)
(1.51)
H,(r
L
)= _ 271: v(rL).
1-1-1
One can see, that (1.49) and (1.52) together gives (1.42).
(1.52)
36
Chapter 1
,ex'
,+,'2='V2+HOr
Figure 1.1. The sketch of the magnetic fluid droplet
37
Chapter 1
Conclusions
The boundary integral equation technique allows to formulate the free moving
boundary problem for a ferrofluid droplet in an external magnetic field in terms of
surface values for both velocity and magnetic field. To solve obtained boundary
integral equations, some approximation technique should be used, as well as the
boundary contour itself should be described in a some approximation. The
approximation technique and approximation errors are the subject of the next Chapter
of the present work
The essence of the direct magnetic field problem formulation, presented here, is
to obtain both magnetic field components on the interface of a droplet by solving two
corresponding boundary integral equations. Two kinds of the indirect formulation are
derived here using either magnetic charges or equivalent currents on surface of the
droplet These two indirect formulations are mathematically completely equivalent to
the direct formulation, but the numerical approximation of them, discussed in the next
chapter, could give different levels of approximation errors. It is found that the normal
field component is proportional up to constant factor to a surface charge density but
the tangent component to a surface current density.
The principal difference of the indirect magnetic field problem formulation from
the direct one is that the indirect formulation allows to obtain by a simple integration
the one of two field components when another one is already found
38
Chapter 2
Approximation of boundary integral
equations
2.1. Interpolation of a planar boundary
The two-dimensional boundary may be represented by a planar line. To set up a
boundary element representation, the planar line are traced by a set of marker points
r., i=l ,...,N, numbered in the counter-clock wise direction according to Fig.2.1. In the
numerical algorithm, a contour consists of its marker points and the interpolation
functions between nodes. Cubic-spline interpolation is chosen to provide a smooth
boundary with continuous first- and second-order derivatives [44]. At every time
instant spline approximation is calculated in two steps [45]:
1. Calculation of boundary curve representation by two parametric cubic-spline
functions: x=fx(P), y=fy(P). Initially the "perimeter" length Pi for N-sided polygon
with marker points as vertexes are taken as the values of the parameter p at marker
points, starting with vertex i=l (Pl=O). Then cubic spline coefficients are calculated
for a closed contour, accounting for the periodicity constrain. According to [7I], a
cubic-spline interpolation provides the unique curve, the shape of which is
independent from the choice of Pi'
39
Chapter 2
2. Calculation of natural curve parameter: the arc length for a curve is calculated
P
using the integral s(p)= f~X'2~)+ y'2~)dt. Since the cubic-spline interpolation
P,
is unique, the natural parameter values Si=S(Pj)at corresponding i-th marker points
are unique, too. After the integration the new cubic-spline functions x=fxCs),y=fis)
in dependence on the natural parameter s are recalculated.
40
Chapter 2
2.2. General remarks about discretization of
boundary integral equations
Once the contour of integration is traced by marker points and cubic-spline
interpolation, the approximation of the unknown boundary velocities and magnetic
field components on the boundary inside the droplet could be performed. In the every
element of the contour the local basis functions for unknowns is introduced and
globally unknown values are described by global basis functions which are built from
the local basis functions multiplied by coefficients and added together. The next step
is the approximation of boundary integral equations for given set of boundary
elements. The realisation of this step in so called Boundary Element Method, widely
known as BEM. The accuracy and the numerical stability of obtained numerical
scheme depends on the choice of the approximation technique. Two governing
approximation techniques applied to compute the coefficients of the local expansions
are either the collocation method, or the method of weighted residuals [30,93,85].
The last of two methods are more general and in fact includes the first one as
particular case. The main idea of the collocation method is the requirement to satisfy
the boundary integral equation in N selected points, leading to the set of N linear
algebraic equations. In the method of weighted residuals the integral equation is
multiplied sequentially by every of N weighting functions in order to obtain the set of
N linear algebraic equations after an integration along the boundary contour. For the
method of weighted residuals different choices of the weighting functions lead to
different schemes with varying degrees of complexity. Identifying the weighting
functions with the global basis functions we obtain Galerkin's method, while
identification of the weighting functions with delta functions having poles at selected
points over the boundary contour gives the collocation method. In [85] the Galerkins
method is recommended only for problems with notable geometrical simplicity due to
"increased computational requirements". In fact, obtained set of algebraic equations is
harder to derive but once it is derived, calculations by computer can have the same
level of complexity. In engineering boundary element methods almost all the codes
are based on collocation methods. It is known that the right choice of the collocation
points is significant for the stability and convergence of the approximation [30].
41
Chapter 2
Preparing computer code for magnetic field calculations it was found that the
collocation method gives the less accuracy than the Galerkin's method at the same
CPU time consummation. Hence in the present work the Galerkin's method is used to
derive sets of linear algebraic equations for the both magnetic field and creeping flow
equations, using pyramidal functions (see Fig.2.2) [33] defined as
o
() (S-Si_l)/(Si -Si_l) Si-l <S~Si'<p S =
I (Si+! - S)/(Si+! - Si) Si < S ~ Si+!'
o Si_1 < s.
where S is the natural boundary contour parameter and s, are its values at marker
(2. 1)
points on the contour. For convenience substitutions of indexes like SN+j=Sj' j=O, ±1,
±2 ... are used in general formulae.
42
Chapter 2
2.3. Equations for a magnetic field
The derivation of the set of linear algebraic equations for the integral equation
for the tangential magnetic field component (see the equation 1.42 in Chapter 1)
H (s)= _2_(H t)-~ I-t-11V H (S,)Ys(X - x')- x,(Y - Y') ds'
I 1-t+1 0 7t 1-t+1 L I (x-x'Y+(y- Y'Y)
consists from the following steps [33]:
(2.2)
1) unknown function HI (s) is expressed in terms of pyramidal basis functions <P;(s)
using values Ht,i of the magnetic field tangent component at marker points as
expansion coefficients:
N
HI (s) = I H,,; <pieS) (2.3)
;=1
2) the values of the ratio
R(s s') = y,(x - x')- x,(Y - y')
, (x-X')+(Y- Y'Y)}x; + y;
at marker points are taken as
(2.4)
Ys (k Xx(k) - x (i)) - x, (k )(y(k) - y(i))
---------------------------- i =t- k;
[(x(k) - x(i)y +(y(k) - y(i)Y ]Jx~ (k) + y; (k) ,
1 Y'5 (Ox, (i) - XIS (i)y, (i)
2 ~(x;(i)+ y~(i))2
3) the integral equation (2.2) is multiplied by a basis function <Pk (s) and integrated
( 2 .5)
i= k.
along the boundary contour:
N ')1<Pk(S)IH,,; <p;(s)ds = ---1<pk(sXHox Xs + HOY y,)ds-
L ;=\ I-t + 1 L
~~J<pk(S J'Ji:H" <p,(s') yet - ~')- x,V ~ ~} dS']dS
7t I-t + I 1 L L ;=1 X - X ) +(y - y )
Here and further it is taken into account that Jx; + y~ == 1 for the natural parameter s.
(2.6)
The integration of the left-hand side of the equation (2.6) gives
43
Chapter 2
N N
1(j)k(S)I n., (j)i(s)ds = I H',i1(j)k(S)q>i(S)dS =
L i=l i=l L
NI u., (~i-l 8i,k+i + ~i 8i,k-1 + 2(~i_1 + ~i)8i,k )/6
i=1
(2.7)
Here
Ss, = Si+l - Si·
Use of the approximation inside the interval [Sj,Si+I]
rei + 1)- rei) .r ~-----
s Ss.
1
(2.8)
for an integration of the first term in the right-hand side of the equation (2.6) results in
J (XH H )d H xk+! - Xk_1 H Yk+1 - Yk-l'j(j)kS oxxs+ OyY, s= ox 2 + OY 2 .
L
JPv N :To evaluate the second term j(j)k (s1f~ H,) (j)i (s') R(s, s')ds' ds in the right-hand
(2.9)
side of the equation (2.6) the integral in square brackets is evaluated by trapezoidal
rule:
1(j)i(S')R(s,s')ds' ~R(S,SJ!1si-12+ ~i ,
L
and the outer integral also evaluated by trapezoidal rule gives:
(2.10)
JPV N ]!1s +~.!1s + ~ .q(j)k(s1f~Hliq>i(S')R(s,s')ds'dS~Rki 1-12 I k-'2 k.
L
Thus the discrete analogue of the equation (2.2) is
(2.11)
~ A+H . = (\k~' - Xk_1)Hox + (Yk+1 - Yk-I )HoyL... kf ',I I ' k = 1,... ,N,
i=1 ~ +
where
(2.12)
(2.13)
Iki = (!1si_18 i.k+I+!1si 8i.k-I + 2(!1si-J+ !1si)8 i.k )/6. ( 2 . 1 4 )
In similar way the discrete equation for the normal component of a magnetic field
(1.43) is derived arriving at
(2.15)
44
Chapter 2
The definition of A;; is already given in (2.13).
Separation of the external field components
Due to the linearity of the field equations, the solution of them could be split in two
parts: two fields created separately by Hox and HOY,which are described by variables
HII i = U X i H, x + U Y i HOY'.. . (2.16)
(2.17)HI,i = ~X.i Hox+~y,i HOY·
The corresponding equations are easy to obtain from (2.12),(2.15):
k=I, ...,N, (2.18)
k=l, ...,N, (2.19)
k=l, ... ,N, (2.20)
k=I, ...,N. (2.21)
This approach is used to obtain the time-averaged field term H; + 11 H~ in the
effective surface force in Chapters 3 and 4, where a 2D ferrofluid droplet in a high-
frequency field is considered.
The shape periodicity of the droplet
In the case of the periodical shape with K periods, the angle occupied by the complete
period is YI=2n/K. If the number of marker points per period is N, then between the
marker point i and the corresponding marker point i+jNp are j full periods. To derive
transformation formulae, which bound together these two points, the magnetic field
components Hox' and Hoy·in coordinates X' ,Y' rotated by the angle Yj=jYlwith respect
to X,Y coordinates, are introduced (see Fig.2.3):
Hox' = Hox cos Yj + HOYsin Yj ,
HOY'= HOYcosYj - Hoxsin Yj •
Due to the periodicity
(2.22)
(2.23)
H ." =Uv' Hox'+uy HOY'.11.1+}lV f' ,\ .J ,I (2.24)
45
Chapter 2
Collecting together (2.22)-(2.24) and accounting for (2.16) gives
a x i+J"N Hox + a Y i+J"N HOY='p 'p (2.25)
a X,i (Hox cos Y ) + HOYsin y})+ a Y,i (Hoycos Y ) - Hoxsin y}
The separation of field components yields
{
a \"i+J"N = a x i cos Y J' - a Y i sin y J' ,~'P' •
aY,i+}Np =aX,i siny} +aY.i cosy}"
In a similar way
(2.26)
{
p X,i+}Np = ~X,i cosy} - PY,i sin t ..
PY,i+}Np =PX,i siny} +PY,i cosy;"
(2.27)
Thus the account for the periodicity of the shape gives two sets of linear algebraic
equations with matrices 2Npx2Np, implemented in Chapter 3. The values of unknowns
ax, ay, Px, Py depend only on the geometry of the problem and from )..I.. Thus these
values are a useful tool to perform calculations for an arbitrary external field. Once
calculated, they allow easy to obtain magnetic field on the boundary according to
(2.16),(2.17).
The accuracy of the magnetic field calculations
The accuracy of the magnetic field calculation here is tested by few examples for the
elliptic shape of the droplet. The exact field solution inside the ellipse is well known
[65]:
H-H a+b H a s b- ox ---+ Oy ---a + )..I.b b + )..I.a
Here the major semi-axis a is orientated in X-axis direction, b stays for minor semi-
(2.28)
axis of the ellipse. To check the accuracy, both analytical and numerical values for the
non-dimensional geometry-dependent magnetic field characteristics ax, ay, Px, Py (see
formulae (2.16), (2.17)) are plotted versus the contour arc length s in Fig.2.4 and
Fig.2.5, number of marker points N=200, magnetic permeability )..1.=15,a ratio of
semi-axes a1b=16. The difference between Fig.2.4a and Fig.2.4b is that in Fig.2.4a the
marker point distribution on the contour with equal arc lengths between them is used,
but in Fig.2.4b the distribution is dependent on the local curvature: in the places with
46
Chapter 2
large curvature the density of marker points is larger in such a way that the distances
between marker points are proportional to the curvature radius. To prevent too high
accumulation of marker points in some places causing absence of them in other
locations the upper and lower limits of distances between marker points are
introduced. Such a curvature-dependent distributions allows essentially improve the
field calculation accuracy in the regions of a contour with large curvature (for
example, the tips of a 2D droplet). The variable ax and hence the normal magnetic
field component on tips exhibit an impulse-like increase, which is a source for the
saw-tooth type oscillations (see Fig.2.4). In Fig.2.4b the implementation of a
curvature-dependent marker point distribution suppresses these oscillations, for ~x
they are insignificant, but for ax there is still presence of them in very tip of an ellipse.
In fact, the saw-tooth type oscillations about the exact solution are eventual origins for
breakup of the numerical algorithm: an artificially elevated field value causes the
elevation of a surface force leading to the formation of a sharper tip and thus
increasing again the elevation of calculated field. In Fig.2.5 the plot of variables ay, ~y
versus the contour arc length displays, that the field component which is perpendicular
to the direction of a droplet elongation, plays less role for surface force. Other point is
that the accuracy for these variables are higher but the comparison with equidistant
distribution (not shown here) shows that there is no real improvement in the present
case for ay, ~Y'
47
Chapter 2
2.4. Equations for the motion of a free boundary
To make discretization of the boundary integral equation for velocities on the
boundary (see the equation (1.28))
1- A r-v»:
V (ro)- -- Jv i (xv; (r, rO)nk (r}il(r) =
J 2n(l+A) L
_ 1 J~J:(r)Gij(r,ro}il(r)
2nll1 (1 + A) L
the collocation technique [30,9330] is applied, nevertheless this approach is similar to
(2.29)
that of a magnetic field discussed above. The difference from magnetic field
equations appears in treatment of singularities: both the logarithmic singularity of
G ij (r, ro) and the singularity of Tijk (r, ro) require special treatment at the singularity
point r=ro. The equation (2.29), in fact, represents two coupled equations, because
both components of the velocity vector, Vx and vy are implemented in the right hand
side of it. The discretization of this equation by N boundary elements (marker points)
leads to the set of linear algebraic equations of order 2N x2N.
For the regular boundary element the conventional trapezoidal rule is used. To
evaluate the term Vi(r)Tijk (r, rO)nk (r) at singularity r=ro, the relations
xes) - x(so) = x,(so )(s - so)+ x,s(so )(s - SoY /2 + o«s - soY ,
nx(s) = Y,(s) = Y,(so) + y,Jso )(s - so)+ o«s - soY ,
n,(s) = -xJs) = -x,(so) - x,,(so )(s - so)+ o«s - soY
are used in infinitesimal neighbourhood of So arriving at
X,XXk
Vi (r )Tijk (r, r0 )nk (r}is = -4 I ~ Vi (r)nk (r}is =r
-4(x, v x + Ys V r )is x,,) ds x
[(y, + y"ds Xx,ds + x" ds2 /2 )- (x, + x s,ds V sds + YIS ds2 /2) J
ds
=
ds4
-2x,,) (x, vx+ y, vYXx,y" - y,x,.,}is.
Therefore the approximation of the integral in the left-hand side of the equation (2.30)
(2.30)
gives
48
Chapter 2
PrYol.Jvi(r)Tijk(r,r(p)}k(r)dl(r)~ -D;i -. =
L
N xj(m) - xj(p)-22: 4 (~r(m,p)v(m))(~r(m,p)n(m)XLUIII_l + LU".)-
111=1 l~r(m,p)1
m.",p
xs)p)(rs(p)v(p))(xsYss - YsXssXLUp-l + LUp)
(2.31)
~r(m,p) = x(m)-x(p), &/m,p) = x/m)-x/p).
Here ro in (2.29) corresponds to the p-th marker point. For singular boundary element
approximation both the result of the exact integral
CJ ) (In C 1)(a+bx)lnxdx =ac(lnc-l)+bc- ---
024
and the property that surface forces have only normal component are used. It leads to
(2.32)
the following approximation formula:
J f xx)tlf(r \ -8ij lnr + ~/ dl(r)~ F; =
L
I[~f(m)lnl&(m,p)l- &/m,p) ~fk(m) &k(7'P)] x (LUIII_1+ LUm) +
m=l 1&(m,p)1 2
lJI:;tp
±~fk(P)[~P lnl&(p + l,pt + LUp_1lnl&(p, p -It - 3(LUp_1 + ~p)}-
», I ( ) LUp ( )-----;f- ~fk P - 1 - 4 ~fk P + 1
(2.33)
Here
&(m,p) = x(m)- x(p), &/m,p) = xi(m) - x Jp).
Inserting (2.31), (2.33) in (2.29) we arrive at the set of linear algebraic equations
2SI Dij Vi = r; i= 1,2, ... ,2N , (2.34)
i=l
where
V; = vx(j), j = 1,2,... ,N,
V/_v = v y(j), j = 1,2, ... ,N,
1- Ie •
Dij = 2n(l + Ie) Dij'
49
Chapter 2
1 •F=-----F.
I 21t1l1(1+A) I
After solution of the set (2.34) the shift of the i-th marker point during the time step
!'i.t in normal to the contour direction is obtained by
!'i.r( i) = (n x (i) V (i) + n y (i) V (i + N) ~t n( i) (2.35)
The accuracy of the boundary velocity calculations
The accuracy of the obtained boundary velocities was tested by the dynamical
simulation of the droplet with small elliptical perturbation from circular shape: the test
of droplet contraction due to surface tension. The decrement of the perturbation
measured by non-dimensional major semi-axis of the ellipse is obtained in [33] from
a" -1 _ a; -1 ( t)-----exp ---
a aD 1 + A
and is equal to 1/(1+A). In Fig.2.6 the results of this test are shown for five different
(2.36)
values of A=O; 0.5; 1; 2; 5. The equidistant marker point distribution was used,
number of marker points N=100, time step !'i.t=O.01.Solid lines represent results from
the formula (2.36), by dotted lines results from numerical simulation are shown. Small
shift between solid and dotted lines is caused mainly by initial values of perturbation,
which is too large to obtain an almost identical fit by (2.36). Nevertheless the fit is
rather good, and for t>2 solid and dotted lines have nearly the same inclination with
respect to the time axis, thus showing excellent agreement for perturbation decrement
between the analytical solution by small perturbation theory and the numerical
simulation started from the initial state with slightly elliptic shape.
The conservation of fluid volume is implemented in boundary integral equations by
use of appropriate Green's functions. The accuracy of the approximation of them is
directly connected with volume conservation during the numerical simulations. Two
simulation tests were carried out to check the numerical accuracy of volume
conservation:
1. Static field test: static magnetic field, characterised by Bm=15, 1l=15, number
of marker points N=200 with curvature dependent distribution of them, default
time step ~t=O.O1. Transition motion of a droplet was simulated starting from
50
Chapter 2
a circular shape. The volume conservation error was 0.25% during 990 time
steps, time interval t,-to=9. 9.
2. Rotating field test: rotating magnetic field, characterised by Bm=105, 11=5,
QH=0.8, number of marker points N=200 with curvature dependent
distribution of them, default time step ~t=0.01. Transition motion of a droplet
was simulated starting from a steady state shape. The volume conservation
errors were:
2.1. 3.4% after 500 steps, time interval t 1-to= 1.14;
2.2. 7.2% after 1000 steps, time interval tTto=2.34;
2.3. 20% after 2710 steps, time interval trto=6.29.
These two tests shows that volume conservation was almost good. the overall
performance is close to that of [97], where a ferrofluid droplet was considered in the
static field, the errors were less than 0.1% over several hundred time steps. In the
cases of rapid motion conservation was poor, with errors as 1% over 10 time steps
[97], and the drop dimensions were rescaled in order to correct the volume.
Throughout the present work forced volume conservation was used via rescaling of
droplet dimensions after every time step, because the change of droplet dimensions
leads to different effective time scaling unit and to different magnetic Bond number.
thus changing the conditions of droplet behaviour. To improve the accuracy of
numerical simulations, the time step was adjusted in every time iteration in such a
way that for every marker point displacement during current time step was limited by
20% of distance to the nearest neighbour point.
The accuracy of the surface force approximation was tested by checking magnetic
field threshold in high-frequency rotating magnetic field (see Chapters 3 and 4). To
obtain it the inverse value of a decrement for elliptic shape perturbation is plotted
versus magnetic Bond number Bm. The straight line plotted through the obtained
points should intersect the zero-line at the threshold value of Bm. Results are shown
in Fig.2.7a (11=10) and Fig.2.7b (11=25). The error is smaller than 0.5% in both cases.
The equidistant marker point distribution was used, number of marker points N=200.
time step ~t=0.0 1.
51
Chapter 2
;. 11ut-
o d .~.
"" J..1lin-~ 0 N,
Figure 2.1. Approximation
of a 2D droplet boundary
by marker points
1
o o
Figure 2.2. The defmition of
the pyramidal function
I
Figure 2.3. Periodic symmetry of a 2D droplet
52
Chapter 2
(a) Equidistant marker point distribution
.IJ.bJ!JJ./1a,--
0.5 Nurn. Anal.0ססoo ~x --_ •
0.0
I
I
I
I
I
-0.5 ~
Jl=15, a/b=16, N=200
o 4 8 12 16
Contour arc length
(b) Curvature-dependent marker point distribution
0.0
Nurn. Anal.
0ססoo ~x --- •
0.5
.IJ.bJ!JJ./1a--x
Jl=l5, a/b=16, N=200
-0.5
o 4 8 12 16
Contour arc length
Figure 2.4. The magnetic field characteristics ax, ~xversus
the contour arc length;
(a): the equidistant marker point distribution,
(b): the curvature dependent marker point distribution.
Definitions of ax, ~x: Hn=axHox+a;Ioy Ht=~xHox+~)HoY
53
Chapter 2
Curvature-dependent marker point distribution
0.10
0.05
Num. Anal.
[JJD[IJ ~y - - _.
0000<> uy--
0.00
-0.05 ~=15, a1b=16, N=200
-0.10
o 4 8 12 16
Contour arc length
Figure 2.5. The magnetic field characteristics uy, ~y versus
the contour arc length, the curvature dependent marker point
distribution. Definitions of uy, ~y: H,=uxHox +u~oY, Ht=~xHox +~yHoY
o Accuracy test
...............................~::?
N=lOO
tlt=O.Ol
A. = TJin!TJex
-- Small pert.
theory
..............Num. simul.
-3 o 1 3 42
Time t
Figure 2.6. The small perturbation decrement test:
the contraction of the droplet.
54
Chapter 2
(a) Field threshold (J.l= 10)
20 25 30 35
Magnetic Bond number H2RJa
(b) Field threshold (J.!=25)
7
N=200
~t=O.Ol
.D
16 18 20 22 24
Magnetic Bond number H2RJa
Figure 2.7. The magnetic Bond number threshold value test
in high-frequency magnetic field.(a) J.!=10, (b) ).1=25.
55
Chapter 3
I-ligh-frequency rotating magnetic
field: the energy approach.
3.1. Shape generation for energy calculations
The behaviour of a magnetic fluid droplet under the action of the high-frequency
rotating magnetic field is considered under the assumption that the characteristic time
of the droplet shape relaxation is much larger than the period of the rotating field and
thus the time averaging with respect to the rotating field period is possible. The
subject of studies is the calculation of the equilibrium shapes, characterised by an
absence of flow if the eventual internal rotation of ferroparticles of a magnetic fluid is
neglected. Under such assumptions the equilibrium shape corresponds to the
minimum of the total energy of the droplet, formed by the sum of a magnetic energy
and a surface energy. Further in this Chapter by magnetic field strength is understood
the amplitude of a rotating field.
Thus the equilibrium shape at given magnetic field value, characterised by
magnetic Bond number Bm=H2R/cr, could be found from some initial shape, for
example, by the variational technique, varying the shape of the droplet. In fact, such a
minimisation of the total energy of the droplet changing its shape could be rather
computer time consuming, since in every iteration the magnetic field have to be
calculated for the new shape of a droplet. The variational technique is used, for
example, in [27],[29] to calculate the equilibrium state of sessile ferrofluid drops in
the case of non-linear rnagnetisation of a ferrofluid, applying the finite element
method to calculate a magnetic field.
57
Chapter 3
Concerning the time dependent magnetic field amplitude, the energy approach
to shape calculation could be used only to calculate equilibrium shape at every time
instant, if the inertia of the system plays no role in the current process. One of the
processes without account for inertia is the creeping flow, considered here. The time
history of the transition motion in the case of the time-dependent magnetic Bond
number thus is substituted by "stroboscopic" shape series. Here it is proposed to
detect these series of droplet's shapes in the external homogeneous magnetic field
from the set of families of symmetric shapes, generated a priori for different numbers
of spikes. The main point is how to generate the families of shapes. In the paper [7*]
(included in the present Chapter) these families are obtained from the conformal map
of the circle in the w plane with radius r >
W-(Il-I)
z = w+ --, (3.1 )n -1
transformed in order to obtain more elongated spikes and widen in such a way the
families of shapes. The number of spikes is given by n. Two control parameters, 8 and
E (see [7*] for more details), allow to control the curvature radius at the tip of every
spike and the length of the spikes. The expression (3.1) gives the contour in a-plane
described by the following parametric equations after the substitution
w = (l + 8) exp( hp ) , i denotes the imaginary unit:
f cos(n -l)<p I
x = (I + <5\ cos q:> + (n -1)(1 + 8 )" )
{
sin(n -1)<p '1
y = (l + <5 sin q:> - ( X )" }n-l 1+8
q:> E [O,2n J
In order to widen the families of shapes, the curves are at first transformed to the
(3.2)
polar coordinates (p = p(8, q:», a = «(8, <p)) and then the shape is transformed again
according to the following power:
( pea )J"pl(a ) = Pmin -.-Pmm
Here Pmin is the minimal distance of the contour to the origin of the coordinates and
(3.3)
E >0 is a control parameter by change of which arbitrary spike length could be
obtained. After application of the rule (3.3) the droplet is rescaled back to the
58
Chapter 3
"volume" of the circular droplet, characterised by the magnitude of the non-
dimensional are Sc=1t enclosed by the boundary contour of the droplet.
