\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)). 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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}