\documentclass[twoside]{mhd}
\mhdhead{40}{1}{1}
\title{Labyrinthine pattern formation in disordered system of the
magnetic fluid drops: numerical simulation}
\author{I. Dri\c{k}is\inst{1}\inst{,2}, J.-C. Bacri\inst{2},
A. C\={e}bers\inst{1}}
\institute{ Institute of Physics, University of Latvia,
Salaspils-1, LV-2169, Latvia
\and
Universit\'e Paris 7, Denis Diderot,
Unit\'e de Formation et de Recherche de Physique (case 70.08),
2 place Jussieu, 75251, Paris Cedex 05, France
and\\
Laboratoire des Milieux D\'esordonn\'es et H\'et\'erog\`enes (case 78)
Universit\'e Paris 6, 4~place Jussieu, 75252 Paris Cedex 05, France}
% ------------------------------------------------------------------------
\begin{document}
\maketitle
% ------------------------------ Abstract --------------------------------
\begin{abstract}
Development of the labyrinthine patterns arises in a wide variety of the
systems. Here the labyrinthine pattern formation is studied in the
disordered system of the magnetic fluid drops. By the numerical simulation
it is illustrated that in the present case labyrinthine pattern formation
occurs due to the fingering instability of the separate droplets and mutual
repulsion of the arising stripes. The structure factor of the highly
disordered system of the magnetic fluid stripes is calculated, and it is
shown that the characteristic distance between stripes coincides fairly
well with the period of the equilibrium system of the infinite straight
magnetic fluid stripes.
\end{abstract}
% --------------------------- Introduction --------------------------------
\section*{Introduction}
Formation of the labyrinthine patterns is common for the wide variety of
the physical systems \cite{SeulAndelman:Science:95:267}. Here the
labyrinthine pattern formed by the magnetic fluid in the plane
layer under the action of the normal field is studied
\cite{CebersMaiorov:MaGyd:80:1,Rosensweig:JMMM:83:39,Bacri:Recherche:87:192,%
LangerGoldsteinJackson:PhysRevA:92:46,Elias:PhysFrance:1997:7}.
At present moment several scenarios of the labyrinthine pattern formation
and transformation are known - connected region division by the intruding
magnetic fluid fingers \cite{CebersMaiorov:MaGyd:80:3}
(in the case of the labyrinthine patterns formed by the phase separated
magnetic fluid stripe splitting leading to the formation of the hexagonal
array of the microdroplets is observed \cite{BacriSalin:JPhysLet:82}),
multiconnected
labyrinthine pattern formation by fingering instability of the magnetic
fluid droplets under their mutual repulsion
\cite{BaryahtarKhizenkovDorman:Sverdlovsk:83,%
HongJangHorngHsuYaoYang:JApplPhys:97,JangHorngWuHongChiuoYang:Timisoara:98},
similar mechanism is observed also in amphiphile monolayers
\cite{SeulSammon:PhysRevLett90,ToAkamatsuRonddelez:EurophysLett:21}, inertial
regime of the multiconnected structure formation due to the collision of
the magnetic fluid jets arising at the abrupt increase of the magnetic
field strength \cite{MaiorovCebers:MaGyd:81:4}, due to the transformation
of the regular pattern of the straight stripes to the labyrinthine one by
the chevron instability \cite{Flament:EuroLett:96:34}. The last mechanism
as shown in \cite{Seul:PhysRevLett:92,Seul:PhysRevA:92} can control also
the labyrinthine pattern formation of the stripe domains in the thin films
of the ferromagnetics.
Structures similar to the labyrinthine structures of magnetic fluids
are observed also in the magnetorheological suspensions
\cite{Grasselli:Physique:94}. What is common
between those patterns formed under quite different conditions remains
rather unclear. Here we are studying by the numerical simulation
technique the multiconnected labyrinthine pattern formation by the
disordered set of the magnetic fluid droplets. The structure factor of
the arising system is determined and it is found that the characteristic
distance between the stripes of the highly disordered labyrinthine pattern
coincides quite well with the period of the equilibrium structure of the
infinite system of the straight magnetic fluid stripes. That gives the
opportunity to obtain important physical information about the system
also by the study of the rather irregular labyrinthine patterns.