At given magnetic Bond number control parameters are varied and the local
minima are found. Thus, tracing the energy minima for some magnetic Bond number
interval, the behaviour of the droplet in time-dependent magnetic field is detected
without real account on transition phenomena. The shapes of droplet are found as the
equilibrium shapes and so they correspond to the stationary state that establishes when
transitions decay down at the constant magnetic field. In general this approach could
be extended to the time-dependent magnetic Bond number, if the rate of the Bond
number changes is small enough for the transition decay.
Some shapes, generated by transformation of (3.1) are juxtaposed in Fig.3.l and
Fig.3.2 with shapes from dynamical simulation from Chapter 4, regarded as almost
"exact" ones, since there are no constraints made concerning the shape of a droplet.
Fig.3.1 displays that the analytically generated shapes can not describe very well both
the curvature radius at the tip and the shape of the "body" of the droplet where the
"star-arms" come together. Thus the analytically generated droplet in comparison with
the one from dynamical simulation has either too sharp tips or the "necking" in the
places where arms join together. In fact, for Fig.3.1(b) this "necking" effect for shapes
generated by formula (3.1) and the consequential shape transformation, may cause
magnetic field defects leading to the essential change of the magnetic energy. Fig.3.2
displays that even small spikes could not be described very exact by the present shape
generation technique. Nevertheless the present shape generation is still the best found
one and allows to obtain important results, as it follows in [7*]. Improvement of the
shape generation technique should lead to the more accurate results but not to some
qualitatively new effect.
59
Chapter 3
Dynamic simulation: Bm=200, Jl=25, N=350
(a)
.-Simulation
~~ .-E:1.58, 8:0.04_-=====;;;.--~ '-E-0.18, 0-0.02----
-8 o 8
(b)
E=3.64,0=0.021
tSimulation
-6 o 6
Figure.3.l: Analytically generated shapes juxtaposed
with a dynamically simulated one: (a) 2-spike shapes,
(b) 3-spike shapes.
60
Chapter 3
Dynamic simulation: Bm=45, ~= 15, N=450
r,
\ "~'. \"":'" "-. .. ,. .. \•..• .... ". .
•..•. ',', ----- ,..... ,,- » E=3.56.8=0.162
.... · 'Jl. .. ::::::'.., "" .': ..' \ E=2.68,8=0.128
~ E=1.36,8=0.06
Simulation
I
-4
Ia I4
Figure.3.2: Analytically generated shapes juxtaposed
with a dynamically simulated one.
61
Chapter 3
Included paper:
I.-C. Baeri, A.Cebers, S.Laeis, R.Perzynski
Shapes of 2D magnetic fluid droplets in a rotating magnetic field
Magnitnaya Gidrodinamika, 1995, N3 (accepted to publish)
63
Chapter 3: included paper
Shapes of 2D Magnetic Fluid Droplets in a Rotating
Magnetic Field
J.C.Bacri (*), A.Cebers (**), S.Lacis(**),R.Perzynski(*)
(*) Universite P.M.Curie, 4 place Jussieu, 75252 Paris, Cedex 05, France
(**) Latvian Academy of Sciences, Institute of Physics, Salaspils-l, LV-2169, Latvia
Abstract
The families of equilibrium figures of 2D magnetic fluid droplets in
high-frequency rotating magnetic field are found numerically. The
magnetic field energy is calculated by a boundary integral equation
technique for nonlinear transformed and smoothed hypocycloidal curves.
Comparison between families with different number of spikes shows that
the lowest energy corresponds to the "two-spikes" shape. A minimum
energy calculation shows that for a fixed number of spikes, these spikes
arise at a given threshold in magnetic Bond number. Increasing magnetic
Bond number, the spikes sharpens. Finally some conclusions concerning
the discrepancy with the experimental results are pointed out.
l.In troduction
Droplets of magnetic fluid (MF) under the action of an external magnetic field show
rather intricate behaviours [1-3]. Phenomena of particular interest are arising with MF
droplets under the action of a rotating magnetic field [4]. Several different shape
transformations are observed experimentally and explained theoretically by a linear stability
analysis [4,5]. Among them, the sequence of shape transformations "oblate-prolate-oblate" is
followed by the arising of the "star-fish" configurations. In [4,5] an analysis of the fastest
growing mode of a 2D magnetizable liquid cylinder under the action of a transversal rotating
64
Chapter 3: included paper
magnetic field is considered. A good agreement with the dependence of the number of the
"star-fish" arms as function of the magnetic field strength is obtained. Nevertheless from a
theoretical point of view the transitions between "star-fish" configurations with different
number of arms, observed in the experiment [4], are remaining unexplained. In the frame of
the 2D model mentioned above, we study the transitions between configurations with
different numbers of arms through an analysis of the MF droplet energy in high-frequency
rotating fields. We also look for the transitions between different possible configurations
from an energetical point of view. An analysis, based on the choice of particular families of
shapes reflecting the formation of sharp tips at the development of the MF instabilities [6,7],
is undertaken in this paper. From the calculation of energy minima of shape families with
different number of tips, we conclude that deepest minimum corresponds to the two-spikes
shape of 2D MF droplet. It is in good accordance with the results of numerical simulations of
the hydrodynamics of a 2D MF droplet [8] but not with the experimental observations of an
increasing number ofarrns of the MF "star-fishes" with the field strength [4]. Thus the results
of the present work show that, for the full understanding of the observed transitions between
different "star-fish" configurations, the theoretical model must evidently account for 3D
effects and possibly internal rotations and shear flows.
The numerical method of the present work is based on the boundary integral equation
technique which has been used previously for the calculation of the elongation of the MF
droplet in constant field [9] and the study of 2D droplet behaviour under the action of the
magnetic field and shear flow [10].
2. Numerical algorithm
The approximate shape of a 2D droplet can be found from an energy minimum analysis,
taking into account both magnetic and surface energies. In this case transient fluid motion and
internal rotation effects are neglected. The method described below allows, to find the shape
corresponding to a minimum of the total energy of droplet from a given family of shapes and
at a given magnetic field strength:
65
Chapter 3: included paper
( 1)
Here tilde denotes dimensional form of energy, c is for surface tension, L is the perimeter of
the droplet, 1\1 is the magnetization of MF inside the droplet, and flo is the external magnetic
field. The family of shapes is given a priori and the figures of equilibrium are obtained within
some approximation, its accuracy depends on a successful choice of the family of shapes.
All along the paper, reduced quantities without dimension are used: non-dimensional
energies are obtained with respect to the surface energy of a circular droplet E; = 2rrcrR,
magnetic field with respect to external field Ho and non-dimensional values of droplet size
with respect to the radius R of an unperturbated droplet. As a result, the figures of equilibrium
H2Rare constructed as a function of the magnetic Bond number Bm = _0_ .c
Thus the non-dimensional surface energy is:
LE --
s - 2rrR· (2 )
The non-dimensional magnetic energy for the linear part of the MF magnetization curve
(magnetic permeability ~ = const) in a homogeneous external field is:
ll-l I--Em = ---2 Bm HHodS.100 s
Last relation accounting for divH = 0, can be rewritten as
11-1 J(--)
E/II = -100 2 Bmg Hor I}{"dl.
r
(3)
The boundary integral equation technique [I 0] is used to calculate the normal component of
the magnetic field strength H" on the droplet surface (the normal points outside of the
droplet):
H I = _2_H_ox__ 3x + _2_H_oy__ 3y +~_(~_-_l)~ H (1')1 xu.r, __ dl_'_
"r (1l+1)3n (~+l)3n rr(1l+1)r" r ' ~x/+y/
K(ll')= x,(y' - y)- y,(x' -x)
, (y' _ y)2 + (x' _ X)2
( 4 )
As it follows from relation (3), for energy calculations, the equation for the normal field
component must only be solved.
66
Chapter 3: included paper
For that, at first, the boundary contour is approximated by a finite number of marker
points connected together by interpolating cubic spline functions. The arc length between
marker points depends on the local curvature of the contour and the periodicity of the contour
is taken into account. The boundary integral equation (4) is solved by an expansion of the
normal component in pyramidal functions and using Galerkin's method. The pyramidal
( 5 )
An approximation technique is described in [10] for the case of equal arcs lengths. In the
present work this approximation technique is used for a non-uniform distribution of the
marker points as described above. It leads to the following set oflinear algebraic equations:
~ {'A = ~ (' [R -I l= HOY (Xi+1 -Xi-\)- HOX(yi+1 - Yi-JL J k ik L J k ik ik II + 1 '
k=1 k=l r-
H,ll)Jxi + yi = 1(1)= IJ;<i'i(l),
( 6 )
i=1
I; -1;-1
6
li+1 -I;
Iik = 6
1;+1-li_1
i=k+l
i=k-l
30, i::/=.k -1 /'d ::/=. k /\ i::/=.k + 1
R = R* (J.!-l)(tk+] -lk-IYJi+I-l;-I)
ik ik 4rt(Il+1) ,
y,(iXx(i)- x(k))- x,(i)(y(i)- y(k))
(x,(i)-x/k)J +&,(i)- y,(k)JR: =
YII (i)x,(i) - XII (i)Y,(i)
2(xi (i) + yi(i))
i= k
i= k
Derivatives along the boundary contour are calculated by differentiating the interpolating
cubic spline functions. Since the shapes observed in experiments [4] have an azimuthal
periodicity, the numerical code can be made more efficient, accounting for that. In the case of
67
Chapter 3: included paper
K periods and Np markers per period, their coordinates for each period can be calculated
according to:
Yi = Yk cosy j + Xk siny j' ( 7)
i= k+ j Np, k = 1,2,... ,Np
Y j = jT, j = O,l, ... ,K - ,
2nwhere T = K is the period of the pattern.
Equation (6) taking into account the periodicity can be rewritten as follows. Let us
introduce the new field variables a; ,a: :
(8)
It is possible to obtain the variation of the variables «. ,«', along the period of the contour
from the conditions of periodicity and uniqueness of the solution. Solution f for marker
point i = k + j N p :
_ k+jNp k+jNp
h+jNp -«. Hox +ay HOY
in the system of coordinates (x',y') rotated by the angle Yj with respect to the system (x,y),
can be rewritten as:
( 9)
Expressing the field components in the system (x',y') by the relations:
Hox' = Hox cosy J + HOY siny j
and
HOY' = -Hox siny j + Hoy cosy j
from (9) due to the independence of Hox' Hoy we obtain:
k+jN r k k .ax ==axcosyj-aysmyj,
k « jN k k •ay· P ==ay cosy j+ax smy J ( 1 0 )
Inserting relations (8-10) into the equation (6) we have a new set of equations which can be
separated in two sets of size Np because of the independence of Hox, Hoy. As a result we
have:
68
Chapter 3: included paper
~f:l ( n n' '\A Yi+1 - Yi-l . 12 NLL\.ux cosy) -u y smy ) pi,m)Np = ----, 1=, ,..., p ,
,1=1 )=0 fl + 1
IN~ I. n . n '\A Xi+I-Xi_l. 12 N\Ux SIUY j +a y cosy j pi,n+}Np = ---, 1= , , ..., p.,,=1 )=0 fl + 1
( 11)
( 1 2 )
To solve the set of 2Np equations (11,12) for 2Np unknowns u;' , a _:~a Gaussian elimination
method is used (it is checked that iterative methods give worse accuracy). Found values of
u;' , u; allow to calculate magnetic energy (3). For that the integral in expression (3) is
approximated by the trapezium approximation formula:
fl -1E =---Bmx
m 167t2
xI~ [Hox (x" cosy) - Y" siny ) )+ HOY G" cosy) + «; siny } )]x
"=~H)=o c " ". ,\, H c " I/')~x~ ox\.ux cosy j -uy smy j r: or\.uy cosy j +ux smy) ~x
1"+1 - 1"_1X 2Jx; (i) + Y; (i)
( 1 3 )
In the case of a high-frequency rotating field (Hox = cosror , Hoy = sinrot), shape
changes during a time period are negligible if ro»1/1 for both characteristic times 1" of
shape changes due to - surface tension 1" = 11R and - magnetic forces 1" = 112, 11 is the MF(J n;
viscosity. Then time averaging can be done. As a result for magnetic energy we have:
II -1 IN ( ) 1 -IE r'__ B K 1/ 1/ 1/+1 ,,-I
«> 642 m x"ux+Y"u.v J?' 2·7t ,,=1 X,-(I)+Y, (I) ( 1 4 )
3. Families of the figures of equilibrium
The family of shapes, taken for the energy minimization problem, are build up looking
for the conformal map of the circle in the w plane with radius r >
w-(,,-l)
n -1 ( 1 5 )z=w+
It gives the contour in z plane described by the following parametric equations:
- ( cos(n -1)<p Jx - r cos <p + ( )"n -1 r
_ { . _ sinen - 1)<p J .Y - Sill <p ( ) nn-lr
( 1 6 )
69
Chapter 3: included paper
For r = the relations (16) gives hypocycloidal curves which have n singular points at
cos zxp = (n=3 deltoid or Steiner curve, n=4 astroid) [13].
The choice of the value r = 1+8 in (16) allows to smooth out the singular points and for the
curvature of the curve at tips (cos nq> = ) we now have a finite value which, at 8 < < 1, can be
taken as
The family of shapes corresponding to the n-spike "star-fish" configurations is constructed as
follows. For the radius of the curvature at the tip rc =n82 the value S 21n (8 =s/n) is
chosen. For definite values of S = n8 = ~ nrc, curves corresponding to the parametric
representation (16) are constructed. For each value of S , the curves with the equation in polar
coordinates p = p (a) are transformed according to the following power:
I (p(a )JEp (a)= Pmin -.-
PmlO
( 17)
Here P min is the minimal distance of the contour to the origin of the coordinates and f: a
parameter which varies from 0 to some limiting value determined by the accuracy of
calculations (elongated "star-fish" arms with sharp tips require too many marker points for a
reasonable accuracy). After all these transformations the area enclosed by the contour is set
back to nR2 by a linear scaling. At first the area of transformed contour is calculated as
N
s, = ~ ~ (XiYi+l - Xi+1Yi
and thus the scaling factor is obtained as
r; = ~n/Sc·
The family of shapes constructed for n=3 and 6 at S = 0.1 and S = 0.4are shown in Fig.I.
One can see that tips at S = 0.1 are much sharper.
The numerical calculations of energy minimization, according to the relations (11,12,14)
for the more realistic "star-fish" configurations (17), are carried out for 4 fixed values of S
(0.1 ;0.2;0.4;0.8). The values of the parameter f: , corresponding to the minima of the droplet
energy (1,2,14), are thus found for different numbers of spikes. In Fig.2,4a,5,6,7 an energy
70
Chapter 3: included paper
normalization is carried out to have a better representation: the total energy of circular droplet
Eco being subtracted from the energy of the droplet.
In Fig.2 the calculated energy minimum is presented as a function of the magnetic Bond
number for S = 0.2. Each curve in Fig.2 corresponds to a given number of spikes on the
droplet surface. Small overshots above the energetical level of circular droplet are connected
with hysteresis phenomena and are discussed in details below. From this picture several
important conclusions can be drawn. As it can be seen from Fig.2 energy curves for different
numbers of spikes do not cross each other. That means that it is difficult to expect from the
energetical point of view in the framework of 2D model transitions between "star-fish"
configurations with different number of spikes. One can mention that it is not the case, for
example, for 2D polarized droplets in amphiphile mono layers [14], where an intersection
between energetical curves corresponding to the configurations with different number of
lobes, has been found, if the shape is given in polar coordinates as
( 1 8 )
where n denotes the number of lobes, and S is the relative amplitude of the undulation. This
shape, for the present problem, gives worse results (larger energy) than smoothed
hypocycloidal curves and therefore is not discussed.
Besides that, as one can see from Fig.2, the elliptic configuration has always a smaller
energy than any configurations considered even including a "star-fish" configuration with two
spikes S 2: 02 (it is not exactly true in the case of S = 0.1, see Fig.S).
It should be also pointed out that this conclusion is in agreement with the results of the
direct numerical simulation [8] of the dynamics of the 2D magnetic fluid droplet in a rotating
field. It shows that, starting from a randomly perturbated state and going through intermediate
stages with a definite number of the spikes, the system ends up with the 2 spikes
configuration. That means that the physical explanation of the observation in experiment [4]
of transitions between configurations with different number of "star-fish" spikes, must be yet
found. Another issue which should be mentioned concerning Fig.2, concerns the first order of
the transition to "star-fish" configurations characterised by same hysteresis phenomena. These
71
Chapter 3: included paper
a plateau and the droplet shape slides down along the energy curve to the state of smaller
energy which corresponds to an elongated shape.
Decreasing the magnetic Bond number, the droplet conserves its elongated shape at
subcritical conditions till the second critical value of Bm corresponding to the turning point
in Fig.3. The energy minimum of an elongated shape disappears (Fig.4b) and the droplet
restores its circular shape with a jump-like decrease of its energy (FigAa).
The influence of the spike shape on the energy minimization calculations is illustrated for
a two-spikes "star-fish" in Fig.5; for a comparison, the energy of the elliptical droplet is also
shown. The first kind transitions to "star-fish" configurations are shown by vertical lines. The
threshold value is a function of the parameter S which controls the sharpness of the spike. It
can be concluded that for almost all the magnetic Bond numbers the elliptical shape has the
smallest energy in comparison with different two-spike configurations.
As one can see from Fig.6, the same conclusion can be drawn also with respect to the 4-
spike "star-fish" configurations. It is interesting to note that the values of the magnetic Bond
number, for the first-kind transition threshold to the "star-fish" configuration, decrease as the
parameter S increases. Transitions between curves corresponding to the different values of
the parameter S are also possible. So according to Fig.6, "star-fish" configuration arises at
the threshold with rather rounded small tips which are growing sharper and sharper as the
magnetic field strength increases.
The behaviour of the MF droplet described above for different numbers of spikes
(n=2,4,6) is summarized in Fig.7 for S = 0.1,0.2,0.4,0.8. For each number of spikes, the curve
corresponds to the minimal energy from the set S = 0.1,0.2,0.4,0.8 at each Bm. For n=4 this
choice of the curve of minimal energy is illustrated by the path A-B-C-D-E-F -0 in Fig.6 and
Fig.7. Discontinuities for a given "star-fish" configuration correspond to a transition to a state
of smaller energy, that is here to a lower value of the sharpness parameter S . In reality the
curve obviously must be smooth due to the continuous variations of the parameter S . Fig.7
shows that despite of the free variations of the parameter S , it is only the two-spike
configuration, or even the elliptic one, which have the smallest energy. That means that for a
74
Chapter 3: included paper
pure 2D MF droplet, in the rotating magnetic field, a transition to a two-spike structure must
take place. Numerical simulation results of the hydrodynamics of the MF droplet which will
be published elsewhere [8] are in fair agreement with that conclusion. That means that
physical mechanisms leading experimentally to the increase of the number of spikes with the
increase of rotating field strength, includes 3D effects or viscous flow around the droplet: tip
effects are more important at 3D than at 2D.
4.Conclusions
1. A 2D numerical analysis of MF droplets of different numbers of spikes shows that the
equilibrium shape with two-spikes corresponds to the lowest energy.
2. Along the branch of equilibrium figures with a given number of spikes, a calculation of
energy minimum shows that the spikes are sharpening as magnetic Bond number increases
beyond a definite threshold.
3. The present calculation shows that 2D figures of equilibrium with a "star-fish"
configuration can exist evidently only as transient states. A further confirmation of this point
by a direct numerical simulation of the magnetic fluid droplet hydrodynamics is highly
desirable.
4. Discrepancies between the present results and experimental observations are presumably
connected with the fact that a 2D model is rather incomplete for the understanding the MF
droplet behaviour in high-frequency rotating magnetic fields.
Acknowledgments
This work was supported by "Le Reseau Formation Recherche n090R0933 du Ministere de
l'Enseignement Superieur et de la Recherche" of France. Two of us (A.Cebers, S.Lacis) are
thankful to International Science Foundation for financial support of the research in terms of
long-time grant LBGOOO.
75
Chapter 3: included paper
References
1. Bacri lC., Salin D. Instability of ferrofluid magnetic drops under magnetic field
JPhys.Lett.,1982, v.43, N17, pp L649-L654
2. Cebers A., Maiorov M.M. Magnetostatic instabilities in plane layers of magnetic fluid
Magnitnaya.Gidrodinamika (in Russ.), 1980, Nl, pp.27-35
3. Langer S.A., Goldstein R.E., Jackson D.P. Dynamics of labyrinthine pattern formation in
magnetic fluids. Phys.Rev.A, 1992,v.46,N8,pp 4894-4904
4. Bacri J.C., Cebers A.O., Perzynski R. Behavior of a magnetic fluid microdrop III a
rotating magnetic field Phys.Rev.Lett., 1994, v.72, N17, pp 2705-2708
5. Cebers A.O., Lacis S. Magnetic fluid free surface instabilities in high frequency rotating
magnetic fields. Brazilian Journal of Physics, (to appear.)
6. R.E.Rosensweig. Ferrohydrodynarnics. Cambridge University Press, 1985, 344p
7. Hao Li, Hasley T.e., Lobkovsky A., Singular shape of a fluid drop in an electric or
magnetic field. Europhys.Lett., 1994, v.27, N8, pp575-580
8. Bacri J.e., Cebers A., Lacis S., Perzynski R. (to be published)
9. Sherwood lD., Breakup of fluid droplets in electrical and magnetic fields JFluid Mech.,
1988, v.188, pp.133-146
10. Cebers A. Numerical simulation of ferro fluid drop dynamics in static and rotating
magnetic fields Magnitnaya. Gidrodinamika (in Russ), 1986, N4, pp.3-1 0 (Y~cneHHoe
MO,Qen~poBaH~e ,Q~HaM~K~ xanna HaMarH~~~BalO~e~C51 )K~,QKOCT~ B
nOCT051HHOM ~ Bpa~alO~eMC51 MarH~THbIX nonsx)
11. Boudouvis A.G., Puchalla J.L., Scriven L.E. Magnetohydrostatic equilibria of ferrofluid
drops in external magnetic fields. Chern. Eng. Cornm. 1988, v.67, pp.129-144
12. Boudouvis A.G., Scriven L.E. Sensitivity analysis of hysteresis in deformation of
ferrofluid drops. .LMagn.Magn.Mater. 1993, v.122, pp.254-258
13. Zwikker e. The advanced geometry of plane curves and their applications, Dover
Publication, Inc., New Yark, 1963, 299p
76
Chapter 3: included paper
14. Vanderlick T.K., Mohwald H. Mode selection and shape transitions of phospholipid
monolayer domains - J Phys. Chern., 1990, 94,pp.886-890
FIGURE CAPTIONS
FIGURE 1
Contours constructed for different values of spike number and parameters £ and S.
FIGURE 2
The total energy of the 2D droplet E, - Eca versus the magnetic Bond number for s=0.2
(number of spikes n=2, ...,6) and for ellipse, fl=25.
FIGURE 3
The large semi-axis a = adim / R of the ellipse in the rotating magnetic field as a function of
the magnetic Bond number.
FIGURE 4a
The total energy of the ellipse E; - Eca in the rotating magnetic field as a function of the
magnetic Bond number, ).1=25.
FIGURE 4b
The total energy E; curves of the ellipse in the rotating magnetic field versus its large semi-
axis value a = adim / R, fl=25. Bm=23.97 corresponds to the threshold and Bm=20.7 to the
turning point.
FIGURE 5
The total energy E, - Eca as a function of the magnetic Bond number, for a 2-spike
configuration: various curves correspond to the ellipse and to shapes with s=O.l, 0.2, 0.4,
0.8.
77
Chapter 3: included paper
FIGURE 6
The total energy E, - Eco as a function of magnetic Bond number, for a 4-spike
configuration: various curves correspond to the ellipse and to shapes with S=O.l, 0.2, 0.4,
0.8.
FIGURE 7
The total energy E; - Eco as a function of magnetic Bond number. The curves are the paths of
minimal energy for ellipse and for the 2,4,6-spike "star-fish" configurations. Minimization is
done for S=O.l, 0.2, 0.4, 0.8. Arrows show the direction of hysteresis curves.
78
Chapter 3: included paper
E=O,1,2,3,4
e=O,0.5,1,1.5,2
e=O,1,2,3,4
Fig. 1
79
Chapter 3: included paper
o e 2 3 4 5 6
-12 o
·······2
--- 3
- 4 - ellipse
20 40
_.-5
-6
60
Bm
80
e
100
Fig.2
6
3
5
4
2
1
o 20 40
Bm
60 80
Fig.3
80
Chapter 3: included paper
0.2
0.1 Jl=25
0.0
0pJy -0.1
Iy
pJ -0.2
-0.3
-0.420 21 22 23 24 25 26
Bm
Fig.4a
0.6
-0.2
1 2 3 4 5 6a
Fig.4b
81
Chapter 3: included paper
2-spike instability
0 0.8 0.4 0.2 0.1
--_ .. 0.8
o -2 0.4o
~ I 0.2o~
-4 _._ ..- S=O·l
ellipse
-6
0 20 40 60
Bm
Fig.5
4-spike instability
0.8 0.4 0.2 0.10
- -- 0.8
-2 0.40(JOJ 0.2I ---------(J
~ -4 .......... __ ..._-- s=O.l
ellipse
-6
0 20 40 60 80
Bm
Fig.6
82
Chapter 3: included paper
o
o -2
~u
I
~u
-4
-6 o
-.-._.-.-.6
---- 4
------- 2
ellipse
20 40 60
Bm
80 100
Fig.?
83
Chapter 3
Conclusions
1. The results obtained from the energetical approach confirms existence of the magnetic
field threshold in respect to any perturbation from circular shape in 2D in the high-
frequency rotating magnetic field. Along the branch of the equilibrium figures with a
given number of spikes, a calculation of energy minimum shows that the spikes are
sharpening as magnetic Bond number increases beyond a definite threshold thus
causing the concentration of the magnetic field in these spikes, specially if a magnetic
permeability has as high value as ~:=:::25considering the concentrated phase.
