Magnetically induced interfacial instabilities in a ferrofluid annulus

We investigate the flow of a viscous ferrofluid annulus surrounded by two nonmagnetic fluids in a Hele-Shaw cell when subjected to an external radial magnetic field. The interfacial pattern formation dynamics of the system is determined by the interplay of magnetic and surface tension forces acting on the inner and outer boundaries of the annulus, favoring the coupling of the disjoint interfaces. Mode-coupling analysis is employed to examine both linear and weakly nonlinear stages of the flow. Linear stability analysis indicates that the trailing and leading annular boundaries are coupled already at the linear regime, revealing that perturbations arising in the outer interface may induce the emergence of deformed structures in the inner boundary. Moreover, second-order weakly nonlinear analysis is utilized to identify key nonlinear morphological features of the ferrofluid annulus. Our theoretical results show that linear, $n$-fold symmetric annular patterns having rounded edges are replaced by nonlinear polygonal-like shapes, presenting fairly sharp fingers. It is found that, as opposed to the linear patterns, the nonlinear peaky structures reach a stationary state, characterized by a growth saturation process induced by nonlinear effects. Furthermore, the response of the ferrofluid ring to changes in the thickness of the annulus, in the relative strength of magnetic and surface tensions forces, as well as in the magnetic susceptibility of the ferrofluid material, are also discussed.

Figure 1

Schematic of the three-fluid, magnetic field-induced flow in a Hele-Shaw cell. The cell contains an initially circular ferrofluid ringlike structure of radii $R_1$ and $R_2$ (dashed circles), and viscosity ${\eta }_2$, surrounded by inner and outer nonmagnetic fluids with viscosities ${\eta }_1$ and ${\eta }_3$. The system is subjected to an in-plane external magnetic field $\mathbf{H}$ pointing radially outward, which deforms the inner and outer boundaries of the ferrofluid annulus (solid curves). The interfacial perturbation amplitudes in the deformed ferrofluid structure are denoted by $\zeta =\zeta (\theta ,t)$ and $\varepsilon =\varepsilon (\theta ,t)$, where $\theta$ is the azimuthal angle.

Figure 2

Linear growth rates ${\lambda }_i(n,t)$ with $i=1,2$, as a function of the azimuthal wave number $n$ for $R=0.3$ (top panels) and $R=0.9$ (bottom panels), and three values of time: $t=0$ [(a) and (d)], $t={10}^{-3}$ [(b) and (e)], and $t=2.2\times {10}^{-3}$ [(c) and (f)]. In addition, we set $N_B=136$, $\chi =0.8$, $A_{12}=\mathcal{A}=1$, $A_{23}=-1$, and $\sigma =1$.

Figure 3

Snapshots illustrating the typical purely linear evolution of the ferrofluid annulus patterns for $R=0.75$ (top panels), $R=0.92$ (middle panels), and $R=0.95$ (bottom panels). Each row corresponds to a different time evolution, and the values of time $t$ utilized are the following: (a) $t=7.0\times {10}^{-3}$, (b) $t=8.0\times {10}^{-3}$, (c) $t=8.6\times {10}^{-3}$, (d) $t=9.5\times {10}^{-3}$, (e) $t=1.1\times {10}^{-2}$, (f) $t=1.15\times {10}^{-2}$, (g) $t=1.2\times {10}^{-2}$, (h) $t=1.4\times {10}^{-2}$, and (i) $t=1.52\times {10}^{-2}$. Moreover, these patterns are generated by considering $N=80$ $(n,2n,3n,\cdots ,80n)$ participating cosine modes, where $n=8$ is the fundamental mode. All the other physical parameters are identical to those utilized in Fig. 2.

Figure 4

Snapshots illustrating the typical weakly nonlinear evolution of the ferrofluid annulus patterns for $R=0.75$ (top panels), $R=0.92$ (middle panels), and $R=0.95$ (bottom panels). Each row corresponds to a separate time evolution, and the snapshots are taken for times (a) $t=7.0\times {10}^{-3}$, (b) $t=8.0\times {10}^{-3}$, (c) $t=1.7\times {10}^{-2}$, (d) $t=9.5\times {10}^{-3}$, (e) $t=1.1\times {10}^{-2}$, (f) $t=3.4\times {10}^{-2}$, (g) $t=1.2\times {10}^{-2}$, (h) $t=1.4\times {10}^{-2}$, and (i) $t=3.4\times {10}^{-2}$. These weakly nonlinear patterns, which have been generated by employing our second-order mode-coupling scheme, should be contrasted with the equivalent linear structures depicted in Fig. 3. All physical parameters and initial conditions used here are identical to those utilized in Fig. 3.

Figure 5

Plots of the cosine amplitudes of the outer $\left[{\overline{a}}_n\left(t\right)\right]$ and inner $\left[a_n\left(t\right)\right]$ interfaces as a function of time $t$, corresponding to each time evolution presented in Fig. 4. Amplitudes for modes $n$, $2n$, $3n,\cdots$, $10n$ are shown. Column (a) is related to the top row of Fig. 4. Likewise, columns (b) and (c) correspond to the middle and bottom rows of Fig. 4, respectively.

Figure 6

Representative weakly nonlinear ferrofluid annulus patterns for (a) $N_B=62$ and $\chi =1.3$, (b) $N_B=78$ and $\chi =1.3$, (c) $N_B=62$ and $\chi =1.7$, and (d) $N_B=78$ and $\chi =1.7$. Here the patterns are plotted for $R=0.96$, $A_{12}=\mathcal{A}=1$, $A_{23}=-1$, $R_2=1$, $R_1=R$, $\sigma =1$, and $t=0.1$. The number of resulting pointy fingers formed in each pattern is (a) 7, (b) 8, (c) 9, and (d) 10.

Figure 7

Representative weakly nonlinear ferrofluid annulus patterns obtained by considering the nonlinear coupling of all sine and cosine Fourier modes in the interval $2\le n\le 80$ and two different sets of random initial phases. Each row of this figure corresponds to a particular set of random initial phases, and the patterns are plotted for $R=0.75$ and $t=7.75\times {10}^{-3}$ [(a) and (c)] and $R=0.92$ and $t=9.1\times {10}^{-3}$ [(b) and (d)].