% ---------------------------- Governing -------------------------------
\section{Governing equations}
The ferrofluid droplets are trapped between two parallel glass plates
separated by distance $h$. The magnetic field is applied
perpendicularly to the plates.
In the absence of an applied field, the equilibrium shape of the droplet is
determined solely by the surface tension $\sigma$ and is clearly a circle.
When the magnetic field is applied, the droplets exhibit fingering instability
leading to the formation of labyrinthine pattern. The repulsive
magnetostatic interactions between droplets there act as a source of initial
perturbations. Beside them droplets exhibit a magnetostatic repulsion forces
from each other that lead to the slow increase of the distance between
droplets and expansion of the pattern. During this expansion process some of
droplets reach the perimeter of Hele-Shaw cell and form magnetic fluid
border around it. The repulsive border-droplet forces now stop the
expansion and pattern can reach equilibrium state.
In our numerical experiments circular Hele-Shaw cell is used which is
surrounded by infinite continous layer of the magnetic fluid with the
magnetization equal to the magnetization of the droplets. That layer is
playing the role of the border.
The equations of motion of the magnetic fluid in Hele-Shaw cell
neglecting wetting phenomena is derived in \cite{Cebers:MaGyd:81:2}:
\begin{equation}
\alpha\bvec{v}+\bigtriangledown \left( p+p_m \right) = 0, \qquad
\alpha = \frac{12\eta}{h^2}, \qquad
\Div{\bvec{v}}=0,
\label{Labyrinth:GovEq:1}
\end{equation}
\noindent where $p_m$~-~effective pressure due to the magnetic interaction
forces. The pressure on the free boundaries of droplets $\Sigma_j$ is
determined by Laplace law
\begin{equation}
\left. p\right|_{\Sigma_j} = \sigma k,
\label{Labyrinth:GovEq:2}
\end{equation}
\noindent where $k$ - curvature of the interface. The dynamics of droplets
can be treated as a hydrodynamicaly independent because the viscosity of
surrounding liquid (air, water) is usually negligible. Droplets interact
with each other only magnetically.
The magnetostatic interactions between droplets can be calculated under
assumption of an uniform magnetization of the magnetic fluid domains
$$
\bvec{M}\left(\bvec{H}\right) = M\left(H_0\right)\bvec{e}_z,
$$
\noindent but the magnetic field in the magnetic fluid is
$\bvec{H}=H_0\bvec{e}_z+\bvec{H}_d$. Two different approaches can be used for
calculation of demagnetization field $\bvec{H}_d$. In ``charge'' formulation
$\bvec{H}_d$ is created by fictious magnetic charges on the upper and lower
boundaries of the plane layer formed by droplets. From the other point of
view - ``current'' formulation - $\bvec{H}_d$ is created by azimuthal currents
along the free boundaries of droplets. Both approaches are equivalent
\cite{JacksonGoldsteinCebers:PhysRevE:94:50}. The magnetic energy of the
droplet in the ``current'' formulation is expressed as follows:
$$
E_m=2\pi M^2V-\oint d\bvec{l}\oint
f\left(\frac{\left|\bvec{\rho}-\bvec{\rho}'\right|}{h}\right)\, d \bvec{l}',
$$
\noindent where $f(\xi)=\ln\left(1+\sqrt{1+\xi^2}\right)-\ln\xi+
\xi-\sqrt{1+\xi^2}$. The magnetic pressure is calculated from the
variation of the energy functional
$$
\delta E_m = -h \oint p_m \delta\zeta_n\,dl,
$$
\noindent where $\delta\zeta_n$ is the normal component of the Lagrange
displacement, what neglecting not important in the present case constant
gives \cite{Dicstein:Science:93:261}
\begin{equation}
p_m\left(\bvec{\rho}\right) = \frac{2M^2}{h}\sum\limits_j\oint\limits_{\Sigma_j}
\frac{\bvec{n}'\left(\bvec{\rho}-\bvec{\rho}'\right)}{\left|\bvec{\rho}-\bvec{\rho}'\right|}
\left(\sqrt{1+\frac{h^2}{\left|\bvec{\rho}-\bvec{\rho}'\right|^2}}-1\right)\, dl',
\label{Labyrinth:GovEq:3}
\end{equation}
\noindent where $\bvec{n}'$ is the outward normal to the droplet $j$
interface $\Sigma_j$.