2. A 2D numerical analysis of MF droplets with different numbers of spikes shows that
the equilibrium shape with two-spikes corresponds to the lowest energy and thus the
transition from 2-spike shape to the shape with larger amount of spikes does not take
place.
3. The present calculation shows that 2D figures of equilibrium with a "star-fish"
configuration can exist evidently only as transient states. A further confirmation of this
point for number of spikes n>3 is given in Chapter 4 by a direct numerical simulation
of the magnetic fluid droplet hydrodynamics.
4. Discrepancies between the present results and the experimental observations are
presumably connected with the fact that a 2D model is rather incomplete for the
understanding the MF droplet behaviour in high-frequency rotating magnetic fields.
That means that physical mechanisms leading experimentally to the increase of the
number of spikes with the increase of rotating field strength, includes 3D effects or
even the viscous flow around the droplet.
85
Chapter 4
High-frequency rotating magnetic
field: dynamic simulation.
In the previous chapter the equilibrium shapes were analysed from pure
energetical approach. In such a way it is possible for the given magnetic Bond number
to detect which shape from some a priori given set has the energy minimum. The
advantage of this method is that for linear magnetisation it is necessary only once to
calculate geometry and magnetic permeability dependent field variables ax, uy (see
formulae (11),(12) in [7*]), and afterwards only simple calculations (see the formula
(14) in [7*]) are needed to get the magnetic energy of a droplet. Thus by limited
calculations the great variety of shapes could be analysed for the energy minimum for
arbitrary values of Bond number. The drawback of the energy approach is that the
shapes should be given a priory, thus limiting the possible shapes by that ones,
generated in some fashion. Thus it is very hard to study the transitions, where non-
symmetric shapes appear, because these non-symmetric shapes are almost impossible
to predict a priori.
A direct dynamical simulation of the surface forced driven motion of the droplet
in high-frequency rotating magnetic field is presented in this chapter. The dynamical
simulation is in some aspect more computer time consuming method, but the power of
it is stated by the absence of constraints on the shape of the droplet. The essence of the
droplet behaviour is already described in the previous chapter, here the behaviour of
the 2D droplet in the high-frequency rotating magnetic field is studied in more details
using the dynamic simulation as an observation instrument, and, in particular, most of
87
Chapter 4
attention is devoted to the transitional motion of a droplet. Transitions of a 2D droplet
in the high-frequency rotating magnetic field could be classified as:
• transitions just beyond the magnetic Bond nwnber threshold;
• transitions from the perturbed shape back to the circular one;
• transitions between shapes with different nwnbers of spikes.
4.1. Theoretical predictions
Transitions which appear at the magnetic Bond nwnber threshold, where the
shape of a 2D droplet is near to the circular one, could be studied using the small
perturbation theory. The simplest case, the linear small perturbation theory, gives
predictions about the critical magnetic Bond nwnber and about the rate of the growth
of perturbations.
In [35] the infmite cylindrical volwne of magnetic fluid (2D droplet) is considered
under the action of the rotating in a plane normal to its axis magnetic field. It is
assumed the characteristical time of the droplet shape relaxation is much larger than
the period of the rotating field and time averaging with respect to rotating field is
possible. Under these asswnptions the stability of the droplet's shape is considered
with respect to n-lobe perturbation given in polar coordinates (r,~) by
r( ~) = R + all cos(n~), ( 4. 1 )
where R is the radius of unperturbed 2D droplet. The differential equation for
perturbation an development is obtained on the basis of the Cauchy-Lagrange integral
[35,25], taking into account kinematic condition and the time averaged magnetic field
strength, which is calculated in linear approximation of small perturbation theory:
di a; cr {< Bm (~-IYJ--+-·n(n-l n+l)---- a =0.dt' pR3 211: ~+IY n
The solution (4.2) defines an exponential dependence on time for perturbation
(4.2)
a =a nexp( co t), thus providing the dimensional growth increment of the surface
perturbation co:
2 c , {Bm ~ -IY ~co = -3 . n\n -1 - 3- (n + 1) .
pR 2n (~+l) (4.3)
88
Chapter 4
From this expression the threshold value of magnetic Bond number with respect to n-
lobe perturbation is found:
Bm(n) =2rr/n+I)~+IY. (4.4)
cr \ ~ -lY
This value in the case of n=2 is in very good agreement with numerical simulation
(see Chapter 2). According to (4), the perturbation with n lobes may appear only if
magnetic Bond number exceeds the corresponding threshold value Bm~;). According
to Fig.4.1(a), the dependence of the threshold value on the magnetic permeability is
not particularly pronounced for ~>20. Quite different is the situation in the case of
small values of a magnetic permeability (u« l 0), decrease of the magnetic
permeability, what physically can be caused by the dilution of the ferrofluid, leads to
the drastic increase of the critical value of the magnetic Bond number (Fig.4.1 (b)).
More information about perturbation evolution could be obtained analysing the
expression (4.3) with respect to "the most unstable mode", which has the greatest
growth rate at given magnetic Bond number Bm. This "most unstable mode" is given
b I· f h . a0)2y so ution 0 t e equauon -- = 0 :an
(2n-l)Bm'-~n2 -1)=0, (4.5)
Bm' = Bm ~ -1): = H~R ~ -1): ' (4.6)
2n (~+1) 2ncr (~+1)
adding the condition that 0)2>0 for the found value of n at given Bm. The largest root
of the equation (4.5)
Bm + ~ Bm .2 - 3 Bm + 3n. = ---------- 3
satisfies that condition for all n,>1. The fairly good approximation of is given by
(4.7)
2Bm·n, ~-- (4.8)3 2'
according to Fig.4.2, the absolute difference even at n.=2 is only 0.06 (3%) .
•According to (4.3) if Bm value is beyond the magnetic field threshold for mode n=2
• •(Bm >3), then the set of competing modes from n=2 to n=Bm -1 co-exists, but the
highest grows rate has mode n•. It is illustrated in Fig.4.3 where a square of a
89
Chapter 4
modified perturbation decrement (j)2pR3/cr is plotted versus n at fixed value of
•modified Bond number Bm =16. Unstable modes are for 2Sn<14, but the highest
grows rate has mode n=10 (from formula (4.8) n.=10.17).
4.2. n-Iobe perturbation decrements in the case
of creeping flow
In order to derive perturbation decrement in the case of a viscous 2D ferrofluid
droplet placed in a surrounding non-magnetic viscous fluid in a presence of high-
frequency rotating magnetic field, the problem of the creeping flow is solved at given
perturbation rate ~r = L cos(n~) on the surface of the circular boundary at the
restriction that ~. The value n= I corresponds to the pure translation motion
vx=const (check (4.14)-(4.17)), there is no change of nether magnetic nor surface
energies, the rate of energy dissipation is zero. Solution of biharmonic equation
t1(t1<1» = 0, (4.9)
where t1 is Laplace operator and <1>is a stream function, is used to calculate velocity
components in polar coordinates (r,~):
18<1> 8<1>
v r = -; 8~' v <l> = - 8r .
unknown coefficients for
(4.10)
The the general solution of (4.9) are found from the
following three constraints.
1. Continuity of velocity vector at boundary:
v/''1 = vexl .r=R r=R
2. Equality of tangent stresses on boundary:
(4.11)
i"l ex 1cr til r=R = cr til r=R· (4.12)
3. The constraint of the perturbation shape:
v I = i: I = L cos(n~ .r r=R '-:>rr=R (4.13)-Since in the present case a velocity field is caused only by perturbation ~, the
azimuthal component of perturbation ~¢' could be found from velocity field:
~q. = vq..
90
Chapter 4
The general solution of (4.9) is found by substitution of <Din (4.9) by the
trigonometric series with respect to the azimuthal coordinate ~:
<1>=II!II(r)cos(n~)+gll(r)sin(n~).
II
Application of constraints (4.11 )-(4.13) gives the velocity field ViII inside the droplet
and the velocity field vex outside the droplet:
L I I )"-1 I )"+1]
V~I =2cos(n~1 (n+1~~ -(n-1~~ ,
ill _ L . (th{ ( l~r)"-I (n-1Xn+2)(r)"+I]v --smn,!, -n+ - +-----
<I> 2 R n R '
L I I R)"+I I R)"-I]v;x=-2cos(n~1(n-1~-; -(n+1~-; , (4.16)
V;'~-~Sin(n+n-tr' _(n+l~n-2tr1 (4.17)
Th f""' . elf!l - 1 H2 . !' d fr h S k .e e tective pressure P = P - -- IS 10un om t e to es equation8n
(4.14)
(4.15)
. 2 () IIelf_ In -1 r
Pill - PO.ill - 2r1;n R -n- R cos(n~ ),
elf _ 2 L n2 - 1( R) II ( th)
Pex - PO,ex + 11ex --- - cos n'!' •R n r
Thus the pressure difference on the surface of the circular droplet r=R is
(4.18)
(4.19)
~ _ elf elf _ ~ 2( ) L n2 - 1( R) II (th)
P- Pex - Pill - Po + 11ex + 11 in R-n- -; COS n'!',
where ~Po = PO,ex - PO';II .
(4.20)
Th 1 t f vi ~rr -- 2n ov r on the surface r=Re norma componen s 0 VISCOUS stresses v •I or
are zero according to the expressions
(4.21)
(4.22)
91
Chapter 4
The dynamic boundary condition on the free boundary r is expressed in terms of
normal forces:
{ _eif eif "t (ex in 2 (M)2 IJ.- 1 H2 d' J'Pex -Pin Ar = ann -ann + rt n +--g;- -a rv n r·
Without perturbations the boundary condition yields
(4.23)
IJ.-1 «: X )2 2) aSp; =--\\j-t-l Hn .+H --.8n R
Magnetic field solution for circular shape of a 2D droplet in polar coordinates is [35]:
(4.24)
H=2Ho(cos(nHt-~)e +sin(nHt-~)e~. (4.25)1J.+1 r
Here e r' e~are unit vectors in radial, resp., azimuthal directions. Time averaging gives
( )) 4 H~H- = 0.t+1Y' (4.26)
I( )2) 2 H~\ Hn = 0.t + i)' . (4 . 2 7 )
Inserting (4.26) and (4.27) in (4.25) gives pressure difference on the boundary of
circular 2D droplet in high-frequency rotating field:
IJ. - 1 H~ aSp; = - - - -. ( 4 . 2 8 )
IJ. + I 4n R
In linear approximation small perturbation theory applied to (4.20) gives In
addition to (4.28) the first-order terms containing Land L, thus enabling to obtain a
non-dimensional perturbation decrement, denoted here by ~co:
L = wrro L, Lex exp(wtlro , LO=T]exRJa : time scaling unit. To obtain additional
terms of (4.23) in comparison with (4.24), (4.28), the perturbation of a magnetic field
and a boundary contour curvature should be found. It is convenient to use Lagrange
variation ~L (variation due to Lagrange displacement) technique [25,32,42,88] to
derive the first variation of boundary conditions on shifted boundary, expressed by
physical variables at unperturbed boundary placement FR. Variation of an effective
pressure and viscous stresses are already given by (4.20)-(4.22), since there is no flow
in equilibrium state:
92
Chapter 4
~ r{ efl elf ) f. ex in)~_
L L\pex - Pin - \cr nn - o nn ~ -
( efl(l) _ elf(l») _(crex(l) _ in(l) 'iI =\?ex Pm r:R \: rr rr rr A'lr:R (4.29)
L n2 -1(R)n2(llex+ T1;n)--- - cos(n~)R n r
The magnetic field variation is found in [35], solving Laplace equation inside
and outside the circular 2D droplet and applying first variations of boundary
conditions for magneto static potential:
(\fI(I) _ \fI(l») = (Il-lf 8\f1~~)~ )
\ I/l ex r:R -\ 8r r ,
r:R
( 8\f1i~~)_ 8\f1;;») = ll-l(~(8\f1i~0) )JIl 8r 8r R2 8~ 8~ ~r
~R ~R
Magnetostatic potential \fILO) of unperturbed droplet according to (4.25) is [35]:
(4.30)
(4.31)
2H2\fILO) = __ 0 r cos(nHt - ~). ( 4 . 3 2 )
Il + 1
First variation of magneto static potential inside the droplet \fIi~~) is found in [35] by
technique described above, arriving at
\fIi;,I) = 2 LHo fl.-12 (!-) 11-1 cos(QHt - ~ + n~ ).(fl+l) R
Magnetic field intensity components could be
potential by the following derivatives:
(4.33)
found from magneto static
H = 8\f1 H -!8\f1 ( 4 . 3 4 )
r 8r' ~ - r 8~
The Lagrange variation of magnetic field term in (4.23) due to Lagrange displacement
IS
~ L [CIl-IXHn)2 + H21 =
~ -l)2(H(O)n(O) XH(O)(~ Ln)+ nlO)(~ LH))+ 2(H(0)(~ LH))l = (4.35)
~ -1)2Hr(H(O)(~ Ln)+ erH(I»)+ 2(H(0)H(I) )1·
A Lagrange variation of normal vector ii and a variation of divii in Cartesian
coordinates are found according to formulae from [42] :
93
Chapter 4
(4.36)
[aaiJ _ a ( ) aai aSk~L - -- ~La ---.(]xi (]xi I (]xk (]xi
In polar coordinates expressions (4.36) and (4.37) result in
(4.37)
(4.38)
(4.39)
Now expression (4.35) could be rewritten explicitly, inserting result of (4.38),
what leads to
/ rr. . X )2 2]) 4 LH~ (11-1 Y ( ) (
\ ~ L L\P- - 1 Hn + H r = -R- ~+ 1) n -1 cos n~ • ( 4 . 4 1)
The time-averaged boundary conditions (4.23), accounting for (4.31), (4.39) and
(4.41) results in equation
• 2L n -12(11e.< + 11in )- -- cos(n~) =R n
LH~(Il-IY (n -1)cos(n~)- La (n2 -1}:os(n~)
2rrR (Il + ly R2
Thus the non-dimensional perturbation decrement cD is
(4.42)
94
Chapter 4
~ n(Bm* - (n + 1))
m=-----, (4.43)2(1 + "-Xn + 1)
where Ic=11ij'llex is a ration of viscosities but Bm* is already defined in (4.06).
Magnetic field threshold with respect to n-Iobe perturbation (4.13) is given by the
same value of critical magnetic Bond number
Bm*(I/) =(n+l) ~ Bm(n) = 21t(n+l)(f.t+1Y . (4.44)
cr cr (f.t - IY
as in (4.4) for potential flow. Comparison of (4.43) with (4.3) displays the basic
difference between these two models. The expression (4.3) corresponds to ideal flow,
the system with inertia but without a friction, therefore there are only two choices with
respect to n-Iobe perturbation: either the exponential growth of perturbation (m2>0,
anccexp(mt)) or the oscillations about the equilibrium state (m2<0, anccexp(imt)). The
expression (4.43), in contrary to (4.3), represents the highly damped system without
inertia and oscillatory motion is damped in time interval that is much smaller than
period of oscillations, thus only the perturbation due to the action of some force on the
boundary could take place. Nevertheless, due to the same treatment of magnetic field
and surface tension, in both cases magnetic field threshold has the same value. In the
case of ideal fluid pure mathematically the lower limit of the perturbation growth
spectrum is n=l due to multiplier n-l in (4.3) (see Fig.4.3) but in the case of creeping
flow the lower limit for n always should be 2 due to the physical properties of the
perturbation (4.13). The positive values of cD in the formula (4.43) at n=l ifBm* is
large enough is the artefact since equation (4.42) at n= 1 is always satisfied and thus
have no roots. Beside that, as it is already mentioned above, "number of lobes" n= I
have no real physical sense.
The threshold value with respect to 2-lobe perturbation is the absolute threshold
for any perturbations from circular equilibrium state, and thus determines the upper
limit of the perturbation growth spectrum. The most unstable mode n:. in the case of
. fl 'd' c d ideri . acD 0 h ~. ak fVISCOUS U1 IS roun consi enng equation - = , were co IS t en roman
expression (4.43):
95
Chapter 4
n' =JBm' -1 = Bm (!-t-l~ -1.
v 2n (!-t+l)
In Fig.4.4 the most unstable mode lobe numbers are plotted versus magnetic Bond
(4.45)
number Bm for three values of magnetic permeability !-t=5;15;25. These three curves
display strong dependence of droplet behaviour on its magnetic properties, so the ratio
of magnetic field strengths, needed to have the same most unstable mode two cases:
!-t=5and !-t=25, have the magnitude 1.63. The same ratio 1.63 stays for corresponding
threshold values of !-t=5 and !-t=25. The increase of magnetic permeability causes
increase of the number of lobes for the most unstable mode at the same magnetic
Bond number, the limiting curve at !-t=oois plotted in Fig.4.4 by the dashed curve. To
illustrate the dependence of instability on magnetic permeability, in Fig.4.5 the
characteristic multiplier J(!-t- 1Y / (!-t+ ly is plotted versus magnetic permeability
!-t.
Including characteristic multiplier J(!-t- ly / ~ + 1Y in the modified Bond
number as it is defined by (4.6) allows to obtain general properties of the most
unstable mode for an arbitrary value of !-t.In Fig.4.6 the spectra of unstable modes are
plotted in the case of the creeping flow according to the relation (4.43). In comparison
with the potential flow (see Fig.4.3), the most unstable modes are shifted to the
smaller values, for Bm*> 10 even few different modes have about the same
perturbation decrement, thus the initial state plays an important role to the formation
of the dominant number of lobes.
The limiting curves plotted in Fig.4.7 represent the threshold of magnetic Bond
number and n=2. The crossing of n=2 with the threshold curve at Bm*=3 gives the
first limit: if Bm*<3, droplet stays circular; the crossing of n=2 with most unstable
mode certifies that eventual transition to larger number of spikes than 2 can appear
only ifBm*>9.
96
Chapter 4
4.3. Dynamical simulation of the 2D droplet in
the high-frequency rotating magnetic field
The dynamic simulation for 2D droplet is carried out for the time averaged
surface forces. The averaging is done with respect to external field rotation, thus the
droplet motion is still time-dependent as well as the amplitude of a magnetic field,
expressed in terms of the magnetic Bond number Bm. The field threshold test with
respect to the droplet perturbation from the circular shape is already shown in Fig.2.7.
Here the test of the small perturbation growth according to the expression (4.43) is
presented in FigA.8. All calculations were carried out for periodic perturbations with
an amplitude Qn=O.OI(see expression (4.1)). Time step was taken ~t=0.01, number of
marker points N=300, magnetic Bond number Bm=100 (Bm*=12.52) at magnetic
permeability fl=25, the ratio of viscosities A=l. The perturbation increment was
calculated for the corresponding amplitude of the n-th harmonic in the Fourier
spectrum for a radial perturbation of the shape. The calculated points, shown by
circles in Fig.4.8, are in very good agreement with the theoretical curve, shown by the
solid line. The small discrepancies are easy to explain by the competition of the most
unstable modes with other ones.
In FigA.9 the perturbation growth and the corresponding Fourier spectra of
perturbations are shown for three cases. Initial states were generated by random
perturbation, realised in the following way: the circle in 60 equidistant points was
shifted in the radial direction. The values of shift were taken as random number from
0.0 to 0.03·R, R being circle radius. The magnetic Bond number Bm=lOO together
with fl=25 corresponds to the limit of maximal allowed mode number n=12 (see
FigA.8). Thus the n-lobe perturbations are allowed only for n=2 .. 12. According to
FigA.8, most unstable mode is n=3. Shape growth in FigA.9 displays, that modes
n=2;3;4 already dominate. Since spikes are not situated symmetric and have the
triangular shape, than in Fourier spectra appear higher harmonics and their growth
rate is approximately the same as for most unstable mode.
The hysteresis phenomenon for 2D droplet in the rotating field is displayed in
Fig.4.10. The solid curve represents the elongation of the elliptic droplet in time-
97
Chapter 4
averaged rotating field (see [7*]), The pointed curve shows the behaviour of the 2D
droplet obtained in the dynamical simulation without any shape constraint. This result
is obtained by the piece-wise linearly time dependant magnetic Bond number. The
value of it is at first kept constant (Bm=25, segment t=0 ..50), then Bm is linearly
increased from Bm=25 to Bm=27 in next 20 time units (segment t=50 ..70).
Discrepancies could be caused by the shape difference. Decrease of Bm to the value
Bm=20 in segment t=70 ..120 causes a contraction of a droplet. The tendency to
extend for a droplet at just after t=70, when the value of Bm already decreases,
displays that the transient motion was not finished and at t=70 droplet was still not
close enough to equilibrium state. Obviously the rate of Bm decrease was all along the
curve too large, thus causing very different slope of the dotted curve in comparison to
solid one. Without the constraint of the elliptic shape the behaviour of the droplet
could be different from the one of elliptic droplet, but it seems that Bm=20 is already
below the turning point (for ellipse it is about 20.7), since further decrease of the field
causes smooth transition back to circular shape. The more exact "fit" to the transition
curve of the elliptic droplet could be obtained by decreasing the rate of the change of
Bm in sequential time intervals, but is should cause significant computer time
Increase.
The transient stages of the 20 magnetic fluid droplet are already described in
[8*] (this paper is included in the present Chapter). In the addition to the paper, here
in FigA.ll the energies Em' Es, E, and the length r of the major spike are plotted in
dependence on time t. The thick solid line represents the total energy Et=Em+Es, the
dotted line: the magnetic energy Em' the dashed line: the surface energy Es' the thin
solid line: the length r of the major spike, measured from the mass centre of a droplet.
These results correspond to the evolution of a droplet's shape in Fig.la of [8*]. The
curve of the length r of the major spike has an approximately constant inclination
during the time interval 0<t<0.6. The shapes of the droplet display that it is an initial
time during which the droplet "finds the favourite growth direction". It means that
strong competition between the spikes takes place, those ones who are larger and
sharper can concentrate more magnetic field thus stimulating a further growth of
themselves and suppressing the evolution of smaller spikes. Thus at t about 0.3 the
98
Chapter 4
"weakest" spikes start to contract and non-reversible transitions to smaller number of
spikes take place, since decreased value of the magnetic surface forces cannot balance
the surface tension established by the curvature of a spike. Thus due to the lower local
value of a magnetic Bond number the smallest spikes contract, causing further
decrease of a magnetic field until these spikes are completely disappeared. So during
the time interval 0.6<t<2.5 the decrease of the smaller spikes and the accompanying
concentration of a magnetic field in the larger spikes causes increase of the growth's
rate of last ones. At ~5 the droplet has established elongated shape with a very small
"side-spike", which start to disappear (see Fig.la of [8*]). At this time moment the
growth of the droplet elongation slows down, surface energy E, has already reached
the plateau, all changes of the total energy E, are due to a change of the magnetic
energy Em' In the present graph Em has noisy character, because the number of marker
points N=350 is still not sufficient to have good accuracy for a magnetic field
calculation at extremely sharp spikes. Nevertheless this approximation error could be
regarded as some perturbation of the droplet shape which cause oscillations about the
stable state, but do not produce the deviation of the mean value of the magnetic
energy. Thus, in Fig.I b of [8*], the three-spike shape appears to be stable with respect
to such a perturbation.
In [8*] the existence of the metastable three-spike shape at Bm=45, 1-1=15IS
proved as well as the transition to the two-spike shape. This transition appears at the
decrease of the magnetic Bond number Bm down to the value Bm=36 thus
eliminating the energy barrier which have suppressed the 3-spike shape from
transition to the 2-spike one at higher value of Bm. The corresponding energy curves
are plotted in FigA.12 and FigA.13. The analysis ofFigA.12 displays that the 3-spike
metastable state is stable with respect to symmetric perturbation, the decay of this
perturbation, given by initially over-elongated spikes, have exponential character. The
juxtaposition of the horizontal guidelines with the corresponding energy curves in
FigA.12 shows that final stage of the transition after t=20 has very small changes of
energies in comparison with the initial stage at t<20. The decrease of the magnetic
Bond number Bm below the magnetic field threshold with respect to 3-spike
perturbation (Bm~;) = 37.5) causes the transition to 2-spike shape, illustrated in
99
Chapter 4
Fig.4.13. The energy curves show that after initial decay of the transition to the
smaller length of spikes (0<t<25) the transition to 2-spike shape occurs during the
rather long time scale 25<t<90, because it is caused by very [me interplay of magnetic
and surface energies. Then, after t=90, during the short time, about 5 time units, the
rest of the smallest spike is taken in by the droplet.
In Fig.4.14 the results of the numerical simulation of a droplet with the 4-spike
initial shape are presented. Evidently the number of spikes, larger than 3 is critical
with respect to the necking effect which takes place between two groups of spikes,
since the number of spikes 4 is the smallest one for a formation of two groups of
spikes, every of which have at least two spikes. The magnetic Bond number Bm=60 in
Fig.4.14 is above the threshold value (4.4) with respect to a small 4-lobe perturbation:
Bm~:) = 46.9 at !l=15. The scenario is similar to the one ofFig.4.13. At the beginning
the spikes elongate close to the equilibrium state for the symmetric 4-spike shape,
afterwards the small perturbations, which are always present due to numerical
truncation errors, causes the development of an asymmetric shape with a neck
between two pairs of spikes (time interval from t=8.74 to t=41,40). After that this
neck starts to elongate, simultaneously two opposite spikes decrease, other two
increase (see a sketch inside the shape at t=53.19).
100
Chapter 4
3456789
Magnetic permeability 11
Figure 4.1. Dependence of magnetic Bond number on the value
of magnetic permeability ).1. (a): selected modes n=2,3,4,6,8, 10
at 1<).1<50, (b): the behaviour of the critical Bond number at
low values of ).1.
