Ferrofluid patterns in Hele-Shaw cells: Exact, stable, stationary shape solutions

We investigate a quasi-two-dimensional system composed of an initially circular ferrofluid droplet surrounded by a nonmagnetic fluid of higher density. These immiscible fluids flow in a rotating Hele-Shaw cell, under the influence of an in-plane radial magnetic field. We focus on the situation in which destabilizing bulk magnetic field effects are balanced by stabilizing centrifugal forces. In this framing, we consider the interplay of capillary and magnetic normal traction effects in determining the fluid-fluid interface morphology. By employing a vortex-sheet formalism, we have been able to find a family of exact stationary $N$-fold polygonal shape solutions for the interface. A weakly nonlinear theory is then used to verify that such exact interfacial solutions are in fact stable.

Figure 1

Representative sketch of a rotating Hele-Shaw cell setup containing a viscous ferrofluid droplet subjected to an applied radial magnetic field $\mathbf{H}$ produced by anti-Helmholtz coils. The outer fluid is nonmagnetic and has higher density. The cell rotates around the $z$ axis with constant angular velocity $\mathrm{\Omega }$. The directions of the electric currents $I$ in both coils are also indicated. In this configuration the magnetic field destabilizes the interface, while surface tension and centrifugal forces stabilize it.

Figure 2

Plot of the commensurability parameter $\delta$ as a function of the integrating constant $a$ for $c={\chi }^2{N}_B=30$. The values of $a$ for which $\delta =0$ (there are 14 of them) determine commensurable stationary shapes.

Figure 3

Commensurable solutions found by using the data presented in Fig. 2. Each shape corresponds to values of $a$ for which $\delta =0$. From left to right we have $a=-0.3288,-3.8877,-8.7297,-13.0185,-16.4536,-19.1599,-21.3043$ for the top row and $a=-23.0462,-24.4995,-25.7647,-26.9147,-28.0621,-28.9931,-29.8200$ for the bottom row.

Figure 4

Typical nonperturbative, stationary shape solutions for a magnetic fluid droplet subjected to rotation and radial magnetic field, considering three values of $c=10,20,30$. For each value of $c$ the different $N$-fold patterns ($N=2,3,4$) are obtained by taking increasingly larger absolute values of the constant of integration $a$ (in a given panel the magnitude of $a$ increases from left to right).

Figure 5

Comparison between exact (solid curves) and WNL (dashed curves) solutions for the steady interface shape for $c=10$ and a number of fingers $N=3,4$.

Figure 6

Time evolution of an initially circular droplet perturbed by a small $n=4$ Fourier mode, for $c=10$ and $0\le t\le 1.35$, in equal time intervals $\mathrm{\Delta }t=0.15$, where darker color curves mean larger values of time. The interface profile for $t>1.35$ is indistinguishable from the one shown at $t=1.35$.

Figure 7

Time evolution of the cosine perturbation amplitudes $a_N\left(t\right)$, $a_{2N}\left(t\right)$, and $a_{3N}\left(t\right)$, where $N=4$, for the evolving interface depicted in Fig. 6. It is clear that all amplitudes eventually tend to stationary values.