On the basis of the energy calculation for a periodic magnetic liquid
stripe pattern at its equilibrium state the period of the structure $l$ and
width of the stripe $2d$ can be calculated \cite{Cebers:JMMM:95:149}.
For fixed magnetic fluid volume fraction $\Phi=2d/l$ in the pattern the
equilibrium period can be found from the equation
\begin{equation}
\frac{\pi^2 h^2}{l^2} = \frac{Bm}{2} \int\limits_0^{\pi h/l}
y \ln\left(1+\frac{\sin^2\pi\Phi}{\sinh^2 y}\right)\, dy,
\label{Labyrinth:GovEq:4}
\end{equation}
\noindent where magnetic Bond number $Bm=2M^2h/\sigma$ measures the
ratio of the magnetic and capillary forces. Numerical values of the ratio
of the structure period $l$ to the thickness of the layer $h$ are presented
in the Fig.\ref{Labyrinth:Fig:StripeTrheory}.
For irregular labyrinthine patterns it is possible to measure some average
width of magnetic fluid stripes and averaged distance between them.
Characteristic period can be obtained by application of 2D Fourier
transform. Theoretical results of course are absent due to the complexity
of pattern. We can make attempt to find out how close they are to the values
determined by the relation for the structure period of the infinite system
of the straight stripes given by Eq.~\ref{Labyrinth:GovEq:4}.
% ------------------------------ Numerical ----------------------------------
\section{Numerical methods}
Equations Eq.~\ref{Labyrinth:GovEq:1}-\ref{Labyrinth:GovEq:3} can be rewritten
in dimensionless form introducing following scalings: for distance
$L=1/5R$ where $R$ is the radius of the circular Hele-Shaw cell, for capillary
pressure $p_0=\sigma/L$, and time
$\tau=\alpha L^3/\sigma$. As result dimensionless equations are as follows
($\tilde{h}=h/L$, $\tilde{k}=Lk$, tildes further are omitted)
$$
\bvec{v}=-\bigtriangledown \left( p+p_m \right), \qquad
\Div{\bvec{v}}=0, \qquad \left. p\right|_{\Sigma_j} = k
$$
\noindent and
\begin{equation}
p_m\left(\bvec{\rho}\right) = \frac{Bm}{h^2}\sum\limits_j\oint\limits_{\Sigma_j}
\frac{\bvec{n}'\left(\bvec{\rho}-\bvec{\rho}'\right)}{\left|\bvec{\rho}-\bvec{\rho}'\right|}
\left(\sqrt{1+\frac{h^2}{\left|\bvec{\rho}-\bvec{\rho}'\right|^2}}-1\right)\, dl'
\label{Labyrinth:GovEq:6b}
\end{equation}
The free interfaces of droplets $\Sigma_j$ are discretized into a set of
marker points and the evolution of droplet shapes is associated with
the moving of these points. As a magnetic fluid droplets are treated as
hydrodynamically independent, the evolution of every droplet is calculated
by boundary integral equation technique \cite{CebersZemitis:MaGyd:83:19} with
adaptive choice of number of marker points \cite{CebersDrikis:MaGyd:96:32}.
It is convenient to split the magnetostatic pressure term into three parts
by appropriate choice of the integration contours in \ref{Labyrinth:GovEq:6b}
$$
\left. p_m\right|_{\Sigma_k} = p_m^k + p_m^o + p_m^b,
$$
\noindent where $p_m^k$ is magnetostatic pressure created by droplet
itself ($j=k$), $ p_m^o$ - magnetostatic pressure created by other
droplets ($j\ne k$), $p_m^b$ - magnetostatic pressure created by the fluid
layer that surrounds the Hele-Shaw cell. In our numerical calculations for
saving of the computer time $ p_m^o$ and $p_m^b$ are calculated only at
each 16-th time step or in the case when the number of markers in the
contour $k$ changes.