(a)
eCO
(b)
8CO
1-0 300
Q.)..0a::st::
""d 200
t::oCO
o
n=2,3,4,6,8, 10
500
o
o 10 20 30 40
Magnetic permeability 11
50
n=2 ..l0
400
o
1 102
101
Chapter 4
*t:=
r./). 3o..0o-~o
~ 2..0
S;::s
Z
102
4
1
Approximation of most unstable mode
Exact curve
I
2
Approximation
3 4 5 6
Magnetic Bond number Bm *
7
Figure 4.2. Approximation of the most unstable mode
Chapter 4
500
400
300
b.............
cia 200Q.
N8
100
-100
Spectrum of unstable modes
a .............
\
\
\
a 4 6 8 10 12
Number of lobes n.
14 162
Figure 4.3. The spectrum of competing modes at Bm*=16
for a potential flow
103
Chapter 4
16
*s:= 12
00<l),£Jo-Co+-; 8o
~(]),£J
8~ 4Z
500 1000 1500
Magnetic Bond number H2R/cr
Figure 4.4. The most unstable n-lobe mode versus magnetic
Bond number in the case of creeping How
a 2000
104
Chapter 4
1.0
0.6
0.4
0.2
0.0 o 10 20 30 40
Magnetic permeability J.l
50
Figure 4.5. The influence of the magnetic permeability
to the n-Iobe mode formation: the multiplier (((IJ.-I)/(!J.+ l)yt5
versus the magnetic permeability IJ..
105
Chapter 4
Definitions:
*8
10
....
ii/
: ... .'t,
!:/....if /
2 :: ..J..g ••.... /: ..' 1*.. I :.. X
/ .
8
6
4
o
o
-2
lc::- Bm *= 16
............2 144
Number of lobes n
Figure 4.6. The spectrum of competing modes at Bm*=6; 11; 16
for a creeping flow
106
Chapter 4
12
10
s:: 8rJJ(j,),.0
0•....•
t..+-< 60
l-;
(j,),.0 4S;::i
Z
2
0
Circular
droplet
,/
,/
/'
,/Elongateddroplet
i,/
n-Iobe perturbation (Transition to larger,/,threshold at ,/! amount of lobes
given Bm* ,/ i becomes possible<:/ !
,/ !
,/ I',/ !
t/ I _+--"-------------~...------:',": ------------- ~n=2
"<~'-<---1------C- Most unstable mo~e
-:.,' : :
,0~ i i~,,"".
o 2 4 6 8 10 12
Modified magnetic Bond number Bm*
14
Figure 4.7. Transition diagram
107
Chapter 4
2.0
8 1.5..•...s::
(l.)S 1.0
ol-o
U.g 0.5
s::o.~ 0.0-eB -0.5o0..
-1.0
2
Bm= 100, Jl=25
Bm*=12.52
A=1
4 166 8 10 12 14
Number of lobes n
Figure 4.8. The spectrum of competing modes at Bm= 100,
A=1, solid lide: the creeping flow theory, circles: BEM
simulation
108
(a)
Chapter 4
Shape evolution
(b)
Shape evolution
(c)
Shape evolution
.g 0.2
.~0...
~ 0.1
Q) 0.5
'"0
.~ 0.40...
~ 0.3
0.2
0.6
0.5
Q) 0.4'"0
.~- 0.30.-
~ 0.2
0.1
0.0
OJ
o t=O.O
II t=0.5
• t=1.0
....- t=1.5
-<1- t=2.0
-,..... t=2.5
--- t=3.0
0.0
o 2010 155
Mode number n
0.7
0.6
o t=O.O -+_. t= 1.5
II t=O.5 -<1- t=2.0
• t= 1.0 - ,..... t=2.5
--- t=3.0
0.1
'"f \ /
I • ~.JI" \ ..- •....•...f \ I '»---..v' \ I ....-.-.--- .• -.
I ~-..... ------..
J ..•.....•.....• .....•-+-•....•...+-+- .•....•...•...•...•......•....- -
('I e 8'"1""1 eel I I I I I I I I I I ••
o 5 10 15 20
Mode number n
0.0
o t=0.0 .._ t=1.5
II t=0.5 -<1- t=2.0
• t= 1.0 -,..... t=2.5
--- t=3.0
/, \ ,. ~\ ~ / Y y ••......•~ ..•..,...-,..........•• ...••
I )'1 -I "¥1./ \ )4. JL.... .-. ...•.....•- •..•...• - •....•....•....•...•/ \;/ ~/ .•..
I ....-; •••.
I:~ "\ •....•..•.•.•..0'...•..•.•.•.•.....•.. +- .•...•....•.....•.- ...-•...•....•...•
J... A.'
.' ~ ~.. A • • A £ A A • A A A A A A A
a a c DOD Dca a ace D D DaD D a
~ 0 0 0 0 0 a 0 a 0 0 0 000 0 0 0 0 0 0
o 5 10 15
Mode number n
20
Figure 4.9(a)-(c). The amplitudes of modes versus a mode
number n: the mode competition for a droplet with an initially
perturbed surface. Bn1=100, ~=25, ~t=O.Ol, N=200. The
initial perturbation given by radial displacements from a
circular shape in 60 equidistant points, generated randomly
with the maximal value 0.03.
109
Chapter 4
(d)
Shape evolution
(e)
Shape evolution
(t)
Shape evolution
I.2
1.0
(1J 0.8-e.a.- 0.6-a
~ 0.4
0.2
0.0
D t=1l.0 5 25t=1l5 ..+- t=l. •..• t= .
: t=1:0 - ••.. t=2.0 ...•... t=3.0
---lI. ''''~x 'r~~""''''''''''---1\."/
II • /.............r .•.....•.~ __ .•_.~ .•_ •....•..• -... ...•Y,' ' ..•..• / •.
IoJ /1A. ••••• _ ••••••••.••••••.••.••• ~ .••••••.•.••••.••••••••••••••••••.•• +-+
~ • A A A A A A • & A • • • 6 • A • • • A
a cae c c DaD D D D D D C D D D D C
o 5 10 15
Mode number n
20
0.7
0.6
0.5
(1J
'"d 0.4.B-a 0.3E<C 0.2
0.1
0.0
0
,•.'..-.-•.•....•...•-.--- ..•- ..--..------
5 10 15
Mode number n
20
0.7
0.6
(1J 0.5
'"0.a._ 0.4-a
~ 0.3
0.2
o t=O.o .-•. - t= 1.5
a t=O 5 -.- 1=2.01\ ....• .&t=1·0 ·_t=2.5~ / -.." ?,,/ 'Y.... . __ t=3.0I V \.1' .•..•..~
I
I •.
.• " ,",,-_ /Ji-.-.. ...•...•...•_•....•....•_ ...•...-.......
I 'rl ...•
I,,~'.•...,•....•....•.....,....•....•...•...-.._ .....•.....-..---
~ 8 • A 8 B • A B B B B B B B B B B B B •
0.1
0.0 o 5 10 15
Mode number n
20
Figure 4.9(d)-(t). The amplitudes of modes versus a mode
number n: the mode competition for a droplet with an initially
perturbed surface. Bm= 100, ~=25, L1t=O.O 1, N=200. The
initial perturbation given by radial displacements from a
circular shape in 60 equidistant points, generated randomly
with the maximal value 0.03.
110
Chapter 4
\:$ 4
t'-l0-
~
Io-S 3(l)
t'-l
~o
o~ro
::E2
5
-- Ellipse, analytically
...... Num. simulation:
),,=1,N=200, ~t=O.05,
&-=0.01
00o·00o·00
00
00
00t=120000•••••""•••""o""""""""""•"•""o"o"t=190
J.l=25
··········t=70......•... : ...... r~50
o···o··o···~,,,,,,,,
,,
et=O
20 22 24 26
Magnetic Bond number Bm
28
Figure 4.10. The elongation of the 2D droplet in a rotating
magnetic field and the transition back to circular shape
at the decrease of the magnetic Bond number below
the turning point
1
18
111
Chapter 4
10
E
~ -10bI).
~
il)l::=
r:Ll -20
-30
-40
0.15 0.34 0.45 0.84 Two-spike shape
19.77,, 8.0, ~,,, il)-t, 7.0 ~,,, .-, c,,,, srx, 6.0 ~0.....-,-.cd
5.0 8
il)..c=4.0 ~
~0
3.0 ..c=bo
2.0 l::=il)
~
1.0
t 20
Figure 4.11. Surface, magnetic and total energies, and the length r
of the major spike versus time 1. The solid line: the total energy
Et=Em+Es, the dotted line: the magnetic energy Em' the dashed line:
the surface energy E, the thin solid line: the length r of the major
spike, measured from the mass center of a droplet
112
o
I I I I
I ~---------- I I: :~--:::::::::::::::::::~------ --- -~
r !: ii 1.72 2.46
,II Iiii 1.17 -: 5.24·
l'l .,'" IOil .: ~/3.97 i 7.03
1.:1 I I I I I::1:: : : : I
III I I I I
III I I I I
II, I I I I
I'I I I IE
:::::: s~:~~i-~--f--+--~
I -rr':: i
" ,
" ," ," ,
:: I
i i r~
,,,,,,,,,
Bm=200
~=25
N=350
~t=O.Ol
9.69,,
11 73
12.73:
,,, ,+----.....-._-: :,
: rJ~--------~--E.3"
s
o 5 10
Tin1e t
15
Chapter 4
The metastable three-spike shape
\ ~~
0.0 4.8
E 4 " i'~ E3 ,-- s-------l------.::-~-~-~-~-~-~.,..-----.-----.-.----~-.,,,,
,,,,,,,,,,,
,
?
36.519.8
2
1>--en 0~
C)s= -1~
-2
-3
-4
,,,,,,,
,
I : I, ,, , ', ,-E~-l---::.-.~~::~:.-.:~~:. ·-- ~;;··-·-f..'11· ••••••••••••••• • ••~··~···- ••• ~ •• ~ ••. -~--
f··· ,........ :s·- :,-5 o 10 20 30
Time t
40
Bm=45
~=15
N=450
~t=O.Ol
4.4
~
4.2 C)~.-c,
4.0 sr:
~03.8 .~roS3.6 C)..c
3.4 ~~0
3.2 ..c~bO
3.0 s=C)
~
t 2.8
50
Figure 4.12. Surface, magnetic and total energies, and the length r
of the major spike versus time 1. The solid line: the total energy
Et=Em +Es, the dotted line: the magnetic energy Em, the dashed line:
the surface energy E, the thin solid line: the length r of the major
spike, measured from the mass center of a droplet
113
Chapter 4
'Transition frorn a three-spike
shape to a two-spike one
Bm=36
~=15
N=450
~t=O.OI
o 70 80
3 I : :E ' ', '--" : :
I
": : :' E, i I Ii! .,....2 -. I : I : I :
I
- - -- - ---- - - -- ------ ----:-:-~- ~ -~----. --_._-~._-~--~-- .•••. - .•• ---~;-- --- i
I I r I r
I I I I I
I I I I •
I 1 I I I
I I I I I
I I I. I
I I I I I
I 1 I I I
I 1 I. I
I I I. '
I I I I
I I I I
I I I r
I I I I
I I 1 I
I I I I
I I I I
I I 1 r
I I 1 I
" ,, ,, ,, ,, ,, ,: '
105
1
,,,,,
, ', '...-r---- ..~-, --------....
I I - .••" I: : :.... ;
I I I a. I: i: ;
I I I I
I : 1 :
-1
-2 ,-----E--- ------:.-.-.~:;;~.-.-.~;;;;;...m······· :••••••••• i..- :,-3
o 20 40 60
Time t 80
100
3.4
~
(])3.2 ~0-c,
003.0 ~o
o~ro2.8 E
(])
2.6 .s
C+-1o
2.4 .s
eos:=
2.2 j
2.0
Figure 4.13. Surface, magnetic and total energies, and the length r
of the major spike versus time 1. The solid line: the total energy
Et=Em +Es, the dotted line: the magnetic energy Em' the dashed line:
the surface energy E, the thin solid line: the length r of the major
spike, measured from the mass center of a droplet
114
Chapter 4
N=360, L1t=0.05, &=0.01
Brn=60, J.l= 15
t=8.74
t=53.l9
t=41.40
c::::-====:::::::--=-_- _- _- _- _- _- _-t==6=4.=4~
c::: t=78.06
I
-5
Scale
Io I5
Figure 4.14. Transition from the four-spike shape to the two-spike one
115
Chapter 4
Included paper:
Bacri J.C., Cebers A., Lads S., Perzynski R.
Numerical simulation of the transient stages of 2D magnetic fluid
droplet in high-frequency rotating magnetic fields
Magnitnaya Gidrodinamika, 1996 (accepted to publish)
117
Chapter 4: included paper
Numerical simulation of the transient stages of 2D
magnetic fluid droplet in high-frequency rotating
magnetic fields
J.C.Baeri (*), A.Cebers (**), S.Laeis(**),R.Perzynski(*)
(*) Universite P.M.Curie, 4 place Jussieu, 75252 Paris, Cedex 05, France
(**) Latvian Academy of Sciences, Institute of Physics, Salaspils-I, LV -2169, Latvia &
Latvia University, Fac. Phys. Math., Raina boul. 19, Riga, LV-1586, Latvia
Abstract: It is found by numerical simulation that 2D magnetic fluid droplets in
high-frequency rotating form shapes with 2 or 3 spikes. Number of spikes depends
on both an initial shape of the droplet and a magnetic field strength. It is found
that 3-spike shape of the droplet exists as metastable state and an energy barrier
suppresses the transition to 2-spike shape which corresponds to the energy
minimum. Results obtained dynamical simulation are in good agreement with the
energy minimum calculations carried out previously. In the case of the initial
shape with some perturbations, development of spikes and transition to smaller
amount of them are observed. At increase of magnetic Bond number the spikes
sharpens. Finally some conclusions concerning the comparison of the results of
numerical simulation with the experimental observations are pointed out.
An experimental study of a magnetic fluid (MF) microdrop in a rotating magnetic field
shows a wide variety of complex phenomena [1]. The complete theoretical description of all
features of corresponding phenomena is yet absent. A linear small perturbation analysis gives
us magnetic field threshold values with respect to n-lobe perturbations [1,2]. It is important to
know if n-lobe shapes exist as stable or metastable configurations. In order to investigate that,
we simulate numerically the behaviour of a MF droplet in 2D. Neglecting inertia terms in the
Navier-Stokes equation one can use boundary integral equations [3] to describe creeping flow
with a free boundary. A high frequency range of rotating magnetic field is considered,
allowing to make time-averaging of forces. Thus here a small oscillatory motion of a surface
of a droplet caused by field rotation is neglected.
Governing equations for fluid flow both inside and outside the droplet are
-Vp + 11V2V+ (MV)H = 0, ( 1 )
div v = O. (2)
Here p is pressure, 11 : viscosity, v : fluid velocity, H : intensity of magnetic field, M :
magnetisation of magnetic fluid. Outside the droplet fluid is non-magnetic, hence
magnetisation M=O and third term in left side of (1) is absent. Corresponding boundary
conditions are
in ex
(Jm=(Jm'
in in ex ex 2 (Mn)2 /R-p +(J,m = -p +(Jnn + rt -(J c. (3)
118
Chapter 4: included paper
Here c stays for surface tension but Rc represents local curvature of the boundary contour of
2D droplet, n : external normal, subscript 'in' denotes region occupied by MF droplet and 'ex'
surrounding fluid.
Further in a paper a non-dimensional form for physical parameters and geometric
dimensions of droplet is used. As characteristic values are chosen: external field strength Ho,
surface tension o , external fluid viscosity llex and unperturbed droplet (i.e. circle) radius R.
Physical processes are characterised by magnetic Bond number Bm = H~R , characteristic
rr
decay time for free surface perturbations 1" = llexR and ratio of viscosities A = llin . So we
cr lla
have non-dimensional coordinates x = x] R, y = y / R , a non-dimensional velocity v = ~ ,R/1"
a non-dimensional magnetic field II= HI H, and a non-dimensional time t = t / 1". For
simplicity further tildes are omitted.
Motion of MF droplet under the action of magnetic field is driven by effective surface
forces assuming that magnetic volume force (MV)H is potential. Action ofthe gravity forces
is neglected. Hence the effective time-averaged surface forces have only the normal
component [3,4] which can be written in non-dimensional form as follows:
I,n = _1 __J.! -_I Bm~( H,;) + (Hn .RL 87t
Here 11 is magnetic permeability of MF. One can see that magnetic field strength values on
the boundary are needed to calculate surface forces. Both normal and tangential magnetic
field strength components Hn, HI on the boundary are found as solutions of the corresponding
boundary integral equations [4,5]:
Hili = 2Hox Ox + 2HOY ay + ~ (J.! -1) f H, (1')\ K(r,r') dl' ,
r (1l+1)on (J.!+I)on rr(Il+1)L I r Jx/+y/
H11 = 2Hox ox + 2HOY ay _ ~ (J.! -1) f H,(I')I K(r,r') dl' ,r (1J.+1) 01 (J.!+1) 01 rt (11+1) L r Jx/ + y/
K'''' -') _ x,(y' - y) - yJx' - x)V r - ---------., (y' _ y)2 +(x' _X)2
Here 1 is a natural parameter: contour arc length.
Approximation technique, applied to the equations (5),(6), is described in details in
[3,4,5,6]. Magnetic field strength is calculated at discrete marker points rio To carry out time
averaging, Hn, HI are expressed by Hox, HOYin every k-th marker point:
HII(rk)= (u~~Hox +U;kHOY)/ JX;(k)+ y;(k),
HI (rk)= (u:UHox +a~kHoy)/ JX;(k)+ y;(k).
Expanding the unknowns a'~:Yk by the series of pyramidal functions similarly to [5], obtained
equations are
(4 )
(5 )
(6 )
119
Chapter 4: included paper
N(/J N(/J 1 ( )"" a Xk A+ = "" a Xk [R + J ]= _ Xi+1 - Xi_1L.... I ik L.... I ik ilc ,
k=1 aYk k=1 aYk Il + 1 Yi+1 - YH
~(a/~JA~ = ~(a:~J[R-J ]= _1 (-&i+1 - Yi-J~L.... /I ik L.... /I Ik,k 1 '
hi a Yk hi a Yk Il + Xi+1 - Xi_1
(7)
(8)
I; -1;-1
6li+1 -I;
Iik = 6
li+1 -li_1
i=k+1
i=k-1
i = k30, i ;;j:. k - 1/\ i ;;j:. k - 1/\ i;;j:.k - 1
R = R' (1l-1)(tk+1 -lk-l YJi+i -1;-1)
ik ik 41t(l.l + 1) ,
YI(iXx(i)-x(k))-x,(i)(y(i)- y(k)) i;;j:.k
(xl(i)-x,(k)j + (y,(i)- YI(k))Ri: = yuCi)x, (i)- XII (i)YI(i)
2(x;(i)+ y;V))
The contour is approximated by interpolating with cubic spline functions coordinates of
the finite number of marker points. First and second derivatives along the boundary contour
(XI' XII> YI, YII) are calculated by differentiating the interpolating cubic spline functions for both
coordinates. For better accuracy of the numerical treatment of the tips of the droplet, which
have large curvature, special non-equidistant distribution of the marker points is used: the arc
lengths between them depend on the local curvature of the contour.
For rotating field we have Hox = cos to, HOY = sintn what lead to
~ ( 2) ( 2)) = ~ Il(a~Yk2 + a ~/} (a:u2 + a ~/Il H" + H, 2 () 2 ( )r=rk 2 x, k + Y, k
Surface motion is calculated using the potential theory of viscous (creeping) flow, described
in [6,7]. The corresponding boundary integral equation for 2D is
vJi) = - (1 ) 1njfs"Gij(i,i')dI' +
21t 1+ A L
1- A J (-')n (-')' {"/IT (- -'}iI'
( ) 'jvj X k X :J s ijk x, X41t 1 + A L
i = k
(9)
(10)
where
G (- -') __ 1: 11-'--1 (x;-x,)(x;-x,)ii' x, X - U'i n x x + 2', li'-il
(X' - X Xx', - x 'Yxk' - xk)T (- -') -4 I , J J };.ijk X,X = 1_' _14 •X -x
Boundary integral equation (10) approximation is similar to [3]:
120
Chapter 4: included paper
NI vm(J)[8~ + Pm,ij]= Qm,;; m = X,Y; i = 1,2,... ,N,
j=l
( 11 )
0, i = i
r., = ~ ~:~ [vk (JXxk (i)- »,Q))] ~s(JXx,(J)- Xs(i))]X,
(xm(J)- xm(i)X!j+l -lj_l)
2Ix(J)-xQt
Qm.i= 1 i[8~Inli(J)-x(i)- (Xj(J)-Xj(i)Rk(J!-xkQ))1 x
21[(1+ AL=, Ix(J)- xCi) )
fsl/(J)nk(J)lj+, ;lj_1 _
± [~l;Gm(i + 1)+111;_Pm(i -1)+ 3(111;+ M;_l Pm Q)]+
G (i)~ [111;lnlx(i + 1)- xCi) + I1/H lnlx(i)- xCi -1)]
rl/(i) = [YIIQ)X"t(i)- XII(i)YI(i) _ ~ Bm€lfall 2 +a" 2h fa' 2 + a' .2)~){X2(i)+ y2(i)
J s I 2 (,) 2 C.) 16 r" Xk Yk F" Xk Yk \: I I'J XI I + YI I 1[
i;/;i
I1lj = Ij+1-Ii"
As one can see, for A=l (llin=lleJ term Pm,ij III (11) disappears and we have explicit
expression for velocity v:
vm(i) = Qm,i; m = x,y; i = 1,2,... ,N ( 12)
If A;/; 1 (llin ;/;llex , then the set of linear algebraic equations (11) must be solved, what
increases computational time considerably.
Two numerical simulations, shown in Fig.1 a,b, represent dynamical developments of the
droplet shape from two different arbitrary initial shapes. These shapes are generated
artificially, by randomly distributed perturbations in selected points joined by lines. As one
can see, random initial perturbations can lead to very different states. Final shapes show that
both 2-spike and 3-spike configurations can exist in time intervals, which are longer than the
characteristic transition time. In both cases N=350 marker points for boundary discretisation
are taken, fl=25 and magnetic Bond number Bm=200. These simulations are extremely CPU-
time expensive since require sufficient refinement of a discrete representation of the
boundary. Numerical simulations clearly show that in 2D after maybe quite long evolution to
final state the 2-spike configuration is realised in high-frequency rotating field. This
conclusion from the results of a numerical simulation is in correspondence with the
conclusions drawn in [4] on a basis of purely energetical grounds: whatever is the magnetic
Bond number 2-spike configuration always corresponds to the lowest energy. From our
simulation results it follows that 3-spike configurations can be observed as quite long-living
121
Chapter 4: included paper
transient states (see Fig. 1b). In that connection the question arises are 3 and more spike
configurations existing as possible 2D equilibrium figures in high-frequency rotating fields.
To check it numerically we have done numerical simulations from symmetric initial states. In
that case we can hope to obtain the metastables states with higher number of spikes than 2.
The results of numerical simulations shown on Fig.2 indeed confirm the existence of
metastable three spike-configuration. On Fig.2a starting from symmetric 3-spike shape at
Bm=45>BIllc(3) [2,4] the establishment of the figure of equilibriums with 3-spikes is
observed. The decrease of Bm below BIIlc(3)=37.5causes slow evolution to 2-spike shape
(Fig. 2b). Thus as it has been already concluded from the energetical calculations the
numerical simulation of the dynamics of the droplet in high-frequency field show that the
ground state corresponds to 2-~pike configuration. Configurations with higher number of
spikes can be obtained from very symmetric initial conditions. We must recognise that this
behaviour is quite distinct from 3D droplets [1], where observed number of spikes grows
proportionally to the square of the magnetic field strength. A 2D model can not describe the
behaviour of the droplet as observed in experiment and at present moment we believe that it
is caused by pure 2D properties of it. Main conclusion is that 3-spike shape of 2D droplet is
stable in high frequency magnetic field, because transition to 2-spike shape is suppressed by
the presence of an energy barrier. These results are in good agreement with the energy
minimum calculations, carried out in [4].
Concerning the behaviour of 2D droplets it seems quite interesting to study the
transitions with different number of spikes at decrease of the frequency of rotating field. In
the case arising oscillations can be large enough to cause the transition over energy barrier
from, for example, 3-spike configuration to 2-spike configuration. We presume to study this
interesting problem in future.
This work was supported by "Le Reseau Formation Recherche n090R0933 du Ministere
de l'Enseignement Superieur et de la Recherche" of France. Two of us (A.Cebers, S.Lacis) are
thankful to International Science Foundation for financial support of the research in terms of
long-time grants LBGOOOand LJQ100.
References
[1] Baed FC, Cebers A. 0., Petzynski R. Behaviour of a magnetic fluid microdrop in a
rotating magnetic field II Phys.Rev.Lett. - 1994. - Vol.72, N17 - PP.2705-2708
[2] Cebers A. 0., Lads 5. Magnetic fluid free surface instabilities in high frequency rotating
magnetic fields II Brazilian Journal of Physics - 1995. - Vol.25, N2 - PP.101-111
[3] Cebers A. II Magnitnaya Gidrodinarnika (in Russ), 1986, N4, pp.3-1 0
[4] Baed I C, Cebcrs A., Lads 5., Perzynski R. Shapes of 2D magnetic fluid droplets in a
rotating magnetic field II Magnitnaya Gidrodinarnika (to appear)
[5] Baed IC, Cebcrs A., Lads 5., Petzynski R. Dynamics of the magnetic fluid droplet in
rotating field II 1. Magn. Magn. Mater. (to appear)
[6] Pozrikidis C Boundary Integrals and Singularity Methods for Linearized Viscous Flow -
Cembridge University Press - 1992. - P.260
[7]Ladyzhenskaya 0.A. The Mathematical Theory of Viscous Incompressible Flow - Gordon
& Breach - 1969.
Captions
122
Chapter 4: included paper
Fig.I. Numerical simulation started from perturbed state, leading to 2-spike and 3-spike
shapes.
Fig.2. Numerical simulation started from symmetric 3-spike shape.
a) stable 3-spike state (Brn>BIl1c(3»,
b) transition of 3-spike shape to 2-spike shape (Bm<BIl1c(3),.
12:
Chapter 4: included paper
Bm=200, J.!=25,N=350
t=2
0
4V:=:===:----t=0
t=O.15~
1=0.34
t=0.84
1=1.17
1=1.72
124
t=309~
t=5 o~==========:::::::::===-
I~:=========~-
t=7.03
t=9.69
1=11.75
Figure 1a
Chapter 4: included paper
Bm=200, ~=25, N=350
t=2.51
t=O 0
t=0.10 0
t=0.25 V
t=0.38 r:;>-
t=0.60 V
t=1.00
t=1.73
Figure 1b
125
Chapter 4: included paper
a)
126
'==:::~-:=jE -==:j
( ~----- ----------I '
I )
36.5
: :
0.04.8 19.8
b)
Figure 2
--r_-_-,.,
t
105
95~ -90
- 80/r - 70
I
..·25
o
Chapter 4
Conclusions
1. The small perturbation linear analysis gives the number of lobes of the most unstable
mode in dependence on magnetic field strength. In the case of the potential flow in 2D
the number of lobes is proportional to the square of the magnetic field strength, in the
case of pure creeping flow it is directly proportional to the magnetic field. The last
relation is proved by the results of a numeric simulation.
2. The threshold value in respect to 2-lobe perturbation (4.4) for the magnetic Bond
number Bm turns out to be the absolute threshold in respect to any perturbation, what
is proved by BEM simulation (see Fig.2.7). If Bm is smaller than this absolute
threshold value, a droplet holds circular shape.
3. It is found that in 2D there are only two stable shapes: 2-spike shape and 3-spike one.
The 3-spike shape exists like metastable state, since 2-spike shape has less energy. The
decrease of a magnetic field below the turning point or, by a sufficiently high
perturbation the transition to 2-spike shape takes place. Evidently there exists no in 2D
metastable shapes with more than three spikes in a high-frequency rotating magnetic
field under assumptions used in the present work.
4. There is no the stable 4-spike shape found by numerical simulations. Evidently a shape
with four and more spikes is unstable in respect to splitting of the "body", where
spikes comes together.
127
Chapter 5
Magnetic fluid droplet in low-
frequency rotating magnetic fields
The behaviour of a ferrofluid droplet in low frequency magnetic field [5*,63*]
have a similar character to one of the motion of bounded spheres in [9,57,101,102].
The essential difference obviously is that in the case of a droplet there is a fluid flow
inside the droplet and the surface tension effects are quite different from the bound
which exist between pair of magnetic holes (microspheres in ferrofluid) in
[57,101,102] or between pair of magnetic particles in [9].
In the low-frequency range the rotating field is given as
{
HOX = n, co.sror, ( 5 . 1 )
HOY = H, SIll rot,
and the magnetic term in the surface forces formula (1.26) becomes essentially time-
dependent.
Thus the BEM method is used in such a way that at first the marker points,
which describe the boundary, are redistributed according to the actual local curvature
of the boundary contour. To prevent the too dense accumulation of marker points in
some local places and rarefaction of them in others, the limits of lover and higher
allowable curvatures for marker point distribution are given as input data for a
computer code. To solve time dependent free moving boundary problem, in every
discrete time instant the magnetic field is calculated according to (2.12) and (2.15) by
the Gaussian elimination method. Obtained magnetic field components are
implemented in (1.26) to calculate a surface force, needed by (2.33) and (2.34). Once
the velocity field at the boundary is calculated, the position of the interface is
129
Chapter 5
advanced by Euler scheme. There are two mechanisms to control time step, initial
value of which is given by default. The first mechanism controls the shift of every
marker point so that it does not exceed the a priori given limit, and time step is
adjusted to satisfy this requirement. The second mechanism controls the time step in
such a way that the shift of every marker point exceeds no 20% of distance to the
nearest neighbour. The calculations performed at A:;t:l have shown that present
method, described in Chapter 2, works up to A=10, but this limiting value already is
critical to perturbation of the droplet shape at tips for a rotational motion, what
obviously is a numerical artefact.
The numerical simulation via BEM indicate two scenarios [5*] of the droplet's
behaviour in dependence on a value of the magnetic Bond number Bm=(Ho)2RJa.
These scenarios are based on the fact that the magnetic torque, acting to the droplet in
a magnetic field has some maximal value, which for fixed shape is frequency
independent. The friction torque, which is present due to the viscous stresses in
creeping flow around the droplet are proportional, in contrary, to the angular velocity
of the droplet's rotation. In high-frequency rotating field, droplet could stay elongated
only if the magnetic Bond number has high enough value. Therefor there are two
scenarios [5*,63*]:
1. "Low-field" behaviour:
If the magnetic Bond number is less than the critical value Bmw the droplet
can not stay elongated in the high-frequency rotating magnetic field; the
extension of a droplet in stationary configurations is diminishing with the
increase of the rotating field frequency, and the maximal phase lag value n/4 is
reached at infmite frequency. The rotation of the droplet's shape in this case is
not identical to the quasi-solid rotation of the droplet. This shape movement is
caused by rather intrinsic flow inside the droplet, which is more like the
travelling wave propagation on the surface of fluid.
2. "High-field" behaviour:
For the magnetic Bond numbers larger than the critical one the maximal phase
lag (~1t/4) is reached already at a finite critical frequency QCRo If the field
130
Chapter 5
frequency QH exceeds QCR' the phase lag between the field and the droplet
increases more and more, exceeds the value n/2, and the droplet moves toward
the field. Thus beyond the critical frequency the motion of the droplet starts to
be "jerky": the rotations alternates with stops and backward motions [5*].
The BEM simulation have proved that the shape of a droplet in the case of
steady motion is near to the elliptic one if the ratio of viscosities A=llin/llex=l. The two
different sets of equations of motion are derived in [5*,63*] and the sufficient
agreement with BEM simulation is found (see Fig.2 in [5*] and Fig.2 in [63*]). On
the basis of these equations the stability of the steady motion of a ferrofluid droplet in
subcritical frequency range is explained in [5*]. To illustrate the "high-field"
behaviour, in Fig.5.1 the time-averaged frequency 0 of the droplet rotation is plotted
versus magnetic field frequency QH. The first part of the figure (QH<QCIJ shows the
steady motion, when the droplet follows the rotation of a magnetic field. The shapes
of a droplet and the orientation of the magnetic field vector indicate how the phase lag
between the droplet orientation and the magnetic field direction increases and the
elongation of a droplet decreases as QH approaches the critical frequency OCR.
Beyond the critical frequency (QH>OcIJ the motion of a droplet becomes "jerky",
what is illustrated by the trajectories of the droplet's tip. Increasing a field frequency
0H, the oscillations of a droplet tip becomes more often but the oscillation amplitude
decreases.
What is the difference between the two sets of equations of motion [5*,63*]?
The first what should be mentioned is that the equations of motion in [5*] are
obtained in more approximate way than ones in [63*]. Nevertheless even such a
phenomenological approach have yielded rather good agreement.
The configuration of the elliptic droplet is described by its large semi-axis a and
the phase lag S of the large semi-axis with respect to the field direction (see Fig.la in
[5*]). The First set is obtained by separation of the motion of the elliptical 2D droplet
in two processes [5*], as it is described below:
(5.2)
131
Chapter 5
· Bm 4t-1Ya2(a4 -1)
cp = hahb - f. y y ) ,8n \a2 +~Aa2~+lAa4 +1
Here h a = cos S, h b = sin S , S being the phase lag between the instantaneous
(5.3)
direction of a field and the orientation of the major semi-axis of an elliptical shape of
a droplet. The definition of term <D(a) is given in [5*,63*] (both papers are included in
the present chapter).
The first process. is an extension-contraction motion caused by the surface
tension and the action of a magnetic field. The change rate of the total energy in this
process is balanced by an energy dissipation in the viscous flow inside and outside the
droplet. The energy dissipation inside the droplet is approximated by the
homogeneous extension-contraction motion, the energy dissipation outside is assumed
to be proportional to the one inside the droplet up to phenomenological parameter p
divided by the ratio of viscosities A. Thus from the energy balance [5*] the equation
(5.2) is obtained. To describe the second process, namely the droplet rotation, the
balance of the viscous and magnetic torques is considered. The friction torque of
viscous forces is determinated by a creeping flow around rotating rigid elliptic
cylinder [5*, 59], and it has to be balanced by a magnetic torque [5*] arriving at (5.3).
To derive the Second set
· 2a4a=------x
n(a4 + 1+ 2'Aa2)
(5.4)
2 a4 + 1 f. 4 )B (II_1)2 a' 2a -4- + A\a -1· -h h ~ __ r____ a -1 (5.5)
cP - a b 8n (a2 + ~Xa2~+1) 2a2 +A(a4 +1)
the virial moment technique [25,42,41] is used to satisfy the dynamic boundary
conditions integrally [63*]. Fluid flow outside the droplet is treated exactly, applying
the 2D analogue of the Jeffrey solution [59], but inside is described in approximation
of constant velocity gradients. The main approximation is made integrally satisfying
the dynamic boundary conditions by virial moment technique. What is the difference
between the boundary conditions, which are satisfied integrally and exactly? The
132
Chapter 5
exactly satisfied boundary conditions should introduce some perturbation from an
elliptic shape of the droplet obtained by integrally satisfied boundary conditions. Thus
the other essential assumption about an elliptic shape of a droplet is related to the
approximation of the dynamic boundary conditions, which is used for both sets of
equations. Numerical simulation of droplet motion is carried out applying Runge-
Kutta 4th order algorithm for both First and Second sets.
The comparison of both sets displays that the equation (5.3) in the First set is to
obtain from (5.5) tending A to infmity. It is quite natural result since the equation (5.3)
is obtained from rigid ellipse rotation. Substitution of the phenomenological
parameter pin (5.2) with the following expression
p > a~a~l ~ a: +o[(±)'J
arrives at equation (5.5). The conclusion could be drawn that the First set of equations
(5.6)
is some simplification of the Second set in the case of very viscous droplet.
The answer to the question, how good is the approximation of the shape of a droplet
by an ellipse, could be obtained by a BEM simulation. In the case of 1..=1, the
juxtaposition of a BEM simulation with results of the First and the Second sets of
equations of motion are given in Fig.5.2. The comparison of all three series of shapes
shows that:
1. The agreement between the "Second set" and BEM simulation is very good up to
t=6.07, when the perturbations from an elliptic shape develops for a droplet, as it is
displayed by a BEM simulation.
2. The shape of a droplet obtained by the "First set" is not sufficiently good at t=6.07,
the difference is already present at t=4.41, because it is not possible to account for a
shear flow inside a droplet by the equations of the First set.
3. After t=8.39 the shape of a free droplet is different in comparison with an elliptic
one and hence both sets of equations of motion are inadequate to describe such a
motion.
Thus, it is proved that equations of motion could be used with good accuracy
only if the shape of the droplet is close to an elliptic one. Other detail to discuss is the
133
Chapter 5
range of A, the ratio of viscosities, for which requirement about near elliptic shape is
fulfilled. Even in Fig.5.2 at t=4.41 the free droplet (BEM simulation) exhibits the
tendency to bending. A more explicit phenomenon is shown in Fig.5.3 where the
shapes of the droplet in the case of a periodic rotation at the ratio of viscosities A=5
are presented. Here magnetic field characteristics Bm=105, ~=5 are chosen to have a
moderate elongation of a droplet. The field rotation frequency n~0.8 is slightly
larger than critical frequency, thus the time interval during which the droplet
unsuccessfully tries to follow the field as well as the backward motion is longer than
for higher frequencies. The time-averaged frequency of a droplet's rotation is
Q = 0.28. It means that the droplet is exposed a time long enough to the
perpendicular magnetic field. The behaviour of the droplet during time intervals
t=13 ..15 and t=19.5 ..22 displays bending instability in the magnetic field which is
orientated approximately perpendicular to its major dimension. How bending
instability appears here? At time moment ~ 12, for example, magnetic field is near to
perpendicular direction and by the shear stresses both tips of the droplet is already
slightly "twisted". Obviously droplet is unstable with respect to the bending instability
in the perpendicular field and if some initial perturbation is present, the time interval
Lit=3 is sufficient to cause such a large effect as it appears from t= 11.78 to t= 15.00.
The next question is what is the role of the A value in this instability? In Fig.5A the
similar results for ),,=1 are shown, leaving others parameters unchanged. The
comparison with Fig.5.3 proves that at the smaller viscosity of the droplet, the shear
flow inside it plays more important role. Thus, rather intrinsic transitions at t~9 and
t~ 16 take place. The nature of these transitions is not yet well studied and it could be
the subject of further research but nevertheless, it is obvious that the more viscous
droplet of Fig.5.3 exhibits tendency to bend instead of changing shape by the way of a
shear flow, because the bending flow requires less energy dissipation inside the
viscous droplet. Another phenomena in Fig.5A which causes interest is the transition
to steady motion with constant phase lag and constant elongation (b/a=3.88) at the end
of simulation (t=30 ..42). This last difference in the behaviour of a droplet at ),,=1 and
),,=5 is evidently the same what is mentioned in [5*]: moderate values of ).. causes
stabilisation of droplet configurations if the phase lag 3 is slightly greater than the
critical value TI/4, but an increase of)" causes the loss of stability. In present case it
134
Chapter 5
means that for 1.=5 droplet can not follow a field rotation. At 1.=1 the transition of a
droplet finishes with the steady motion of it. The droplet follows the field rotation
with a constant phase lag (see Fig.5A).
135
Chapter 5
o
n 0.5
0.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Magnetic field frequency 0H
Figure 5.1. Droplet rotation: the time-averaged
frequency of a droplet's rotation versus
the frequency of a magnetic field
136
Chapter 5
0.0 1.21 4.41 6.07 8.39 10.90 12.9914.07 16.21~~---I~-I t I I t--1 ~
~
~ ~ Equations of motion I -1J I": \ ~ ,~--~ Q O';\~, \
~
~ ~~
Equations of motion II ,'" l'
~
~ ~\<-- ~~~ ~~~, \
\..,
Figure 5.2. Juxtaposition of a BEM simulation with
simulation by two kinds of equations of motion
137
Chapter 5
HG
N=200, ~t=O.Ol
Bm=105, f.l=5, QH=O.8, A=5 )
t=9.60 t=10.24
~ t=15.40
t=17.13
t=10.67
t=15.84
t=17.64
t=18.61 t=19.28
138
t=22.55
t=22.01
Figure 5.3. Rotation shapes of a viscous droplet:
the bending instability in the "perpendicular field"
Chapter 5
Bm=105, ~=5, QH=O.8, A=l G H )
N=200, ~t=O.Ol
t=7.03
t=17.09
t=26.62
t=18.20
t=34.41
t=3,15
t= 10.00
t=15.35
t=23.06
t=38.41
t=24.92
t=42.01
Figure 5.4. Transition to steady state
139
Chapter 5
Included papers:
1.
J.e.Bacri, A.Cebers, S.Lacis, R.Perzynski
Dynamics of the magnetic fluid droplet in rotating field
Journal of Magnetism and Magnetic Materials, 1995, Vol. 149, 143-147
2.
S.Lacis
The equations of motion of the 2D elliptic magnetic fluid droplet in
rotating field
Magnitnaya Gidrodinamika, 1996, (accepted to publish)
141
Chapter 5: included papers
Journal of Magnetism and Magnetic Materials 149 (1995) 143-147
ELSEVIER
M lournal 01
NImagnetismandM magneticmalerlals
Dynamics of a magnetic fluid droplet in a rotating field
J.e. Bacri a", A. Cebers b, S. Lacis ". R. Perzynski a
a Lab. AOMC (Assoc. CNRS), Uniiersite P.M. Curie, 4 place Jussieu. 75252 Paris Cedex 05, France
o l.atcian Academy of Sciences. Institute of Physics. Salaspils-l . L \1-2109, Lau-ia
Abstract
The response of a magnetic fluid microdrop to a rotating magnetic field is studied numerically in 2D by the boundary
element method (BEM), On increasing field frequency, the motion of the droplet goes through a transition from a state
where the droplet follows the magnetic field with a constant phase lag to a state where the phase lag increases in a series of
kinks when the field frequency passes the critical one. The equations of the droplet motion are derived analytically and good
agreement with the BEM is obtained.
II is well known that for a rigid magnetic dipole there is
a critical angular velocity of the rotating field below which
the rotation of particle and field are synchronized [1).
Similar phenomena are observed for a bound pair of soft
magnetic particles [2,3]. The interplay between magnetic
and viscous forces leads to various modes of motion,
which in Refs. [2,3] are classified as (1) steady-state
rotations; (2) 'jerky' (rotations with stops and backward
motions); and (3) localized oscillations. Transitions be-
tween these modes are well described by a single non-lin-
ear equation and depend on the frequency and amplitude
of the rotating field, the fluid viscosity and the magnetic
susceptibility, [t is found both experimentally and numeri-
cally [3] that for a pair of free spheres phase locking takes
place in an elliptical polarized field 72/ flH = [/2, 1/4,
etc., where n is tbe average angular frequency of the
pair-rotation and flll is the angular frequency of the
magnetic field rotation. The magnetic fluid (MF) micro-
droplet in the rotating magnetic field includes a wide
variety of very complex phenomena in the high-frequency
range [4]. In the [ow-frequency range, an elongated droplet
rotates with magnetic field frequency, with the surface
tension playing an equivalent role to that of the soft
binding in a pair of magnetic spheres.
The scope of the present paper is the behaviour of the
MF droplet in the intermediate frequency range of a
rotating magnetic field, studied by a numerical simulation.
In the two-dimensional (2D) case, we apply the boundary
. Corresponding author. Email: jcbac@ccr.jussieuJr: fax: + 33-
1-44273854, Affiliation: Universire Paris VlI.
0304-8853/95/$09,50 © 1995 Elsevier Science BY All rights reserved
SSDI 0304-8853(95)00357-0
142
integral equation technique (BEM: the boundary element
method) considered previously in Refs. [5,6]. We assume
that the magnetic permeability is constant ( J.L = canst), and
that gravity and inertia forces are negligible due to the
very small size of the droplet. Hence the effective surface
forces have only the following normal component:
Here a is the surface tension, R L is the local curvature
radius. and IJ. is the magnetic permeability of the MF.
Both the normal and tangential components Hn, H, of the
magnetic field strength on the boundary are found as
solutions of the corresponding boundary integral equations
[5]:
zn., ilx 2110r ily I (J.L-I)Hnl,,= ---- +---- +----
( J.L+ 1) iln ( J.L+ 1) Oil 'IT (J.L + I)
dl'
X¢.Hn(l') II·K(r. r') ~.
L vx; + v; (2)
2Hox ax 2n.; ily I (J.L - I)H 1,,=----+ . ------
I (J.L+ 1) ill (J.L+ I) ill 'IT (IJ.+ I)
dl'
X¢'H,(I')I rK(r, r") ~'
I. 'Ix; + yi (3)
, x,(y'-y)-y,(x'-x)
K(r,r)= , ,.
(\' - Y r + (x' - x r
Calculated field components are used for effective surface
forces (I). The surface motion is calculated using the
potential theorv of viscous flow, described in Refs. [7,8].
Chapter 5: included papers
144 Ie. Bacri et al. / Journal of Magnetism and Magnetic Materials 149 (1995) 143-147
The corresponding boundary integral equation for the ve-
locity v of the fluid on the boundary of the droplet in 20
is
7/ex - 7/in+-----
4n(7/cx + 1);n)
where
x=x' -x.
TJ stands for viscosity, the subscripts' in' denotes the MF
droplet and' ex' the surrounding fluid. In the present paper
only the results obtained by BEM for the case of equal
viscosities (7/in = 7/e,,)' are presented. In this case Eq. (4)
simplifies to
The boundary contour is approximated by a finite
number of marker points connected together by interpolat-
ing cubic spline functions. As in Ref. [61, the distance of
marker points separation is proportional to the local radius
y
({lrOnt I
.,. .
..... I
" I
/ .._- -,
r- .•..•
I~*tn''t'a ".
(a)
of curvature within limits which prevent the absence of
points at places on the contour with small curvature. This
non-equidistant distribution of marker points gives a better
accuracy at droplet tips where the contour curvature is
larger.
The approximation technique for Eqs. (2), (3). (5).
described in detail in Refs. [5,9J, is based on linear interpo-
lation of the corresponding unknowns (FIn' HI' r) be-
tween marker points along the droplet contour and applica-
tion of the Galerkin method. The singularities are sub-
tracted and integrated analytically. The first and second
derivatives along the boundary contour (x t, X 1/' )-'1' .1'1/)
are calculated by differentiating the corresponding cubic
spline functions. For a rotating field we have Hox =
Ho cos flH I, HoI' = Ho sin flH I. The sets of linear alge-
braic equations, obtained for Eqs. (2), (3) are solved using
Gaussian eliminalion The approximation of (5) leads to a
sum. In each time step, magnetic field components are
calculated and applied for the surface movement velocity
calculation. Once the velocity has been calculated. the
position of the interface is advanced using an explicit
Euler method. After each time step the droplet dimensions
are rescaled to improve the volume conservation.
Further in this paper, a dimensionless form for physical
parameters and geometric dimensions of droplet is used.
The external field strength Hn, the surface tension a , the
external fluid viscosity 1)ex and the unperturbed droplet
(i.e. circle) radius R are chosen as characteristic values.
Physical processes are characterized by the magnetic Bond
number Bm = H(~ R/ a , the surface energy of a circular 20
droplet per unit length of a cy Iinder E; = 2 Ii R c . the
characteristic time interval for droplet motion T = 7/e, R/ (T
x
\
\"\'
(b)
Fig. I. (a) The orientation of a droplet with respect to the magnetic field and laboratorv coordinates. (h) The wave-like perturbation
propagation.
143
Chapter 5: included papers
J.C. Bacri et al. / Journal of Magnetism and Magnetic Materials 149 (1995) 143-147
and the ratio of viscosities A = 11in/ TJe,. So we have
dimensionless coordinates .i' = x / R, Y = y / R, a dimen-
sionless velocity l' = l'T/ R, a dimensionless magnetic field
H = H / Ho' a dimensionless frequency of field rotation
{} = flT and a dimensionless time i= t/ T. All the further
formulae are written in a dimensionless form dropping the
tilde; in special cases the dimensional form is denoted by a
circumflex C ).
The behaviour of the MF droplet in a rotating magnetic
field can be understood on the basis of a simple model,
derived under the assumption of an elliptical shape of the
20 droplet. The configuration 'of the elliptical droplet is
described by its large semi-axis a = Ii / R and the phase lag
{} of the large semi-axis with respect to the field direction
(see Fig. ](a». Let us separate the motion of the elliptical
20 droplet into two processes. The first is an extension-
contraction motion caused by the surface tension and the
magnetic field. acting on the droplet in a direction determi-
nated hy the phase lag {). The total energy (magnetic plus
surface contributions) of the elliptic 20 droplet [9] is, with
respect to the field direction (in a dimensionless form),
2oE( e)
£1=---
or
where
E( e) = {TT v' 1 - ( c sin x)" d x
n
is the complete elliptic integral of the first kind. e2 = I -
b"/o2.
In this process. the total energy changes are balanced
by an energy dissipation in the viscous now inside and
outside the droplet:
dE, dE,E = Ein + (, = .- = a - .dt do (7)
The energy dissipation inside the droplet is approximated
as an homogeneous extension-contraction motion (l'x' =
x'a/o. I,' = -v'a/o, see Fig. Ita) for .r' . .I'). It gives
(8)
For the energy dissipation outside the droplet the following
approximation is used:
(9)
where p is some phenomenological parameter of the model.
144
L45
Collecting Eqs. (6)-(9) together, we obtain the droplet
motion equation for large semi-axis a:
a = __O_2_[_(_JL_-_l)_2 Bmo( cos" {) 0
or(A+p) 16 (a"+JLf
sin"{) )_ cf>(a)] (10)
( JLa" + I)2 04 - I '
1>(a) = «(a~ + l)E(e) - 2K(e»),
where
is the complete elliptic integral of the second kind.
The second process is the droplet rotation, character-
ized by the balance of the viscous and magnetic torques
( Nr + Nm = 0). The viscous forces are determinated by a
creeping flow around the rotating elliptic cylinder, which
could be derived using the method described for a general
ellipsoid in Ref. [I OJ, modifying this method for the case
of 20. The ohtained viscous torque (the dimensional z-
component, per unit length of a cylinder) is
Nr= -2orr,eJi(li2+b2).
It has to be balanced by a magnetic torque
A f' A A "sin2t1AA(JL-I)"(fi2-b2)
Nm = M dS X H = n; -- ab A ( A) .
5 8 (1i+lLb) oJL+b
(12)
(11)
The angular frequency of an ellipse rotation in laboratory
coordinates is (see Fig. ll a) for the definitions of Ifn• If,,)
n = <Pn = <{:II - () = nIl - t~. ( I3)
Collecting together Eqs. (I I )-( IJ) we obtain the dimen-
sionless droplet motion equation for the phase lag {}:
()= nH - ncr sin 2l'J.
where
( 14)
e; (JL_1)2aC(04- I)
Jlcr= 160r (o2+IL)(o}.t+ 1)(0.\+ I)
At low magnetic field rotation frequencies flH" the MF
droplet rotates uniformly with a frequency equal to that of
the field. Steady-state rotations are studied using both the
BEM and the simple model. and the results for A = 1 are
shown in Fig. 2 in coordinates (x', y') which rotate to-
gether with the magnetic field so that the v-axis is pointed
in the field direction (see Fig. ltal). A typical number of
marker points for BEM calculations is N = 200. For the
simple model simulations, the value p = I is used. The
solid curve in Fig. 2 represents the positions of the elliptic
droplet tips for a continuously varying frequency flH. The
end of the large semi-axis with coordinates x~ = 0 sin {}.