Further the numerical algorithm used for the image analysis is described. If
the characteristic function of the droplet $\chi(\bvec{\rho}-\bvec{\rho}_i)$ is
introduced ( $\chi(\bvec{\rho}-\bvec{\rho}_i)=1$ if $\bvec{\rho}-\bvec{\rho}_i$
belongs to the droplet and $\chi(\bvec{\rho}-\bvec{\rho}_i)=0$ otherwise) then
density of the system $\rho(\bvec{\rho})$ and its structure factor can be
expressed as follows
$$
\rho(\bvec{\rho})=\rho_0\sum\limits_{i=1}^N \chi(\bvec{\rho}-\bvec{\rho}_i)
$$
\noindent and
$$
\left\langle \left\vert\rho_{\bvec{k}}\right\vert^2 \right\rangle=
\left(\rho_0\varphi\right)^2 \left\vert\chi_{\bvec{k}}\right\vert^2
\left\vert f_{\bvec{k}}\right\vert^2,
$$
\noindent where $\chi_{\bvec{k}}=\frac{1}{\Delta S}\int\chi(\bvec{\rho})
\exp\left(-i\bvec{k}\bvec{\rho}\right)\,d\bvec{\rho}$; $f_{\bvec{k}}=\frac{1}{N}
\sum_{j=1}^{N}\exp\left(-i\bvec{k}\bvec{\rho}_j\right)$ and
$\rho_0$~-~density of the droplet, $\varphi=N\Delta S/S$~-~the volume
fraction of the droplets, $S$~-~the area of the sample. Thus it follows
that the maximum of the structure factor can arise from two reasons -
from the one side due to the structure factor of the single droplet, from
the other side due to the arising correlations between the droplets at the
development of the labyrinthine pattern. The last reason is prevailing
if the pattern is well developed.
To obtain the structure factor of the labyrinthine pattern, a square
domain with the center at the center of the Hele -Shaw cell is chosen.
The domain is digitized to 2D pixel array $a_{j,k}$ with dimensions
$N\times N$. $a_{j,k}=1$ if the pixel contains magnetic fluid and $a_{j,k}=0$
otherwise. The mean volume fraction of the magnetic fluid in square
domain can be calculated as follows
$$
\Phi_{box}=\frac{1}{N^2}\sum\limits_{j,k=-\frac{N}{2}}^{\frac{N}{2}-1}a_{j,k}
$$
To make 2D function defined by the pixel array periodic and with zero
mean simplest window function
$\cos\frac{\pi x}{L_{box}}\cos\frac{\pi y}{L_{box}}$ ($L_{box}$~-~dimension
of the square domain) is used and pixel array is transformed as follows
$$
\tilde{a}_{j,k} = \left( a_{j,k}-\bar{a} \right)
\cos\frac{\pi j}{N}\cos\frac{\pi k}{N},
$$
\noindent where
$$
\bar{a} = \tan^2\frac{\pi}{2N}\,\sum\limits_{j,k=-\frac{N}{2}}^{\frac{N}{2}-1}
a_{j,k}\cos\frac{\pi j}{N}\cos\frac{\pi k}{N}
$$
Discrete Fourier transform to the array $\tilde{a}_{j,k}$ is applied
$$
c_{j,k}=\sum\limits_{p,q=-\frac{N}{2}}^{\frac{N}{2}-1}
\tilde{a}_{p,q}\exp\left(-\frac{2\pi i j p}{N}\right)
\exp\left(-\frac{2\pi i k q}{N}\right)
$$
\noindent and the structure factor $\left\vert c_{j,k}\right\vert^2$
is determined. To obtain the structure factor of the real patterns obtained
by the numerical simulation the averaging with respect to two
parameters~-~parameter $s$ determining the size of the square domain and
parameter $\beta$ determining its orientation angle is carried out. Value of
the parameter $s$ equal to 1 corresponds to the case when a square domain
just reach the boundary of the Hele-Shaw cell. Numerical tests have indicated
that the calculated value of the labyrinthine pattern period $l$
slightly depends on the size parameter $s$ in rather complicated way.