Chapter 5: included papers
146 IC Bacri et al. / Journal of Magnetism and Magnetic Materials 1.J9 (1995) 143-147
Y'3
5
6
0.0
0.0 0.5 1.0
X'
4
Bm=30
Jl'= 15
1.5 X' 2 3 X' 2 3
(a) (b) (c)
Fig. 2. (a),(b) Steady-state configurations of a MF droplet in a rotating field. (c) Paths leading to the steady state. Results for different field
rotation frequencies. Thick solid lines and triangles: simple motion model: dotted lines and crosses: BEM: dashed line in (a). (h): unstable
simple model configurations. In (c) thin solid lines are simple model paths to steady state. initial state is given by solid circles: dashed lines
are paths calculated by BEM.
y~ = a cos {J is taken as the tip. In Fig. 2(a),(b) the solid
curve is continued by a dashed one. which corresponds to
the unstable simple model configurations. Triangles show
the stable configurations at definite frequencies J2j{' The
steady-state configurations obtained by the BEM at defi-
nite frequencies {}H are shown by crosses, the pointed
curve represents the interpolation of these results for a
continuously varying frequency [lll' In the case of the
BEM the most extended point from the centre of the
droplet is taken as the tip of the droplet, and the phase lag
is calculated for this point. The comparison between the
BEM calculations and the simple motion equations calcu-
lation shows a fairly good agreement. Discrepancies could
be explained by two causes. The first is that the shape of a
droplet in BEM calculations (see also Fig. 3) has more
rounded tips, and hence the droplet has a smaller exten-
sion. The second is that the simple model does not account
for the wave-like perturbation propagation on the free
~
"~
~" / 8--~ '1 ~-:::, CrV"0.0 Ul 4.41 6.07 8.39 10.90 12.9'J 14.07 16.21I I I I I1 (I
~ c~ ~~
'1 10 ~O-~ - ,"
Fig. 3. Series of droplet shapes at fixed time moments for transient motion in laboratory coordinates. The upper series is for the BEM. the
lower for the simple model simulation. Dashed arrows represent the field direction: the length of arrow is 3R.
145
Chapter 5: included papers
.l.E. Bacri et al. / Journal of Magnetism and Magnetic Materials 149 (1995) 143-147
surface, caused by asymmetric stresses (see Fig. ](h». At
low values of the elliptic extension (a < 2) these stresses
cause a motion of the droplet surface which dominates
over the pure rotational motion. Thus the simple model
simulation shows the lack of stable configurations (Fig.
2(a» for large field rotation frequencies when a is small.
Two different types of steady-state behaviour are ob-
served (see Fig. 2(a),(b» depending on the magnetic field
strength. These two types are separated by the critical
value of the magnetic Bond number, which turns out to be
the threshold value of the instability of a 2D droplet in a
high-frequency rotating field with respect to the elliptical
deformations [9]: 8m = 6TI( J1. + ])3/( J1. - ])3. If the
magnetic Bond numberis less than the critical one ce; =
28.14, J1. = 15), the extension of the droplet in stationary
configurations is diminishing with increase of the rotating
field frequency. and the maximal phase lag value TI/4 is
reached at infinite frequency as it is shown in Fig. 2(a).
For the magnetic Bond numbers larger than the critical one
the maximal phase lag ('" TI/4) is reached already at a
finite critical frequency ncr (Fig. 2(b». Phase portraits of
the system are presented in Fig. 2(c). they are in accor-
dance with the BEM simulation. As one can see, the
dynamics of the droplet in the subcritical range of angular
frequencies (n H < flcr) is characterized by the existence
of a stable focus, whatever are the initial conditions. In the
case of anticlockwise field rotation. the tip of the droplet
near the focus rotates clockwise in coordinates (x'. y"),
The imaginary part of the perturbation decrement whenncr is approached is increasing in comparison with the
real part. thus causing the rise of the droplet tip rotation
around the focus.
In Fig. 3 a transient droplet rotation is shown by a
series of droplet shapes and corresponding field orienta-
tions. One can see that there is a fair agreement between
the simple model and the BEM for large droplet extensions
and some discrepancies for small ones. If 8m > 8m then
for large field rotation frequencies ( nH > flcr) the ~otion
of the droplet turns out to be jerky and just similarly to the
case of the two bound spheres [2,3] it can be characterized
as rotation with stops and backward motions. The jerky
rotation of the droplet could be described by average
angular frequency n. Fixing the value of a (A » I. ornH» ncr), the integration of the droplet motion equa-
tions gives [3]
( 15)
The value of a could be obtained from the non-linear
equation (10) by a = O. The results of simple model
simulation are shown in Fig. 4 for two different magnetic
Bond number values (8m = 30. 8m = SOl. As one can see.
the critical frequency increases with the increase of 8m,
Another important conclusion could he drawn ahout the
dependence of the critical frequency on A: finite values of
A cause stabilization of droplet configurations if the phase
146
147
0.4
0.2
0.2 0.6 0.80.4
Fig. 4. Plot of the average angular frequency of the droplet
rotation versus the field rotation frequency. ,\ = cc corresponds to
the analytical solution for fixed a.
lag {) is slightly greater than the critical value TI / 4.
Increasing A (the viscosity of the droplet), the critical
frequency decreases and tends to its analytical value flcr
(14). This effect could be explained by the small perturba-
tions from the steady-state configuration ao, {)n' Denoting
the right part functions in Eqs. (10). (14) by gt a, {).
respectively. II( a. In, the perturbation amplitude depends
on time as exp«ga + ha )t/2). where ga = ng/iJa, hr1 =
iJh/iJ{) at ap. {)". Beyond TI/4. h,1 > (l, thus the stability
criterion is g a < - h,~. From Eq. (10) it follows that
ga a 1/( A + p), hence obviously an increase of A causes
the loss of stability.
Test simulations show that in the case of an elliptic
polarized rotating magnetic field. phase locking is found
like in Ref. 13].
Acknowledgements: This work was supported by 'Le
Reseau Formation Recherche No. 90R09J3 du Ministere
de l'Enseignement Superieur et de la Recherche' of France.
Two of the authors (A.c. and S.L.) are thankful to the
International Science Foundation for financial support
through long-term grant LBGOOO.
References
[I] R. E. Rosensweig Ferrohydrodynarnics (Cambridge Univer-
sity Press. Cambridge, 1985).
[2] G. Helgessen. P. Pieranski and AT. Skjcltorp, Phys, Rev.
leI!. 64 (J 990) 1425.
[3] A.T. Skjeltorp and G. Hclgessen, Phvsica A 176 (I991l 37.
[4] J.e. Bacri. A.a. Cebers and R. Perzvnski. Phys. Rev. Leu.
T!. (1994) 2705.
[51 A Cebers, Magnilnava Gidrodinamika N4 (1986) 3 (in Rus-
sian).
[6] J.D. Sherwood. J Fluid Mech. 188 (I9R8) 133.
[7] a.A. Ladyzhenskaya. The Mathematical Theory of Viscous
Incompressible Flow (Gordon and Breach. New York. 1969).
[sl e. Pozrikidis. Boundary Integral and Singularitv Methods for
Linearized Viscous Flow (Cambridge University Press. Cam-
hridge. 1992).
[91 J.e. Bacri, A. Ccbers. S. Uicis and R. Perzynski. Magnitaya
Gidrodinamika. to appear.
[101 G.F. Jeffrey. Proe. R. Soc. AI02 (1922) 161.
Chapter 5: included papers
The equations of motion of the 2D elliptic magnetic fluid
droplet in rotating field
S.Lacis
Latvia University, Fac. Phys. Math., Raina boul, 19, Riga, LV-1586, Latvia;
Latvian Academy of Sciences, Institute of Physics, Salaspils-l , LV -2169, Latvia
Abstract: Equations of motion for 2D magnetic fluid droplet in low-
frequency rotating field are derived under the assumption of its elliptic
shape. Fluid flow outside the droplet is treated exactly, but inside is
described in approximation of constant velocity gradients. Dynamic
boundary conditions are satisfied integrally by virial moment
technique. Numerical simulation of droplet motion is carried out
applying Runge-Kutta 4th order algorithm. Juxtaposing with the
numerical simulation results of the droplet motion in low-frequency
rotating field by boundary integral equation technique good
accordance is confirmed.
It is well-known that for rigid magnetic dipole there is a critical angular
velocity of the rotating field below which the rotation of particle and field are
synchronized [1]. Similar phenomena are observed for a bound pair of soft magnetic
particles [2,3]. The behavior of the magnetic fluid (MF) microdroplet in the rotating
magnetic field includes a wide variety of very complex phenomena in the high-
frequency range [4].
The scope of the present paper is to derive simple equations of motion for 2D
MF droplet under assumption of its elliptic shape. Simple equations describing the
behavior of 2D elliptic droplet in rotating field are considered previously in [7] and by
comparison with the numerical simulation data it has been found that near the critical
frequency of the locking of the field and droplet rotation quite large discrepancies
exist. Here a new set of equations is proposed taking into account shear flow inside
and outside the droplet. For that exact solution of 2D Stokes equations for the velocity
distribution outside the droplet with homogeneous deformation rate is found. As result
much better agreement with the numerical simulation data on basis of boundary
integral equations method (BEM) is found.
A small size of microdroplet and relatively small characteristic velocities of
flow allows to neglect inertia and gravity terms, concentrating on surface tension and
magnetic forces on the surface of the droplet. Hence the dynamics of a free surface of
a droplet can be described in the framework of the creeping flow. Elliptic
incompressible MF droplet is completely determined by the length of its large
semiaxis a and the angle <p of its orientation with respect to X1ab-axisof the laboratory
frame (see Fig.1). Arbitrary viscosities of fluids inside and outside the droplet are
considered.
Governing equations for a fluid flow inside the droplet under the assumptions,
mentioned above, are:
o (in f..l. -1 H2) ocr;: 0-- p --- +--=,
OX; 8n Oxk
divv=O,
where p is pressure and crik is viscous stress tensor:
(1)
(2)
147
Chapter 5: included papers
«: = 11(8vk + 8vk). (3)Oxi Ox;
Here v is a velocity of a fluid and 11 its viscosity. Here and further superscript 'ex'
stays for the domain outside the droplet and 'in' for the domain inside it.
To derive equations of motion, the virial technique [5] is applied. For that
equation (l) is multiplied by Xj and integrated over domain SID (see Fig.l). Obtained
virial moments are:
Vi} = -fnixj(pill - ).l-l H2)dl + bij r(/ll _ ).l-l H2)dS +
L 81t sin 81t
1nkx jcr;~dl - r a~' dS = a
L sin
(4)
The boundary conditions on a droplet surface are
in ex in ex
V =V , crTl/=aw' (5)ill ill ex ex 2 (M)2 /R- p + a 1111 = - P + cr lin + n n - cr c
where subscript 'n' denotes normal component but subscript 1" tangential one, Rc:
local contour curvature radius, a: surface tension. n is an external normal. Accounting
due to the continuity of tangential stresses for
(6)
one can obtain
(
ex _ ill + ~ H2) + f. in _ ex)_ni P p 8n nk\aik crik -
(in e.T 2 (Mn)2 di ).l - 1 H2) f. ill ex) (7)ni -cr'lll + a,11l + 1t - a IV n + ~ + nk \O'ik - O'ik =
n;( 21t(Mn Y - a div n + ~8: 1H2) •
Applying boundary conditions to expression (4) the virial moment transforms to
Vu = -~ x jn; p" dl +1x jn; (2n(Mn y + ~8- 1H2)dl - O'~ Xjni div ndl +
I. L 1t L (8)
,.[ eXdl s: r( ill ~ -1H2)d'S ill"[{, \.il1Xjnkcr;k +UijSin p -~ -Y] 1\njVi -r».»
To conserve elliptic shape of the droplet a flow inside the droplet is approximated by
constant gradients Yik of velocity field
Vi =y ik xi ,
what leads to constant pressure inside the droplet:
pill = p;' . (10)
A magnetic field inside elliptic cylinder with constant magnetic permeability ).l IS
constant:
H2 = H2 + H2 = H2 ( a + b ) 2 + H2 ( a + b ) 2 (11)
Q h OQ b Oh b'a+).l ).lG+
(9)
148
Chapter 5: included papers
Here H.r H, are magnetic field intensity components in instant directions of axes a,b
of the ellipse (see Fig.I),
For the calculation of the virial moment external stresses on the boundary of
droplet are needed. A creeping flow outside the droplet is treated exactly [6], using
harmonic functions Q( x, y), <D(x, y) :
00 ~ 2 2 )Q(x,y)= fF'(~ ~+-1- dS+F(~)
a +~ b +~p
F'(~)= R(~) = 1/~(a2 + ~Xb2 + ~),
00
<D(x,y)=xy fR3(~~,
p (12)
~Q = 0, ~<D= 0
x2 lHere p is a positive root of the equation 2 + -2-- = 1. Flow velocity and
a +p b +p
pressure can be described in terms of harmonic functions Q(x,y), <D(x,y) (subscripts
x,y of harmonic functions denote partial derivatives):
v;x = T<D.r - S<Dy + All (xQu - Qx) + A12 (xQ,y - Qy )+
A21yQxx + A22yQXY
v:x = T<Dy + S<Dx + A11xQXY + AI2XQI:Y + (13)
A21 (yQXY - o, )+ An (yQ,y - n,)
pet = p~x + 211"X(All o, + (A12 + A21Pxy + Annyl,)
Values of constant coefficients T, S, Aij are associated with definite flow. The values
of derivatives of harmonic functions on the droplet surface in the elliptic coordinates
{
X = c cosh a cos ~ (14)y = c sinh a sin ~
are
<Dx.o =_ ( 2) 2 sinP(2cos2 ~ __ a_)\.a + b ~ a + b
<D\.o = - ( 2 \z...2 cos P(2 sin 2 ~ __ b_)
~+b~~ a+b
4 cos ~ 4 sin ~ 4 .n., =---, a., =--, n"o =--2 sin b cos p,-. a+b' a+b " ~
4(a 2) 4(b '2)Qu,o =-2 ---cos ~, QI'Yo =-) ---S1O ~ ,~ a+b" ho a v b
4b I 2 2 (- 2 2)~Dx:u,o = -Qx,y,o = ~ cos~\-3a + cos ~\3a + b ')
Quv.o = ~ sin ~(_a2 + cos' p(a2 + 3b2))
<1>xx,o = - sin ~ cos ~ ~a2 - cos/ ~~a2 + b2)+ cos" ~(2a2 - 2b2))1 ~ ,
<1>'\,,0 = -2~3(2a + b)- cos ~3a(a + b)(2a2 + ab + b2)+
cos' ~ 6a2 (a + by - cos" ~ 2(a + by (a - b)Y (~(a + bY)
149
Chapter 5: included papers
here
hi; = a' - (a2 - b2)coS2 ~,
subscript 0 denotes function value on the surface (~=O).
Velocity of external flow, given by (13), on the surface of a droplet is equal to
velocity of a flow inside the droplet
v .r = Yxxa cos ~ + Yxyb sin ~
v y = Yyxa cos ~ + Yyyb sin ~
In order to express values of characteristic coefficients of external flow (13) by
velocity gradients Yij inside the droplet both sides of the equation vin = vex from
boundary conditions (5) are multiplied by hi; and the obtained equation is treated like
Fourier series. This procedure gives the following values:
(a + bY(a2 +b2) yyS=------4 '
_ a(hy xy + (2a + b)y.n)A21 - --------- 8
(a+bYy Vl'An = ...-- 8
follows that Yxr = -Y yy' Viscous stresses (3)
(15)
ab(a+b){, )
T = 4 \J xy + Y yx
(a + byy yy
All =- 8 '
A = _ b((a + 2b)y.<y + ay vx)
12 8 '
(16)
From incompressibility divv = 0 it
outside the droplet are
ex ex
0"xx = -0" yy =
211 "X (T<D.u - S<Dxv + A11xQ.m + AI2XQ.uy + A21yQur + AnyQxxy
Here
150
Chapter 5: included papers
rt/2 rt/2
where E(e)= J~l-(e sinxydx and K(e)= JI/~l-(e sinxYdx are the complete
o 0
elliptic integrals of first, resp., second kind, e2 = 1- a2 / b2, 11in , 11e.x: viscosities of
fluid inside, resp., outside the droplet, R = J;;b : a radius of a circular drop shape. A
time scaling unit is orb = Rllex / o , Bm = H~R/ o : magnetic Bond number.
The large semi-axis a of an ellipse and the angle <pof its orientation (see Fig.l)
are chosen as system variables to characterize the state of an elliptic droplet. The
relation between rate of change of a,<p and velocity gradients Yij is established by
considering the evolution of an elliptic droplet during the time interval dt . In a
dimensionless form
Y + a4y;.. A • xv vx (19)a-cry <p- - -- xx' - ""4 1a -
Expressing Yij from (18) and inserting in (19) one obtains the following motion
equations:
;.. 2a4a=------xn(a4 + 1+ 2A(2)
I( _1)2 A[H2 1 H2 1 ] <D(a)]l1-'16 Hma IJi (a'+I-') - IJi (;;21-'+1) -;;'-1
A2 a4 + 1 (A4 )
. HOaHOb Bm ().l-lya2 2a a4 -1 + A~ -1<p - -------- -~---- (21)
- H~ 8n (a2 + 11X(2).l + 1) 2a2 + A(a4 +1)
Results of numerical simulation, applying Runge-Kutta 4th order algorithm are
shown on Fig.2. The main difference of results comparing with [7J is the following: if
a magnetic Bond number (Bm=15, Fig.2a) is below the critical value then critical field
frequency is equal to infinity like it is obtained by BEM simulations [7]. It means that
by field frequency tending to infinity droplet still follows the field and the reason of it
is that droplet extension diminishes and a shape of the droplet tends to unpertubed
one. This phenomenon is not observed using equations of motion, described in [7J. If
magnetic Bond number (Bm=30, Fig.2b) is beyond the critical value then critical
frequency exists exceeding which droplet can not follow the field because friction
forces on an extended droplet in a surrounding fluid are too large. Other evidence of
better agreement of the present motion equations with 2D BEM simulation [7J is
smaller difference between stable states calculated by both method (BEM and the
equations of motion derived here). Main improvement is achieved due to better
representation of the extensional motion of a droplet and due to accounting for
rotational motion caused by a shear flow inside the droplet. Results prove that the
present equations of motion could be used to simulate droplet behavior in a magnetic
field with sufficient accuracy. It is interesting to note in Fig.2b that unstable focuses
beyond critical frequency exist.
This work was supported by "Le Reseau Formation Recherche n090R0933 du
Ministere de l'Enseignement Superieur et de la Recherche" of France and by
International Science Foundation in terms of long-time grant LBGOOO. Author is
grateful to Dr.J.C.Bacri, Dr.A.Cebers and Dr.R.Perzynski for helpful discussions.
(20)
151
Chapter 5: included papers
References
[1] Rosensweig R.E Ferrohydrodynamics - Cambridge University Press - 1985.
[2] Helgessen G., Piersnski P, Skjeltorp A. T. II Phys.Rev.Lett. - 1990. - Vol. 64, N12
- P.1425
[3] Skjeltorp A. T., Helgessen G. II Physica A, 1991, v.176, p.37
[4] Baed 1.C, Cebers A. 0., Perzynski R. Behaviour of a magnetic fluid microdrop in
a rotating magnetic field II Phys.Rev.Lett. - 1994. - Vo1.72, N17 - PP.2705-2708
[5] Cebers A. II Magnitnaya Gidrodinamika (in Russ) - 1985. - N1 - pp.25-34
[6] Jeffrey OF. II Proc. Roy. Soc. - 1922. - A102 - PP.161-179
[7] Baed 1.C, Cebcrs A., Lads 5., Perzynski R. II Abstracts of 7th IeMP -Bhavnagar
- 1995. - PP.187-188
Captions
Figure 1
An elliptic 2D droplet in laboratory coordinates. Axis X,Y rotates synchronously with
magnetic field.
Figure 2
Stationary positions of the droplet tip for different field frequencies. Bm=15 :
magnetic field value below critical one, Bm=30 : magnetic field beyond critical value.
152
Chapter 5: included papers
x
\\
Figure 1
153
Chapter 5: included papers
3.0
2.5
2.0
Y 1.5---.H
1.0
0.0
0.0
154
a)
~=O.l6"
\1
\ I
\
Bm=15
~=15
0.5 1.0 1.5x
BEM
.: stable focus
Figure 2
b)
6
~=.O.008
5
4
Y3
Bm=30
j.!= 15
2
o a 2 3x
Equations of motion
A: stable focus,
.: critical frequency,
0: unstable focus
Chapter 5
Conclusions
1. In low-frequency magnetic fields a ferro fluid droplet exhibits two kinds of a behaviour
in dependence on the applied magnetic field strength. The critical Bond number which
separates these two kinds of behaviour turns out to be approximately the same as the
threshold value with respect to 2-lobe perturbation in a high-frequency field.
2. At lower values of the viscosity of the droplet the critical frequency with respect to an
ability to follow a rotation of a magnetic field is slightly higher than for a rigid droplet.
The configurations of a droplet with the phase lag S larger than n/4 are unstable for
large values of A, but moderate values of A cause the stabilising effect: there exist
steady states with S slightly larger than n/4 (QH slightly smaller than QcJ0.
3. The motion of 2D droplet could be described by a simple set of equations of motion,
given in the present chapter under the name "the Second" set. The use of this set is
limited by assumption of an elliptic shape of a droplet, what means that it is adapted
for moderate ratios of viscosities Iv:::, 1 and the accuracy is better if higher values of the
magnetic Bond number are considered, since at low values the flow can not be
described by constant velocity gradients with a high accuracy. Beyond these limits the
equations could be still used to predict the common properties of the behaviour of a
droplet.
4. A 2D viscous droplet (A~5) exhibits a bending instability in the magnetic field
oriented perpendicular to the major dimension of a droplet. The significant bending
effect is observed in the case when droplet is exposed long enough to such a field, i.e.
if the field frequency slightly exceeds the critical one with respect to the ability to
follow the field. More detailed studies of this effect are highly desirable.
155
Chapter 6
Droplet in low-frequency elliptically
polarised field: mode locking and
Devil's staircase
As it is shown in Chapter 5, the "jerky" motion of a ferrofluid droplet beyond
the critical frequency in a rotating field exhibits a "smooth" dependence on field
frequency QH in the case of "high-field" (see Chapter 5) behaviour, if the polarisation
of a rotating field is circular and QH>QCR' By the "smooth" dependence on QH it is
understood that time-averaged angular frequency Q of a droplet rotation decreases
strongly as QH increases (see Fig.5.l). The consequence of that "smoothness" is the
following: in a rotating field with circular polarisation there is no frequency locking
between QH and Q. This observation is quite natural since there is just one frequency,
namely the frequency of droplet oscillations QD=QW Q, which is present in such
rotation (see the trajectories in Fig.5.1). The frequency Q have mostly mathematical
sense in this case. The field frequency QH excites rotation of droplet, the oscillations
appears by change of the phase lag S which depends not only on QH' but also on the
friction torque. The physical situation changes if the inertia effects are taken into
account: in such a case there is a frequency of a shape oscillations and hence an
eventual frequency locking may take place. In the case of microdroplets inertia plays
no role and all oscillations are damped during a fraction of a self-oscillation period
due to the viscous dissipation of the energy.
The frequency locking is the common phenomenon in systems with two
frequencies. The ferrofluid droplet behaviour could obtain a second frequency by an
157
Chapter 6
elliptically polarised field: it contains the pure rotational component, which is an
origin of the frequency QD=QW Q, and the pure linearly oscillating component,
which has the frequency QH [6*]. The characteristic property of the system like a
ferro fluid droplet is that the response to the oscillating field has frequency 2QH. Thus,
the ferrofluid droplet in the elliptically polarised field possesses the necessary
requirement for a mode locking. The subject of the present chapter is to describe mode
locking for a ferrofluid droplet. Similar phenomena were already observed in
[57,101,102] for a pair of bounded non-magnetic spheres in a magnetic fluid (this
system is called "magnetic holes"). There mode locking ranges are found
experimentally and well described by a single nonlinear equation.
In fact, the mode locking is not a rare phenomenon, it appears in many systems,
where at least two competing frequencies are present [16,80]. The simplest example is
the one observed in the 17th century by Christian Huyghens: two clocks hanging back
to back on the wall tend to synchronise their motion [16]. The two frequencies may
arise dynamically within the system itself or through the coupling of an oscillator to
external periodic force. There is usually some parameter, varying which the system
passes through regimes that are phase locked and regimes that are not. When systems
are phase locked the ratio between their frequencies is a rational number. The width of
the phase-locked intervals depends on the strength of a coupling. A typical example of
mode locking intervals is so called Devil's staircase [16,80]. The Devil's staircase
appears not only in dynamical systems, but also in long-range spatially periodic solid
structures.
Here only some examples, where a phase-locking takes place, are mentioned.
Thus key physiological systems, such as the cardiovascular, respiratory,
neuromuscular, and hormonal systems, display intrinsic oscillatory behaviour [53].
These systems interact with another and the outside environment. Moreover, there are
innumerable feedback loops acting to maintain physiological variables within normal
limits. Another example is the driven van der Pole oscillator [82]. It is one of the most
intensively studied systems in nonlinear dynamics. Van der Pole oscillator is
described by an equation, which in the 1920s was introduced as a model for a simple
vacuum tube oscillator circuit [80]. The well-known example is also the Belousov-
158
Chapter 6
Zabotinsky reaction. It is a complex chemical reaction with oscillatory character
[80,16]. The Rayleigh-Benard hydrodynamic convection systems exhibit similar
properties [60].
Most of information about main properties could be obtained analysing very simple
dynamical systems, called circle maps [3,80], which are mappings in terms of an
angular variable en on the form
( 6. 1 )
The circle map is a discrete analogue of a differential equation
8=0-Kg(en, (6.2)
written in the form
de = (0 - K g(en))tt. (6.4)
Here K and 0 are constants and g is a periodic function, e and en are not the same due
to the different scaling. The winding number is defined as
W 1· en -eo= Im---,
tl""""7etJ n (6.5)
and it is studied in dependence on 0 for different values of K. The comparison of
(6.5) with (6.3) shows that W is the ratio between time-averaged angular frequency of
the system (parameter e) and the driving frequency Q.
If K>O, than mode-locking takes place and results in a diagram W versus Q,
which owns the name "the devil's staircase" due to its fractal properties [80], see
Fig.3 in [6*]. The function W versus Q at K=1 is called a complete devil's staircase,
because all the range of Q completely covered by mode-locking intervals without any
overlapping. Consequently, at K<I, W versus Q is an incomplete devil's staircase. In
the present case mode-locking means, that W is equal to a rational number in some
frequency range. In the case of an incomplete devil's staircase between mode-locking
intervals exist values of Q, at which the W is irrational.