Besides that for the patterns with small values of $\Phi$ and patterns
in their initial stage of development $l$ slightly depends on
the parameter $\beta$. Due to this $l$ is determined after the
averaging of the results with respect to 6 different runs with
$s=0.8, 0.9, 1$ and $\beta=0^0, 45^0$. Illustrating example is given
in Fig.~\ref{Labirinti:Fig:Metode}.
Each column in this figure corresponds to the different pair of the
parameters. The density plot of the absolute values of Fourier amplitudes
$\left\vert c_{j,k}\right\vert$ produces ring (row 3 in the
Fig.~\ref{Labirinti:Fig:Metode}) which indicates about the existence
of some structure period and the rotational isotropy of the labyrinthine
patterns. The radial distribution of the Fourier amplitudes
$\left\vert c_{j,k}\right\vert^2$ is determined according to the
formula:
$$
|c|^2(l) = |c|_F^2\left(\frac{\sqrt{2}sR}{l}\right), \qquad
|c|_F^2(m) = \max_{\sqrt{j^2+k^2}\in [m-0.5,m+0.5[}
|c_{j,k}|^2
$$
The maximum of the averaged radial distribution gives the characteristic
distance of the pattern which as we will show further at enough strong
correlation between droplets coincides with the distance between the
stripes in the labyrinthine pattern.
% ------------------------------ Discussions ---------------------------------
\section{Numerical results of the labyrinthine pattern formation}
Using boundary integral equation method the numerical simulations
of the development of the
labyrinthine patterns from the discrete array of the droplets is carried
out. As initial condition the array of the randomly distributed circular
magnetic fluid droplets with random sizes is chosen. When magnetic field
is applied the droplets with enough big volume (depending on $h$ and $Bm$)
start to deform. This distortion is governed by the magnetostatic
dipole~-~dipole interactions. It should be emphasized that there are not other
sources of the distortion in comparison with the former numerical
experiments with separate droplets
\cite{LangerGoldsteinJackson:PhysRevA:92:46,%
JacksonGoldsteinCebers:PhysRevE:94:50,CebersZemitis:MaGyd:83:19,%
CebersDrikis:MaGyd:96:32}
where initial perturbations since the influence
of the numerical noise is negligibly small is always set artificially.
As an illustrative example the development of the labyrinthine patterns
for $\Phi_{cell}=0.4$, $h=0.1$ and $Bm=1.9$ is presented in
Fig.~\ref{Labirinti:Fig:DevExample}. Here are several issues which should
be remarked concerning the observed labyrinthine pattern formation.
At first let us draw attention to the fact that the labyrinthine patterns
obtained are slightly nonuniform - magnetic fluid volume fraction is
slightly higher at the center of the Hele-Shaw cell and less near the
border as it is illustrated in Fig.~\ref{Labyrinth:Fig:Sadal} where
coarse-grained magnetic fluid volume fraction distributions defined as
\begin{equation}
\phi(m) = \frac{\sum\limits_{\sqrt{j^2+k^2} \in [m-0.5,m+0.5[}a_{j,k}}
{\sum\limits_{\sqrt{j^2+k^2}\in [m-0.5,m+0.5[}1}
\label{Labyrinth:EQ:11}
\end{equation}
\noindent and
\begin{equation}
\Phi(m) = \frac{\sum\limits_{\sqrt{j^2+k^2} < m}a_{j,k}}
{\sum\limits_{\sqrt{j^2+k^2} < m}1}
\label{Labyrinth:EQ:12}
\end{equation}
\noindent are shown. Slight increase of the volume fraction in the center is
connected with the repulsion from the border which in the present case
is taken as infinite magnetic fluid layer surrounding the circular Hele-
Shaw cell. It can be pointed out that some other model of the border can
slightly change this conclusion. As other possible border models we can
indicate the model as infinite layer with magnetization equal to the mean
magnetization of the circular Hele-Shaw cell $\Phi M$ or the model of the
border as stripe with the width equal to the equilibrium width of the
stripe pattern.