An irrational value of W indicates a quasiperiodic motion, which basically can
be thought of as a mixture of periodic motions of several different incommensurable
fundamental frequencies [80]
159
Chapter 6
The phase-locked oscillations are easy to fmd by numerical methods, smce
mode-locking means that small perturbations, caused by numerical errors, lead to the
same mode and same winding number W. The quasiperiodic oscillation in present
case is not possible to detect by numerical methods, because it will be approximated
by the nearest phase-locked oscillation mode.
An interesting property of the devil's staircase is that mode-locking intervals
correspond to a structure, called the Farey tree [74] as it is shown in [6*]. The
dependence of mode-locking on the control parameter K usually is illustrated by the
plot of mode-locking domains in phase diagram K versus Q. The mode-locking
domains in such diagram are shown by so-called Arnold's tongues (see Fig.9 in [6*]).
The structure of the Arnold's tongue is sketched in Fig.6.1 [26]. The domain of the
definite mode locking consists of the entire region between lines GAL' OAR' Below
the critical line K=1 the W is unique, above the critical line there are three important
lines: first the hyperbole-like curve with tip at D where a period doubling takes place
and the point D correspond to the lowest value of K for that. The period doubling in
this case does not change the value of the winding number, because the orbit changes
from P/Q to 2P/2Q, as it is illustrated by an example from the behaviour of ferrofluid
droplet in Fig.6.2. In this trajectory plot two doublets of period doubling are present. It
is typical, that above that hyperbola-like curve there is a complicate set of period
doublings. Second there are in Fig.6.1 two lines gL and gR, corresponding to the
neighbour tongues. Between these two lines the winding number W is still unique, but
overlapping with the neighbour tongue, for example, between OAL and gL, leads to
the effect that in this region it is possible to find a whole interval of winding numbers.
The period doubling scenarios and transition to a chaos are extremely interesting
phenomena, studies of them could explain the behaviour of a ferro fluid droplet in the
"X-region" in Fig.9 [6*]. Until now mostly the properties of the mode-locking domain
for subcritical values of control parameter is considered and summarised in [6*]. As
last, in Fig.6.3 and Fig.6.4 the one example of a motion, which could be chaotic, is
shown by the states at time moment, which correspond to the magnetic field
orientation along X-axis. Thus, these pictures correspond to the all time moments
tj=n-jIQH' Both figures are obtained performing calculation of time-averaged
160
Chapter 6
frequency of a droplet rotation. As result for every point in the diagram some value of
W is obtained. All together more than 100 different values are detected, in both
figures only the six mostly met are shown, the initial states of the rest are shown by
dots. Are they all really different? It is not sure, because if there is quasiperiodic
motion, or motion which is near quasiperiodic, even during rather long time intervals,
essentially different average frequencies could be observed. Here 2000 field periods
are taken for averaging of every point. The parameters Bm=50; m=15, 1=5, n~0.35,
g=0.3 already correspond to the X-region in [6*]. It is obvious that even in the
zoomed area (see Fig.6.4) there is no some concentration of one mode. Further
research of this phenomenon implementing calculation of Lyapunov exponents [1]
should give sure answer about the character of the behaviour of a ferrofluid droplet in
an elliptic field if parameters match the X-region.
161
Chapter 6
K
1
o o 1
Figure 6.1. Anatomy of an Arnold tongue as explained in the text
162
Chapter 6
6
Y
-2
-4
-6
-6
4
2
o
Bm=50, ~=15, A=5, QH=O.37
-4 -2 x6o 2 4
Figure 6.2. The second period doubling:
a doublet of doubled trajectories
163
Chapter 6
o w=0.2777 ...
6.0
. .- ~ .
A w=0.2833 ..
v w=0.2756 .. all tested
initial states
G w=0.2703 ... • w=0.2760 ...
Figure 6.3. Map of the modes obtained from different
initial states in the laboratory coordinates
164
Chapter 6
Bm=50, A=5, J.l=15, QH=O.35, y=O.3
2 y c w=O.2703 ...
v w=0.2756 ..
A w=O.2760 .
o w=O.2777 .
• w=O.2787 .
A w=O.2833 ..
all tested
initial states
1
x
1 2
Figure 6.4. The zoomed area of the modes
obtained from different initial states in the
laboratory coordinates
165
Chapter 6
Included paper
J.C.Baeri, A.Cebers, S.Laeis, R.Perzynski
Mode locking and devil's staircase for 2D ferrofluid droplet in
elliptically polarized rotating magnetic field
Physica D (submitted)
167
Chapter 6: included paper
J.C.Bacri, A.Cebers, S.Lacis, R.Perzynski
Mode locking and devil's staircase for 2D ferro fluid
droplet in elliptically polarized rotating magnetic
field
Abstract. - Numerical studies reveal that dynamics of the magnetic fluid
droplet under the action of elliptically polarized rotating magnetic field
-
can be quite complicate including transition to chaotic behavior. On the
basis of the equations of motion derived by virial method devil's
staircase and its Farey tree structure for the time-averaged angular
velocity of the droplet in dependence on the angular velocity of
elliptically polarized field has been found. Considering mode locking
(Arnold tongues) in dependence on magnetic Bond number multiple
basins of the attraction in different regions of the parameter space have
been established. By numerical simulations their fractal properties are
illustrated. Existence of the period doublings and transition to chaotic
behavior is also predicted.
A system with non-zero magnetic dipole moment in rotating magnetic field
tends to follow the field rotation, until at some critical frequency friction torque can
no more be balanced by magnetic torque [1]: "break-off' takes place and a rotation of
the droplet becomes "jerky". Similar phenomena are observed for a bound pair of non-
magnetic particles in a magnetic fluid [2,3]. The interplay between magnetic and
viscous forces leads to various modes of motion, classified as I) steady-state
rotations; 2) rotations with stops and backward motions; and 3) localized oscillations.
Transitions between these modes are well described by a single nonlinear equation
and depend on the frequency and amplitude of the rotating field, the fluid viscosity
and the magnetic susceptibility. It is both experimentally and numerically found out
168
Chapter 6: included paper
[3] that for a pair of free spheres phase locking takes place in the elliptical polarized
field: Q./Q.H = 1/2, 1/4, etc., where Q. is the average angular frequency of the pair
rotation, and Q.H is the angular frequency of the magnetic field rotation. Recent
studies of a pair of rigid ferrofluid drops in a rotating magnetic field [4] display the
existence of a plateau for the frequency of a pair rotation at high-frequency range. The
chaotic motion of a compass [5] and a permanent magnet rotor [6] is already studied
in oscillating field accounting for inertia of a system since energy dissipation rate is
quite small in these two cases. The behavior of the magnetic fluid (MF) microdroplet
in the rotating magnetic field includes a wide variety of very complex phenomena in
the high-frequency range [7]. In low-frequency range the shape of droplet is close to a
general ellipsoid [7,8], but nevertheless the behavior even at low frequencies could be
rather complex, since a droplet has more degrees of freedom than rigid magnetic
dipole placed in a rotating field. One very important characteristic of such a system is
the extraordinary high rate of the energy dissipation due to the small size of droplets,
hence the inertia play no role.
In [9] we have studied numerically the response of a magnetic fluid microdrop
to a rotating magnetic field in 2D by the boundary element method (BEM). In the
low-frequency range, an elongated droplet rotates with magnetic field frequency.
Increasing field frequency, the motion of the droplet goes through a transition from a
state where the droplet follows the magnetic field with a constant phase lag to a state
where the phase lag increases in a series of kinks when the field frequency passes the
critical one. The equations of the droplet motion were derived analytically and good
agreement with BEM was obtained.
Two different types of steady-state behaviors were observed depending on the
magnetic field strength. These two types are separated by the critical value of the
magnetic Bond number, which turns out to be the threshold value of the instability of
a 2D droplet in a high-frequency rotating field with respect to the elliptical
deformations [10]:
Bm, =61I~+IY /~-IY.
At 11=15 we have Bl1lcr=28.14.
1. "Low-field" behavior: If the magnetic Bond number is less than the critical one
( 1)
tBm, = 28.14, 11 = 15), the extension of droplet in stationary configurations is
169
Chapter 6: included paper
diminishing with the increase of the rotating field frequency, and the maximal
phase lag value n/4 is reached at infinite frequency.
2. "High-field" behavior: For the magnetic Bond numbers larger than the critical
one the maximal phase lag (;:::;n/4) is reached already at a finite critical frequency
It was shown in [9] by linear small perturbations analysis of differential equations for
a motion near the stable point that finite value of viscosity inside the droplet causes
stabilizing effect to the ability of droplet to follow the magnetic field rotation. It
results in time-averaged frequency jump from field frequency QH to some smaller
value just after critical frequency Qcr in the case of the "high-field" behavior.
Increasing ratio of viscosities A=TJin/TJex the critical frequency Qcr slightly decreases
and tends to its analytical value QcrO (see relation (l4)further in the text).
Test simulations have shown [9] that in the case of elliptic polarized rotating
magnetic field phase locking takes place like in [3]. The phase locking and Devil' s
staircase in the phase plane Q.jQH versus QH is the subject of our present studies
reported here.
For these studies improved (to compare with [9]) equations of motion, derived
in [11] are used. The new set of equations describes a 2D magnetic fluid droplet in
low-frequency rotating field under the assumption of elliptic shape and accounts for
shear flow inside the droplet. 2D Stokes flow outside the droplet is treated exactly, but
inside is described in approximation of constant velocity gradients. Dynamic
boundary conditions are satisfied integrally by virial moment technique. Arbitrary
viscosities of fluids inside and outside the droplet are considered. A small size of
microdroplet and relatively small characteristic velocities of flow allows to neglect
inertia and gravity terms, concentrating on surface tension and magnetic forces on the
surface of the droplet. Hence the dynamics of a free surface of a droplet can be
described in the framework of the creeping flow.
Elliptic incompressible MF droplet is completely determined by the length of
its large semi-axis a and the angle <p of its orientation with respect to X1ab-axisof the
laboratory frame (see Fig.l ), To conserve elliptic shape of the droplet a flow inside
the droplet is approximated by constant gradients Yik of velocity field
(2 )
170
Chapter 6: included paper
Thus the equations of motion [11] in non-dimensional form are
. 2a4a=------x7t(a4 + 1+ 2 'A.a2 )
[
(~-1)2Bma(H~; 1 2_H~: 1 2J-~(a)]
16 H 0 (a 2 + I..l) H 0 (a 2 J.l + 1) a-I
(3 )
( 4 )
Here
cD(a)=~a4 +1)E(e)-2K(e)) A=l1iIJ1lex' a=adim/R,
,,/2 ~____ ,,/2
where E(e) = f~I-(e sinxyd-r and K(e) = fl/~l-(e sinxYdx are the complete
o 0
elliptic integrals of first, resp., second kind, e2 = 1- b' / a2 , u: magnetic permeability,
l1in' l1et : viscosities of fluid inside, resp., outside the droplet.
A radius of a circular drop R = J;;b and a time scaling unit 1" h = Rl1 ex / a are
used to obtain non-dimensional form of equations, the interplay of magnetic forces
and a surface tension is characterized by magnetic Bond number Bm = H~Ria.
Main improvement is achieved due to better representation of the extensional
motion of a droplet and due to accounting for rotational motion caused by a shear flow
inside the droplet. Results prove that the present equations of motion could be used to
simulate droplet behavior in a magnetic field with sufficient accuracy.
In the case of elliptic field polarization, instant external magnetic field
components are given by the following relations:
Hx(t)=Hoxcos(nHt, (5)
H y (t) = HOY sin(n Ht).
We introduce parameter y to control magnetic field ellipticity, assuming Hox ;:::HOY:
n., = Ho.J1+Y , ( 6 )
Hoy=Hoh,
H~x - H~yY= 2 2· (7)Hox + HOY
171
Chapter 6: included paper
Advantage of such a definition is that mean square value for external field is fixed by
choice of Ho whatever is the value of y, keeping constant effective magnetic Bond
number Bm = H; Ria:
H2 = (Hox cos(QHt)y + (Hoy sin(QHt)j = ~(H~x + H;y)= H;. (8)
Hence elliptically polarized field is given by its "average" intensity Ho and ellipticity
parameter y. For the physical interpretation of elliptical field polarization we can
divide the field in the pure rotational field Hrat, the components are
{
Hx,r01((): Hoh C~S(QHt)
HY,ro,(()- Hoh sm(QHt)
and a non-dimensional amplitude is hro, =h, and the pure linear-oscillating field
Hose' the components are
{
Hx,ro,(t): Ho(,Jl+Y -h}os(QHt)
H y .rot (( ) - 0
and the non-dimensional amplitude is hOle= .J1+Y - h. Increasing g from 0 to
1, the rotational field amplitude hrat tends to 0, but the oscillating field amplitude hose'
in contrary, increases from 0 to J2 .
The instant projections of external field intensity to directions of both axes of elliptic
droplet Hoa, HOb'implemented in equations (3), (4), are derived as
Hoa(t)= Hx(()coscp + Hy(()sincp, (9)
HOh(t)= -Hr(()sincp + Hy(()coscp,
Here cp IS the angle between the major semi-axis of a droplet and X-axis of the
laboratory coordinates. Let us introduce non-dimensional field components in
directions of semi-axes of the elliptic droplet ha = Hoa/ Ho and h, = HOh/ Ho . Then
h; = [1+ cos 2<pHcos 2cp + Y(cos 2cp H + cos 2cp)+
.JI=Y2 sin 2cp H sin 2cp J2 '
h; = [1- cos 2cp Hcos 2cp + y(cos 2cp H - cos 2cp)-
~1 - Y2 sin 2cp H sin 2<pJ2'
hahb = E- cos 2cp H sin 2cp - Ysin 2cp + ~1 - Y2 sin 2cp H cos 2cpY2.
Here cp H = QHt . In the case of circular field polarization (y=0) we have
( 1 0)
( 11 )
( 1 2 )
172
Chapter 6: included paper
h~Iy=o = [1 + cos 2cpH cos 2cp+ sin 2cpH sin 2cp}'2 = cos? (cpH - cP),
h;ly=o = [1- cos 2cpH cos 2cp- sin 2cpH sin 2cp}'2 = sin 2 (<pH - cP),
hahb Iy=o = [- cos 2cpH sin 2cp+ sin 2cpH cos 2<p}'2 = sin 2(cPH - cP)/2.
Hence the rigid droplet limit A.~OO in the case of circular field polarization IS
analogous to bound pair of soft magnetic particles [2]:
<i> = Ocr sin 2(cpH - cp),
Bm (J.l-Iya2(a4 -1)
Ocr = 16n 02 + /l Xa2/l + lXa4 + 1)"
Time-averaged frequency ° of a droplet rotation is defined as
(13)
( 1 4)
T°= lim ~ Jipdt . ( 1 5 )T~ooT o
An analytical integration for (13) gives the following expression for time-averaged
frequency:
- J 2 2O=OH-"OH-Ocr' (16)
N.B.: the value of a is fixed, what corresponds either to "rigid droplet" approximation
'A» 1 or "high-frequency" approximation 0H»Ocr. Tending 0H/Qcr to infinity we
arrive at approximation for "very high" frequencies:
Q ~ Q;r .
2QH
In [4] it is reported that after a jerky regime, the frequency of the pair rotation reaches
( 1 7 )
a plateau independent of QH' Hence in really high frequencies (QH/Qc?> I0) present
approximations could be to rough to describe the behavior of the droplet, account for
effects like internal rotation could be necessary.
The elongation of a droplet depends on the intensity of applied magnetic field,
i.e. major semi-axis a could be obtained from equation (4) at constraint a = 0 :
(J.l-l)2 Bma[ 1 _ 1 J = <D(a) .
32 (a2 + /l j 02 /l + Ij a 4 - 1
Curves of a versus Bm are plotted in Fig.2. Curves for different /l values certificate
( 1 8 )
that influence of magnetic permeability to droplet extension could be very strong. In
dependence on magnetic permeability the instability in respect to elliptic shape
173
Chapter 6: included paper
perturbation beyond the magnetic field threshold has either the first order (11<10.71)
or the second order (IV 10.71) transition properties [10].
Let us now discuss rigid droplet limit A~OOin the case of elliptic field polarization.
Equation (4) transforms to
<P = Ocr E- cos 2<pH sin 2<p- Ysin 2<p+ .Jl7 sin 2<pH cos 2<p ( 19 )
Though multiplier Ocr is constant in the approximation of rigid droplet, analytical
integration of equation (19) in order to obtain ° is not possible due to strong
nonlinearity. Hence 4th order Runge-Kutta algorithm is applied to solve the system of
equations of motion (3),(4). Time-averaged frequency ° is obtained in the following
I
way: at first numerical algorithm is run some transition time T, in order to obtain from
initial perturbations free state, afterwards this state is taken as initial one for numerical
algorithm. After N field revolutions winding number [12]
1 T,+27tN/D.ff _
W = -- J<P dt = <pH <Po
N 2rrN t: 2rrH,
(20)
is found. Increasing N, averaged frequency 0= WNOH could be found with arbitrary
accuracy, of course, truncation errors as well as accuracy of Runge-Kutta algorithm
should be taken into account. In Fig.3a the winding number wN = O/OH is plotted
versus field frequency 0H, Bm=50, 11=15, A=5, y=0 .15. The solid curve shows the
winding number w obtained increasing field frequency, the dashed one: decreasing it.
There is two new phenomena introduced by elliptic field polarization: mode locking
in certain frequency ranges and overlapping of modes. The first two rather wide mode
self-locking steps Will and wl/2 correspond to the frequency range in which circular
field has stabilization of droplet rotation with field frequency due to finite value of A.
The overlapping of both modes results in mode hysteresis: increasing field frequency
there is jump in phase plane from mode WIll directly to mode WII2 passing over non-
existent modes with winding numbers between 1 and 0.5. Increasing further field
frequency, sequentially mode-locking W1/3, W1/4, WI/5 take place. Here and further
ratios like p/q with integer p,q are used to denote frequency ratios
wp/q = O/OH = rl«. Such a mode locking structure is called DeviI's staircase
[12,13]. Here we have degenerated Devil's staircase since overlapping take place and
winding numbers between modes III and 1/2 are not present. The classical Devil's
174
Chapter 6: included paper
staircase is fractal structure, where sequentially all ratios from Farey tree [12] are
present. Obviously degeneration could be eliminated, diminishing stabilization of
droplet's rotation with field frequency, i.e. increasing /....
Another self-locking modes overlapping takes place for the field frequencies in range
about QlFO.372 ..0.376 where exist modes WIl2 for the increase of a frequency and
sequence of modes without detected mode-locking steps for the decrease of a
frequency. According to Devil's staircase properties, in this last case mode-locking
intervals for frequency could exist, just the width of them are to small to detect by
frequency step i\QlFO.0005.
In Fig.03b the frequency range QlF[0.372 ..0.385] is plotted magnified to compare
with the Fig.3a. The left part of the plot is degenerated due to overlapping of modes
already mentioned above. The noisy character of the lower curve corresponding to the
increase of a field at frequencies about 0.374 could be caused either by not a
sufficiently long averaging time or by the presence of competing modes situated close
each another in the phase space. For frequencies larger than QlFO.377 the character of
the curve changes to the classical devil staircase [13], the winding numbers for mode-
locking intervals correspond very well to the construction of Farey tree [12,13] as it is
shown in Fig.3b above the plot of the winding number versus field frequency. One
can see quite well the self-similarity of the devil' s staircase in that frequency range.
The equation of motion (3),(4) show that the solution of them depends on five
parameters: Bm, IJ., /..., y, QH' Here we do not discuss influence of magnetic
permeability IJ. on the solution, but it should be pointed out that IJ. can not be
eliminated from equations of motion by including it in magnetic Bond number since
there are two terms like (a2 + IJ. and (a\l + I , containing both a and u. In the
present paper we focus on the case IJ.=15 which corresponds to an intermediate value
of IJ., since special effects, caused by different choice of Ilare not yet observed.
The influence of Bm
In [9] it is already shown that behavior of ferrofluid droplet III circular rotating
magnetic field depends on Bond number Bm in such a way that two different
scenarios exist (already described above):
1. "low-field" behavior,
175
Chapter 6: included paper
2. "high~field" behavior.
Similar properties are inherit in the case of the elliptic field polarization. In FigA
mode-locking domains are plotted in phase plain Bm versus QH' One can clearly see
the field threshold, below which only one mode WI/I (droplet follows the field)
corresponding to scenario 1. exist. Beyond the field (magnetic Bond number)
threshold mode-locking domains for different modes appear. Width of them do not
change essentially, increasing Bm, slope of boundaries is nearly constant.
We can point out main sequence of mode locking: Wl/q, where q is an integer.
According to Farey tree law between two modes with winding numbers pl/ql and
p2/q2 there is an other mode (PI+P2)/(ql+CU). In present case it works good just for
second sequence, obtained from the main one, for example (1+ 1)/(2+3) gives 3/5. In
general every next sequence becomes more and more narrow and hence harder to
detect. If the width of the mode-locking plateau becomes smaller than a frequency
step for numerical algorithm, then we "do not see" that mode-locking at all. For the
frequency range from mode WI/3till infinity and Bm larger than 40 the behavior of a
mode-locking seems to be more less regular: mode-locking domain boundaries are
parallel with fixed slope, increasing q, a mode-locking w l/q frequency-width
decreases. For lower frequencies rather wide overlapping domain take place for modes
WIlland wll2' Between modes Wl/2 and w2/5 we have observed frequency interval with
oscillations close to quasiperiodic ones (see further in text for definition of
quasiperiodicity), so for these modes overlapping exists no.
More complex behavior is for Bm just after the threshold till values about 34 and for
field frequencies QH between 0.32 and 0.36. This part of phase space is plotted in
"zoomed" area. Due to finite numerical methods, mode-locking domains W4/5 and W3/4
perhaps are not well detected. More accurate are plotted domains for locking with
winding numbers Will W1/2, w2/3and w3/S' Here for the point (QlFO.335, Bm=30.5) of
the phase space an overlapping for four modes take place. It is illustrated by
trajectories of the droplet's tip in the laboratory coordinates in Fig.5. Trajectories
show rather stable motion without any transitions. The "choice" of the mode, in which
a droplet self-locks, depends from initial conditions. To illustrate it in Fig.6 bassins of
attraction [14] are plotted. This map represents laboratory coordinates at initial (t=0)
moment, at which a magnetic field has only positive X-component according to (5).
176
Chapter 6: included paper
There are 4 bassins of attraction (domains of initial conditions) leading to mode W1/2,
separated by the mode W3/5 on the diagram. Domains of initial conditions for mode 1/1
are concentrated near the corresponding Poincare sections at time moments
~=n*1t/QH' Interesting is the structure of the spiral tails of those domains, it, as well as
initial points for mode W213 exhibit fractal like properties, proved by the "zoom" of the
present picture, shown in Fig.7. In both diagrams mode WIll is shown by empty
circles, W1/2: by crosses, W2l3: by filled circles, and mode W3/5 by small "-" characters.
Main conclusion, which is drawn from both figures, is that the mode could be very
sensitive to the choice of initial conditions, from which it evolves. If the hole map of
initial conditions has fractal structure it should lead to a chaotic motion, since every
small perturbation leads to the transition to a different mode, and thus, to the loss of
the information about initial conditions. In the present case, periodical solutions for
equations of motion show that initial conditions which initiate periodical motion
without any perturbation, are placed in non-fractal regions of map, hence here just the
sensitivity for a transition to some definite mode for the certain ranges of initial
conditions is present.
Another quite important question is how the threshold for magnetic field behaves if
frequency is fixed but the ellipticity parameter y of the field polarization is increased
from zero. Fig.8 shows that increasing y, the changes of the threshold value are near
negligible. It proves the choice of magnetic Bond number: it characterizes the field
threshold. Two curves in the Fig.8 stays for the increase of Bond number (solid curve)
and decrease of Bond number (dashed curve), shift between them displays the
presence of a hysteresis with a magnitude which corresponds quite well to the
hysteresis magnitude in the case of magnetic permeability like !l=15 [10]. If y reaches
some limiting value, a droplet is locked by the oscillating component of the field and
the rotating component is too weak to cause droplet rotation. A tip of a droplet
provides located oscillations without any rotation.
The influence of y
In Fig.8 the influence of y is observed by mode-locking to WO/l> i.e., the mode of
localized oscillations without rotation of the droplet. The critical value of y is about
y=O.3 ..0A (Bm=30 ..140). That corresponds to the amplitude of pure rotating field
177
Chapter 6: included paper
hrot=0.84 ..0.77 and the amplitude of pure oscillating one hcsc=0.30..0.55. So the limit
of y for rotation of a droplet increases rising the value of Bm. It displays that
increasing Bm, the rise of the magnetic torque turn out to be more significant than the
rise of the friction torque and the thus a droplet can perform full revolutions up to
slightly higher limit ofy (see Fig.8).
The traditional representation for the mode-locking phenomena in dependence on
nonlinear coupling control parameter is so-called Arnold's tongue diagram [13,15]. In
Fig.9a the mode locking domains are plotted in the phase plane of the field ellipticity
parameter y versus the field frequency QH for Bm=50, fl=15, A=5. Two modes Will
and Wall predominates in the frequency range QH without any overlapping. The space
between them is filled with Arnold's tongues for other modes at y<0.275, the behavior
of a droplet for y>0.275 is the subject of our studies to be published elsewhere. The
boundaries of the domains Will and Wall are rather smooth, except the single step with
some slope for the mode Will at y about 0.275. This value, y=0.275, turns out to be
near the critical y, beyond which motion of a droplet becomes chaotic and Arnold's
tongues loses their structure. In Fig.9b the region below y=0.3 is plotted in more
details for the same values of passive parameters (Bm=50, fl=15, A=5). Here "Region
X" is that one, results about which we shall publish elsewhere, flat top boundaries of
tongues are not very realistic, detailed studies of them are related to the "Region X"
subject. In the Fig.9b one can see quite well the two sequences of mode-locking: the
main sequence as l/q and the second one as 2/(2q+ 1), q: integer. Beyond WlllO the
structure of tongues becomes too dense to show it, intervals " ... " are used to
substitute Wl/q and W2/(2q+I), q> 10. The unique property of the present system is
overlapping for mode Wl/2: even at y~O complete overlapping takes place for modes
Will and W1/2' The mode-locking domain W213 at y about 0.27 and QH about 0.35 looks
curious but in fact that strange placement of it obviously should be explained by
common properties of Arnold's tongues in the case of the supercritical regime.
Throughout the work the value y=O.15 is used in the cases, when g is a "passive"
parameter (i.e., it is constant). This value, y=O.15, is regarded as a moderate value for
the field ellipticity control parameter.
178
Chapter 6: included paper
The influence of ')..
By increase of').. the ratio of viscosities are changed. Here we discuss almost the case
')..=5, what could be called the case of moderate viscosity of a ferrofluid, if we
compare with [7], where ~100. 2D simulation using the boundary element method
shows [16] that in the case of an arbitrary shape (without the constraint of the elliptic
shape) ferrofluid droplet displays bending instability for A.» I and even for ')..=5.The
results of present studies nevertheless could be applied because for A=5 bending just
slightly eliminates effective extension of a droplet and thus causes a minor decrease of
a friction coefficient. General properties for droplet rotation still stay the same.
In the Fig.lO Arnold's tongues are plotted in the phase plane A (log-scale) versus QH'
The values of passive parameters are Bm=50, fl=15, y=0.15. The overlapping of
modes W1/1 and WI12 is common phenomena up to ')..=115,for larger values of Amodes
do not overlap. Overlapping for modes Wl/2 and W1/3is present just up to ')..::d.5. In
general, increase of ), makes mode-locking width smaller and thus tends to eliminate
overlapping. Nevertheless at A=100 the width of the mode WI12 is still quite
commensurable with distances between this mode and neighbor ones, whether w 1/1 or
w1/3' he second property of A is the shift of mode-locking frequencies into the
direction of a larger frequency, if A increases.
The character of that how the mode-locking width depends on ')..follows from general
mechanism of mode-locking for ferrofluid droplet in the elliptically polarized
magnetic field. The ellipticity of the field polarization, in fact, is the key-parameter for
mode-locking phenomena in for the present problem. Since inertia plays no any role
for oscillations of the droplet due to the small size of it, there must be two oscillatory
motions to have mode-locking. One oscillation frequency quite naturally IS
maintained by rotational component of the magnetic field Hrot. Other one IS
introduced into the motion of the droplet by pure oscillatory field Hose. The sketch of
the mechanism of mode locking in present case is as follows. The Hrot causes whether
the steady rotation of a droplet or the non-steady but periodic rotation with
oscillations if the field frequency exceeds the critical one [9]. In the last case the
frequency of droplet rotation is less than that of the field. The pure linear oscillations
of a magnetic field causes oscillations of the droplet with frequency 2QH in X-axis
direction. The amplitudes of both oscillations strongly depend on A: increasing A,
179
Chapter 6: included paper
amplitudes diminishes. In the limit A~OO (rigid droplet) the elongation of the droplet
is fixed (constant) and just the phase lag between the magnetic field and the droplet
could oscillate according to (19). For a system without inertia the self-resonance is
absent and two "external" frequencies are needed to realize the mode-locking. Hence
the rigid droplet and the droplet in a magnetic field with the circular polarization
definitely do not have mode-locking. The "soft" droplet in the field following regime,
naturally, has the simplest mode-locking WIll' After the exceeding of the critical
frequency the break-off for the droplet takes place and it rotates with some lower
frequency than the magnetic field (see approximation (16)). This is "the clock" which
may, or may not, lock to the another "clock" of pure linear oscillations.
Conclusions
The ferrofluid droplet in elliptically polarized field exhibits Devil's staircase like
mode-locking to the magnetic field rotation frequency. In the limit of the circular field
polarization the mode-locking disappears. In the case of linear field polarization
pulsing oscillations of a droplet takes place. The difference from traditional Devil' s
staircases is overlapping at very small values of control parameter. Thus the
interesting mode-locking system for dissipative system without inertia is found and
described. The transition to the chaos will be described elsewhere.
REFERENCES
[1] R.E.Rosensweig. Ferrohydrodynamics. Cambridge University Press, 1985
[2] G.Helgessen, P.Pieranski, A.T.Skjeltorp Phys.Rev.Lett., 1990, v.64, N12, p.1425
[3] A.T.Skjeltorp, G.Helgessen, Physica A, 1991, v.176, p.37
[4] lC.Bacri, C.Drame, B.Kashevsky, S.Neveu, R.Perzynski, C.Redon - Motion of a
pair of rigid ferrofluid drops in a rotating magnetic field - Progr.Colloid.Sci.( 1995) 98,
pp124-127
[5] V.Croquette, C. Poitou - Cascade of period doubling bifurcations and large
stochasticity in the motions of a compass - lPhysique Lettres 42 (1981) pp L537 -
L539
[6] F.C.Moon, lCusumano, P.J.Holmes - Evidence for homoclinic orbits as a
precursor to chaos in a magnetic pendulum - Physica 24D (1987) pp 383-390
180
Chapter 6: included paper
[7] J.C.Bacri, AO.Cebers, RPerzynski Behaviour of a magnetic fluid microdrop in a
rotating magnetic field, Phys.Rev.Lett. - 1994. - Vo1.72, N17 - PP.2705-2708
[8] ACebers, S.Lacis Magnetic fluid free surface instabilities in high frequency
rotating magnetic fields. - Brazilian Journal of Physics, 1995,Vo1.25, N2 ,pp.l0l-l11
[9] J.C.Bacri, ACebers, SLacis, RPerzynski Dynamics of the magnetic fluid droplet
in rotating field -Journal of Magnetism and Magnetic Materials 149 (1995) 143-147
[10] J.C.Bacri, ACebers, S.Lacis, RPerzynski On the shape calculation of the
magnetic fluid droplet in rotating magnetic field - Magnitnaja Gidrodinamika (to be
published)
[11] S.Lacis Equation of motion of 2D elliptic magnetic fluid droplet in a rotating
magnetic field - Magnitnaja Gidrodinamika (to be published, full text)
[12] F.C.Moon Chaotic and fractal dynamics: an introduction for applied scientists
and engineers, New York: Wiley, 1992, 508p
[13] P.Bak The Devil's staircase Physics Today, 1986, Vo1.39, N12, pp38-45
[14] E.Ott Chaos in dynamical systems New York: Cambridge University Press, 1993,
385p
[15] V.I.Arnold Geometrical methods in the theory of ordinary differential equations
New York:Springer-Verlag, 1983, 334p
[16] J.C.Bacri, A.Cebers, S.Lacis, R.Perzynski - (to be published)
FIGURE CAPTIONS
FIG. 1 - Sketch of the droplet in laboratory coordinates
FIG.2 - Droplet extension in time-averaged high-frequency rotating magnetic field
FIG.3a - Winding number w versus magnetic field frequency QH' Bm=50, f.1=15,/..=5.
Field ellipticity parameter y=O.15. The solid line corresponds to increase of a
frequency, dashed one: decrease of it.
FIG.3b - Zoom from the FIG.3a. Winding number w versus magnetic field frequency
QH' Bm=50, f.1=15,/..=5. Field ellipticity parameter y=O.15. The solid line corresponds
to increase of a frequency, dashed one: decrease of it. Farey tree branches for
construction of observed mode-locking intervals are shown above the plot.
FIGA - Mode-locking domains (Arnold tongues) in phase plane Bm versus QH'
y=O.15, f.1=15,/..=5.
181
Chapter 6: included paper
FIG.5 - Four different modes for the point (QH=O.335, Bm=30.5) of phase plane.
).1=15,A=5, y=O.15.
FIG.6 - Map of initial conditions for four different modes of Fig.5.
FIG.? - Zoomed area from FIG.06.
FIG.8 - Magnetic field threshold versus the ellipticity parameter y of the field
polarization.
FIG.9a - Arnold's tongues diagram in the phase plane the field ellipticity control
parameter y versus the field frequency QH' Mode locking domains for W1l1 and WOIl'
FIG.9b - Arnold's tongues diagram in the phase plane the field ellipticity control
parameter y versus the field frequency QH' Mode locking domains for intermediate y
values.
FIG.lO - The width of mode locking in dependence on droplet viscosity, Arnold's
tongues in phase plane A versus field frequency QH'
182
Chapter 6: included paper
x,
Fig.!
183
Chapter 6: included paper
12 I /II
if 11=15 II .Jl.:"'4 I ./
11=40./: I l10 I /11=10I
/~I
, . I .,/
I //... I
~ I ~...8 I ../C/) I i.- I / .....~ I. Iro . I
../'/I . Ia ! II ......(]) 6 I ............00 I I ~=5I$-< I ,.'0 I ./'. I ;' ~=10"(;j1 I .../ ..'
~
I
4 I /'/ ~=15JI //'I ~=25I
I /, / ~=40,2 ,I !I {l1
101 102 103 104
Magnetic Bond number Bm
Fig.2
184
Chapter 6: included paper
1.0 1/1
I
I Bm=50, J.l=15, A=5, y=O.151A
0.8
~
~ w=O/OH~ 0.6 -+...0 IS II;::j : 1/2l:::: -------«------ ..t.en Il:::: 0.4 \.-""d
l:::: 2/5.-
~
0.2 ' ,1/4 ' ,1/5
0.0
0.30 0.40
Field frequency 0H
0.450.35
Fig.3a
0.50
185
Chapter 6: included paper
0.50
~
[) 0.45
..L:Ja;:j
s=
bO
C.-""0
~ 0.40
0.35
186
1/2 •. 1/3
1/2 t 2/5 •. 1/3
3/7 T 2/5 •. 3/8 r 1/3
5/12 2/5-----.-5113.- 3/8-.--4111
2/5 .7/18.5/13 8/21T3/8 7/19T4/11
2/5.9/23 12/31 11/29 11/30
11/28
1/2
Bm=50, J.l= 15, "'-=5, y=0.15
r<«,
~r. 5/12- - ~"",:T\---
'.',I..••....\. -(_ 2/5
12/31
11/28 --===~- : 5/13
9/23 :---.
7/18 3/8
8/21 :
11/29
0.375 0.380
Field frequency QH
0.385
Fig.3b
Chapter 6: included paper
1:1 & 1:2
2:3 &1:1&1:2
2:3 &1:1
2:3 & 1:2
y=O.l5
A=5
4:5 3:4 3:5
70e
~
~ 60(])..0 1:1a::s 50=""0=0 40co
o.-..•...
(])= 30co
C\l
~ .-
20 4:5 3:4 2:3 3:5
10
0.30 0.35 0.40 0.45 0.50
Field frequency QH
Fig.4
187
Chapter 6: included paper
5
o
-5
-5
188
Bm=30.5, ~=15, "-=5, QH=O.335
1: 1
aa=2.516
uo=2.334
1 :2
aa=2.370
uo=1.413
2:3
aa=2.888
uo=2.308
o 5
Phase portraits of the droplet's tip
Fig.5
Chapter 6: included paper
IRI mode 1/1
mode 1/2
.en mode 2/3
""'~,:-.,mode 3/5
Map of initialconditions, leading to different modes
Fig.6
189
Chapter 6: included paper
1.7
1.6
1.5
1.4
1.3
1.2
1.1
1.0
1.1
190
=~mode
~ 1/1
••••••••••••••••••••••••••••••mode•••••••••••••••••••••••••••••••..............2/3••••••••••••••••••••••••••••••
1.2 1.3 1.5 1.6 1.71.4
Zoomed are from the map of initial states
for different modes of locking
Fig.?
1.8
Chapter 6: included paper
, 140
Bm
120
100
80
60
40
20
0.0 0.1 0.2
Fig.8
1:1 0: 1
0.3 0.4y
191
Chapter 6: included paper
?-
C 0.6.•...•o.-.•...0...----'---'(1)
""d
'""0 0.4.-~
192
1.0
0.8
0.2
0.0
0.0 0.1 0.2 0.3
Field frequency QH
0.4 0.5
Fig9a
.-u.-..•...0..
- 0.15-(1)
"0-(1).-~
Chapter 6: included paper
0.30
0.25
?->. 0.20..•...
0.10
0.05
0.00
0.25
Bm=56
A,-5
0:1
\---~----
\
\
\
\,
\,
\
\,
\\.
\1:1 \\
\
\
\
\
\ ,
\ 'v . ,\ 1:2 "\ ,
\ ,
\ I
\ I\l
2:19
2:17
2: 15
2:13
2:5 2:7 2:9
0.30 0.45 0.500.35 0.40
Field frequency QH
Fig.9b
193
Chapter 6: included paper
Bm=50, ~=15, y=0.15
102
I
Ir
J
J illI
~
I
I 1~1U') I(l) ".- 'I~ J.- JU')
0 I
U 101 IU') I.- I;> I
~ ,
0 ,
0 I.- I
~ •~ I 1:1 & 1:2I
I
I
I
I 1:3100
0.32 0.36 0.40 0.44 0.48
Magnetic field frequency QH
Fig.! 0
194
Chapter 6
Conclusions
Here the behaviour of the 2D ferrofluid droplet in an elliptically polarised
magnetic field is studied by analysis of the equations of motion. These equations are
the set of two first-order nonlinear ordinary differential equations. It is the difference
from traditional approach to mode-locking studies, where mostly "difference"
equations like circle maps are analysed. The ferrofluid droplet in elliptically polarised
field exhibits Devil's staircase like mode-locking to the magnetic field rotation
frequency. It is proved that the sequences of mode-locking obey the Farey tree law. In
the limit of the circular field polarisation the mode-locking disappears. In the case of
linear field polarisation pulsing oscillations of a droplet takes place. The influence of
parameters, such as a magnetic Bond number, a ratio of viscosities as well as an
ellipticity parameter y for field polarisation, on the character of mode-locking is
studied in great detail.
The difference from traditional Devil' s staircase is overlapping at very small
values of control parameter. The mode-locking disappears only in the limit of the
circular field polarisation. Thus the interesting mode-locking system for dissipative
system without inertia is found and described. The transition to the chaos is the
extremely interesting topic, studies of which should be continued to explain the
behaviour of a droplet at large value of a control parameter. The simulation by model
equations have already displayed the period doubling scenario and the transition to
chaotic dynamics.
195
General Conclusions
In the present thesis work, the 2D droplet behaviour in rotating magnetic fields
is studied, assuming linear magnetisation of a ferrofluid and creeping flow inside the
droplet and outside it. To perform this task, different algorithms and corresponding
computer codes are developed, tested and applied. Particularly, the computer codes
are developed, using the boundary element method (BEM), for the following
purposes:
• to solve the magnetic field problem in the case of time-averaged magnetic energy
at given magnetic Bond number and at given shape of a 2D droplet in a high
frequency rotating field and to find the shape with the minimal total energy from
the given set of shapes;
• to solve the unsteady coupled creeping flow and magnetic field problem in the case
of time-averaged magnetic surface forces for a 2D droplet in a high frequency
rotating field;
• to solve the unsteady coupled creepmg flow and magnetic field problem
accounting for instantaneous magnetic surface forces for a 2D droplet placed in a
low frequency rotating field.
The BEM is based on the boundary integral equation technique, which is a
powerful tool to solve the free moving boundary problem for a ferrofluid droplet in an
external magnetic field in terms of surface values for both velocity and magnetic field.
In present work the local curvature dependent, non-equidistant marker point
distribution is applied to solve the magnetic field problem which is very sensitive to
boundary contours with sharp tips. In the Chapter 2 a special technique is proposed to
solve the magnetic field problem in new variables. These variables depend only on
shape of a droplet and its magnetic permeability in the case of the linear
magnetisation. It is shown that the magnetic field problem could be reduced either to a
problem of magnetic surface charges or to a problem of equivalent surface currents,
197
General conclusions
called indirect magnetic field problem formulations. The direct formulation of the
magnetic field problem in terms of tangential and normal field components, with
respect to the boundary contour, provides solution with less approximation error. The
principal difference of the indirect magnetic field problem formulation from the direct
one is that the indirect formulation allows to obtain by a simple integration the one of
two field components when another one is already found.
Beside that, analytical investigations are carried out to get simplified equations
of motion which match numerical simulation data obtained by BEM and allow more
thoroughly investigation of the droplet behaviour that could include even transition to
chaotic dynamics. On the basis of the equations of motion the set of computer codes is
developed implementing Runge-Kutta fourth order method to simulate the droplet
rotation and to perform time averaging of it. This method is able to describe
phenomena of simple periodic rotations, quasiperiodic rotations, rotation with a mode-
locking, and chaotic motion of a droplet. The special technique is used to study
multiple basins of attraction to different phase locking modes.
In general the motion of a droplet in rotating field could be classified in two
cases with respect to the field frequency: the case of a high-frequency field and the
case of a low-frequency field. In the present work the term high-frequency field is
used with respect to the field frequency at which the characteristical relaxation time of
droplet is much larger than the rotation period of a field.
It is found by the energy approach that:
1. There exists a magnetic field threshold with respect to any perturbation from
circular shape in 2D in the high-frequency rotating magnetic field. The increase of
a magnetic Bond number beyond the threshold value causes evolution of spikes on
the surface, which become sharper at further growth.
2. A 2D numerical analysis of MF droplets with different numbers of spikes shows
that the equilibrium shape with two-spikes corresponds to the lowest energy and
thus the transition from 2-spike shape to the shape with larger amount of spikes
does not take place if only energetical arguments are involved.
198
General conclusions
The small perturbation analysis and the direct BEM simulation of a 2D droplet
in the high-frequency rotating magnetic field prove that:
1. In the high-frequency rotating magnetic field a field threshold exists with respect to
any perturbation from circular shape. If Bm is smaller than this absolute threshold
value, a droplet holds circular shape. Increasing the magnetic Bond number beyond
the "absolute threshold", the most unstable mode develops. The small perturbation
linear analysis gives the number of lobes for the most unstable mode in dependence
on magnetic field strength. In the case of pure creeping flow in 2D the number of
lobes is directly proportional to the magnetic field. This relation is proved by BEM
simulation.
2. It is found that in 2D a 3-spike shape is metastable, but a 2-spike shape has the
absolute minimum of energy. The transition to a 2-spike shape from a 3-spike one
takes place either if a magnetic field strength is decreased below the turning point
or by a sufficiently high shape perturbation. Evidently in 2D there exists no
metastable shapes with more than three spikes in a high-frequency rotating
magnetic field under assumptions used in the present work. The simulations of a 4-
spike shape display splitting instability for the "body" of a droplet, where spikes
come together. So it could be concluded that the symmetric shapes with 4 and more
spikes, obtained by the energy approach, correspond to saddle points in a phase
space and thus they are unstable.
It is found by the BEM simulation and the numerical solution of the equations
of motion that in the low-frequency rotating magnetic field the droplet exhibits the
following properties:
1. There are two kinds of a behaviour for a 2D ferrofluid droplet in dependence on the
applied magnetic field strength in low-frequency magnetic fields, called the "high-
field behaviour" and the "low-field behaviour". The critical Bond number which
separates these two kinds of behaviour turns out to be approximately the same as
the threshold value with respect to 2-lobe perturbation in a high-frequency field. If
the magnetic Bond number is larger than critical one, a droplet even at high
frequencies, where it can not follow the field, stays elongated. If the magnetic
199
General conclusions
Bond number is smaller than critical one, a droplet elongation diminishes to a
circle as the frequency tends to infinity.
2. At lower values of the viscosity of a droplet the critical frequency with respect to
an ability to follow a rotation of a magnetic field is slightly higher than for a
droplet with large viscosity in comparison with the fluid outside the droplet.
3. Analytical model is derived, which allows to describe the motion of a 2D droplet
by a set of non-linear equations of motion, if the droplet shape is near the elliptic
one.
4. A 2D viscous droplet (A~5) exhibits a bending instability if the magnetic field
oriented perpendicular to the major dimension of a droplet. The significant bending
effect is observed if the droplet is exposed a time long enough to such a field, i.e.
if the field frequency slightly exceeds the critical one with respect to the ability to
follow the field.
In the case of an elliptically polarised rotating field, the droplet exhibits rather
complex motion, properties of which strongly depend on the ellipticity of the field. By
equations of motion it is proved that Devil's staircase like mode-locking to the
magnetic field rotation frequency takes place. In the limit of the circular field
polarisation the mode-locking disappears. The outstanding property of the present
mode-locking is that the remarkable overlapping of mode-locking intervals is present
even at very small values of control parameter. The simulation by model equations
have displayed the period doubling scenario and the transition to chaotic dynamics.
The bending of the viscous droplet in a rotating field and the transition to the
chaos for a droplet rotation in an elliptically polarised rotating magnetic field are put
in evidence for the first time. The further research in these two directions is highly
desirable.
In short, present studies has proved that many phenomena, observed in real 3D
experiments concerning ferro fluid droplet behaviour, could be explained on the basis
of 2D model, accounting only for the linear magnetisation of a ferrofluid and the
constant surface tension on the interface between a droplet and an external fluid. Some
effects are caused by the 2D geometry but, nevertheless, the obtained results could be
200
General conclusions
implemented to explain the behaviour of so called "liquid bridges" between two
plates, having approximately the two-dimensional geometry. For a more precise
agreement between experiments and numerical simulation results, the 3D simulation
algorithms, which are more computer-time expensive, are supposed to be elaborated
during the further research.
201
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211
Table of figures
Figure 1.1. The sketch of a magnetic fluid droplet.. 37
Figure 2.1. Approximation of a 20 droplet boundary by marker points 52
Figure 2.2. The definition of the pyramidal function 52
Figure 2.3. Periodic symmetry of2D droplet 52
Figure 2.4. The magnetic field characteristics ax, ~x versus the contour arc
length; (a): the equidistant marker point distribution, (b): the curvature
dependent marker point distribution. Definitions of ax, ~x:
Hn=axHox+ayHoy, Hl=~xHox+~yHoY 53
Figure 2.5. The magnetic field characteristics ap ~y versus the contour arc
length, the curvature dependent marker point distribution. Definitions of ay,
~y: Hn=axHox+ayHoy, Ht=~xHox+~yHoY 54
Figure 2.6. The small perturbation decrement test: the contraction of the droplet. 54
Figure 2.7. The magnetic Bond number threshold value test in high-
frequency magnetic field. (a): magnetic permeability 11=10, (b): 11=25 55
Figure 3.1. Analytically generated shapes juxtaposed with a dynamically
simulated one: (a) 2-spike shapes, (b) 3-spike shapes 60
Figure 3.2. Analytically generated shapes juxtaposed with a dynamically
simulated one 61
Figure 4.1. Dependence of magnetic Bond number on the value of magnetic
permeability 11. (a): selected modes n=2,3,4,6,8,10 at 1<11<50, (b): the
behaviour of the critical Bond number at low values of 11 101
Figure 4.2. Approximation of the most unstable mode 102
213
Table a/figures
Figure 4.3. The spectrum of competing modes at Bm*=16 for a potential
flow 103
Figure 4.4. The most unstable n-Iobe mode versus magnetic Bond number in
the case of creeping flow 104
Figure 4.5. The influence of the magnetic permeability to the n-Iobe mode
formation: the multiplier (((~-1 )/(Il+ 1))3)0.5 versus the magnetic
permeability /l 105
Figure 4.6. The spectrum of competing modes at Bm*=6; 11; 16 for a
creeping flow 106
Figure 4.7. Transition diagram 107
Figure 4.8. The simulation of the small perturbation growth 108
Figure 4.9(a)-(c). The amplitudes of modes versus a mode number n: the
mode competition for a droplet with an initially perturbed surface. Bm= 100,
m=25, Dt=O.Ol, N=200. The initial perturbation given by radial
displacements from a circular shape in 60 equidistant points, generated
randomly with the maximal value 0.03 109
Figure 4.9(d)-(f). The amplitudes of modes versus a mode number n: the
mode competition for a droplet with an initially perturbed surface. Bm= 100,
m=25, Dt=O.Ol, N=200. The initial perturbation given by radial
displacements from a circular shape in 60 equidistant points, generated
randomly with the maximal value 0.03 110
Figure 4. IO. The elongation of the 2D droplet in a rotating magnetic field
and the transition back to circular shape at the decrease of the magnetic
Bond number below the turning point 111
Figure 4.11. Surface, magnetic and total energies, and the length r of the
major spike versus time 1. The solid line: the total energy Et=Em+Es, the
dotted line: the magnetic energy Em, the dashed line: the surface energy Es,
214
Table of figures
the thin solid line: the length r of the major spike, measured from the mass
center of a droplet 112
Figure 4.12. Surface, magnetic and total energies, and the length r of the
major. spike versus time 1. The solid line: the total energy Et=Em+Es, the
dotted line: the magnetic energy Em, the dashed line: the surface energy Es,
the thin solid line: the length r of the major spike, measured from the mass
center of a droplet 113
Figure 4.13. Surface, magnetic and total energies, and the length r of the
major spike versus time 1. The solid line: the total energy Et=Em+Es, the
dotted line: the magnetic energy Em, the dashed line: the surface energy Es,
the thin solid line: the length r of the major spike, measured from the mass
center of a droplet 114
Figure 4.14. Transition from the four-spike shape to the two-spike one 115
Figure 5.1. Droplet rotation: the time-averaged frequency of a droplet
rotation versus the frequency of a magnetic field 136
Figure 5.2. Juxtaposition of a BEM simulation with simulation by two kinds
of equations of motion 137
Figure 5.3. Rotation shapes of a viscous droplet: the bending instability in
the "perpendicular field" 138
Figure 5.4. Transition to steady state 139
Figure 6.1. Anatomy of an Arnold tongue as explained in the text 162
Figure 6.2. The second period doubling: a doublet of doubled trajectories 163
Figure 6.3. Map of the modes obtained from different initial states in the
laboratory coordinates 164
Figure 6.4. The zoomed area of the modes obtained from different initial
states in the laboratory coordinates 165
215