The period of the developed stripe pattern can be also estimated from
the graph of the coarse-grained density $\Phi(m)$ shown in
Fig.~\ref{Labyrinth:Fig:Sadal}. $8$ maximums seen in
Fig.~\ref{Labyrinth:Fig:Sadal} for coarse-grained density correspond to the
structure period $l/h=2.1$. It agrees fairly well with the theoretical
value calculated for given volume fraction of the magnetic fluid
$\Phi_{cell}$ in the frame of the model of the infinite straight stripes
$l/h=2.3$ and also with the value obtained with the corresponding
error bar from the numerically calculated structure factor
$l/h=1.8\ldots 2.7$.
During the development of the labyrinthine pattern several interesting
processes are observed which could be mentioned. The tendency of
the nonmagnetic phase to the formation of the 3-fold equilibrium vertex
seen in the upper row in Fig.~\ref{Labirinti:Fig:DetExample} looks like
as ``attraction'' of the repulsing magnetic fluid fingers. During the
droplet fingering phase of the formation of the labyrinthine pattern as
we can see from Fig.~\ref{Labirinti:Fig:DetExample}(b) and (c) the
splitting of the metastable 4-fold and higher order vertices
occurs. It should be mentioned that the well developed Steiner tree phase
as for single droplet \cite{CebersDrikis:MaGyd:96:32} there is not observed
due to the limited space in
the labyrinth. It is curious to mention that for the stripes of the
nonmagnetic phase between the magnetic fluid droplets the development
of the overextension instability can be well registered. At the later stages
the development of the overextension instability is nonlineary suppressed
due to the 4-fold vertex splitting event as can be seen from 4 and 5 row
in Fig.~\ref{Labirinti:Fig:DetExample}. During the labyrinthine pattern
formation the development of the hairpins of the magnetic fluid stripes is
remarked as well (lower row in Fig.~\ref{Labirinti:Fig:DetExample}). Their
formation can be understood as the defect transformation of the stripe
pattern \cite{CebersDrikis:FBP}.
The most important issue which can be studied by the present
numerical simulation of the labyrinthine patterns concerns the period of
the structure as obtained from the calculated structure factor.
Labyrinthine structures are generated from random set of circular
droplets with random sizes at abrupt application of magnetic field.
In that case opposite to the case when magnetic field is increased
slowly multibranched configurations of the droplets are developed.
For structures obtained in such way
the dependence of the distance determined by the wavenumber at which the
structure factor (Fig.~\ref{Labirinti:Fig:PeriodDin}(c)) has maximum
on the time of the development of the labyrinthine pattern
(Fig.~\ref{Labirinti:Fig:PeriodDin}(a)) is shown in
Fig.~\ref{Labirinti:Fig:PeriodDin}(b). Initial values of $l/h$
evidently correspond to the structure factor of the separate droplet
but at later times to the distance between stripes.
The period of lattice that corresponds
to the maximum of $\left|f_k\right|^2$ is determined according to the same
algorithm, when the lattice represents small circles that are placed at the
mass centers of the droplets.
The later as
can be seen from Fig.~\ref{Labirinti:Fig:PeriodDin}(b) corresponds quite
well to the theoretically calculated at the given parameters
$Bm$, $h$, $\Phi_{cell}$ for the model of the infinite system of the straight
stripes. The comparison of the labyrinthine structure period obtained by
the numerical simulation with the theoretically calculated for the
different magnetic Bond numbers is given in
Fig.~\ref{Labirinti:Fig:CompPeriod}. An example of the
equilibrium labyrinthine pattern sequence in dependence on magnetic
Bond number used to obtain the data shown in
Fig.~\ref{Labirinti:Fig:CompPeriod}(b) is given by
Fig.~\ref{Labirinti:Fig:EqPattC}. It should be remarked that the time
dependence and the topological appearance of the labyrinthine patterns
depends on the rate of the magnetic field increase as it is illustrated
by Fig.~\ref{Labirinti:Fig:EqPatt2} - in the case when field is increased
slowly during the evolution of the labyrinthine pattern the structure obtained
is less branched and better ordered. It is quite natural since during the slow
evolution of the labyrinthine pattern it is in fact only first instability
mode of the droplet leading to the elongated stripe shape which develops.
Concerning the data in
Fig.~\ref{Labirinti:Fig:CompPeriod} it should be mentioned also that the
value of the period obtained from the maximum of the structure factor
coincides quite well with the value obtained in so called ``box''
approximation. According to that approximation the width and the length
of the box having the same perimeter and area ($L_j,S_j$) as a droplet are
calculated according to the equations
$$
x_j^W \cdot x_j^L = S_j, \qquad x_j^W + x_j^L = \frac{L_j}{2}
$$
For long droplets ($x_j^W \ll x_j^L$) that gives the simplified expression
$$
2d=\frac{2S_j}{L_j}
$$
By average stripe width in the ``box'' approximation we understand the
mean for pattern value of the box width
$$
\overline{2d}_{box}=\frac{1}{n}\sum\limits_{j=1}^{n}x_j^W
$$
The value of the period according to this approximation if the volume
fraction of the pattern is known can be obtained as
\begin{equation}
l = \frac{\overline{2d}_{box}}{\Phi}
\label{Labyrinth:GovEq:7}
\end{equation}
From Fig.~\ref{Labirinti:Fig:CompPeriod} it can be seen that the value
obtained in the ``box'' approximation coincides quite well with the value
given by the maximum of the structure period. That confirms the fact that
the maximum of the structure factor at the late stages of the labyrinthine
pattern formation is determined by the distance between
stripes.
% ------------------------------ Conclusions ---------------------------------
\section{Conclusions}
\begin{itemize}
\item The distribution of the magnetic fluid volume density according to
the model of repulsive Hele-Shaw cell border as infinite magnetic fluid layer
is slightly nonuniform with the higher magnetic fluid
volume fraction in the centre of the Hele-Shaw cell;
\item The development of the random array of the droplets into the
labyrinth is similar to the development of single droplet. Both
exhibit the same stages of development including vertex splitting events;
\item The channels between the droplets similar to the magnetic fluid stripes
exhibit nonlinearly suppressed overextension instability what leads to the
formation of 4-fold vertex subsequently splitting to two 3-fold vertices;
\item The value of the period of the irregular labyrinthine pattern at its
equilibrium state corresponds very well at the given magnetic Bond number
and volume fraction of magnetic liquid to the period value calculated for
the infinite system of straight stripes.
\end{itemize}
% ------------------------------------ Acck --------------------------------
\section*{Acknowledgements}
This work is partly supported by Ministere de l'education nationale,
de la recherche et de la technologie (reseau formation recherche pays
Europe Centrale et Orientale (Nr 95 P 0057)).
We would like to express our gratitude to thank Dr. L.~Buligins for
providing computer time at the Center of Computational Technologies
of the Department of Physics, University of Latvia.
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\clearpage
% ------------------------------ Figures ------------------------------------
\begin{figure}
\caption[SmecticTheory]{The equilibrium period of magnetic
fluid stripe patterns for different magnetic Bond numbers $Bm$
and magnetic fluid volume fractions $\Phi$ calculated according to the
\protect\ref{Labyrinth:GovEq:4}. }
\label{Labyrinth:Fig:StripeTrheory}
\end{figure}
% ------------------------------------
\begin{figure}
\caption{The algorithm of the determination of the characteristic period of
labyrinthine structure given in the upper row. The parameters are
$\Phi_{cell}=0.4$, $h=0.25$ and $Bm=5.5$. Selected square domains
($\Phi_{box}=0.47$) are in the second row
and corresponding Fourier amplitudes in the third. In the figure at bottom
radial distributions of the Fourier amplitudes and averaged curve are
displayed. The columns correspond to the pair of parameters
($s=1,\beta=45^0$), ($s=0.8,\beta=0^0$) and ($s=0.8,\beta=45^0$)
that determine the size of the square domain and its orientation
angle.}
\label{Labirinti:Fig:Metode}
\end{figure}
% ---------------------------------------
\begin{figure}
\caption{Development of labyrinthine patterns for
$\Phi_{cell}=0.4$, $h=0.1$ and $Bm=1.9$.}
\label{Labirinti:Fig:DevExample}
\end{figure}
% ----------------------------------------
\begin{figure}
\caption{ Magnetic fluid volume density radial distribution $\Phi(r)$
and the density of magnetic fluid volume density $\phi(r)$ for equilibrium
pattern obtained for $\Phi_{cell}=0.3$, $Bm= 6$, $h=0.25$. The relation
between abscissa and arguments of Eq.~\protect\ref{Labyrinth:EQ:11} and
\protect\ref{Labyrinth:EQ:12} is $r=2mR/N$. }
\label{Labyrinth:Fig:Sadal}
\end{figure}
% ----------------------------------------
\begin{figure}
\caption{Development of labyrinthine patterns. The attraction of magnetic
fluid fingers in (a) for $\Phi_{cell}=0.3$, $h=0.25$, $Bm=4.5$. Splitting
of 4-fold vertices in (b,c) for $\Phi_{cell}=0.4$, $h=0.1$, $Bm=1.9$ and
$\Phi_{cell}=0.5$, $h=0.25$, $Bm=6.5$. The development of the overextension
instability of the channels between droplets in (d,e) for
$\Phi_{cell}=0.4$, $h=0.25$, $Bm=4$ and $\Phi_{cell}=0.4$, $h=0.1$, $Bm=2$.
The defect transformation in the stripe pattern (f). $Bm$ varies step-like with
increment $0.5$ from $4.5$ in the first figure till $8$ in the last. }
\label{Labirinti:Fig:DetExample}
\end{figure}
% ---------------------------------------
\begin{figure}
\caption{Numerical simulations of the dynamic of magnetic fluid labyrinthine
pattern for $Bm=4.5$, $h=0.25$, $\Phi_{cell}=0.30$: transient shapes of
labyrinthine pattern (a), characteristic period of pattern, estimated
value of the period (\protect\ref{Labyrinth:GovEq:7}) and the
period of lattice (maximum of $\left|f_k\right|^2$) (b), and radial
distribution of Fourier amplitudes (c). }
\label{Labirinti:Fig:PeriodDin}
\end{figure}
% -------------------------------
\begin{figure}
\caption{The calculated values of labyrinthine pattern equilibrium period
versus the theoretical values for magnetic liquid stripe pattern. There
parameters are $\Phi_{cell}=0.2$ and $h=0.25$ (a),
$\Phi_{cell}=0.3$ and $h=0.25$ (b), $\Phi_{cell}=0.4$ and $h=0.25$ (c),
$\Phi_{cell}=0.5$ and $h=0.25$ (d), $\Phi_{cell}=0.4$ and $h=0.1$ (e),
$\Phi_{cell}=0.4$ and $h=0.5$ (f). }
\label{Labirinti:Fig:CompPeriod}
\end{figure}
% --------------------------------------------
\begin{figure}
\caption{Equilibrium labyrinthine patterns calculated numerically
for $\Phi_{cell}=0.3$ and $h=0.25$ for different magnetic Bond
numbers. The length of the line in the right bottom
corner of each figure corresponds to the calculated value of the
structure period. }
\label{Labirinti:Fig:EqPattC}
\end{figure}
% -------------------------------------------
\begin{figure}
\caption{Labyrinthine patterns calculated numerically
for $\Phi_{cell}=0.3$ and $h=0.25$. Magnetic field increases step-like
during the calculation. The length of the line in the right bottom corner
of each figure corresponds to the calculated value of the structure period. }
\label{Labirinti:Fig:EqPatt2}
\end{figure}
% ------------------------------------------------------------------------
\lastpageno
\end{document}