Fingering patterns in magnetic fluids: Perturbative solutions and the stability of exact stationary shapes

We examine the formation of interfacial patterns when a magnetic liquid droplet (ferrofluid, or a magnetorheological fluid), surrounded by a nonmagnetic fluid, is subjected to a radial magnetic field in a Hele-Shaw cell. By using a vortex-sheet formalism, we find exact stationary solutions for the fluid-fluid interface in the form of $n$-fold polygonal shapes. A weakly nonlinear, mode-coupling method is then utilized to find time-evolving perturbative solutions for the interfacial patterns. The stability of such nonzero surface tension exact solutions is checked and discussed, by trying to systematically approach the exact stationary shapes through perturbative solutions containing an increasingly larger number of participating Fourier modes. Our results indicate that the exact stationary solutions of the problem are stable, and that a good matching between exact and perturbative shape solutions is achieved just by using a few Fourier modes. The stability of such solutions is substantiated by a linearization process close to the stationary shape, where a system of mode-coupling equations is diagonalized, determining the eigenvalues which dictate the stability of a fixed point.

Figure 1

Characteristic exact stationary interface shapes that emerge when a magnetic fluid droplet is subjected to a radial magnetic field in a Hele-Shaw cell: (a) Sixfold concave-shaped, polygonal ferrofluid pattern and (b) MR fluid sixfold pattern-forming structure, presenting convex edges.

Figure 2

Representative sketch of a Hele-Shaw cell of gap thickness $b$, located at the mid-distance between two anti-Helmholtz coils. The cell contains an initially circular magnetic fluid droplet of radius $R$, surrounded by a nonmagnetic fluid with trifling viscosity. The in-plane radial magnetic field $\mathbf{H}$ is produced by the coils. The directions of the electric currents in both coils are indicated. In this framework, the magnetic field destabilizes the interface, while surface tension and viscoelastic effects stabilize it.

Figure 3

Representative weakly nonlinear, ferrofluid patterns (top panels) considering the coupling of two Fourier modes $n$, and $2n$. The values of the magnetic Bond number and participating modes are (a) $N_B=50$, $n=7$, and $2n=14$; (b) $N_B=105$, $n=10$, and $2n=20$; and (c) $N_B=175$, $n=13$, and $2n=26$. The corresponding time evolutions of the cosine amplitudes $a_n\left(t\right)$ for modes $n$ and $2n$ are depicted in the bottom panels. In addition, we set $\chi =1$, $S_0=S=0$. The values of the ferrofluid droplet unperturbed radius are (a) $R=0.854418$, (b) $R=0.937215$, and (c) $R=0.964060$. The time used to plot the patterns in panels (a)–(c) is $t=0.015$.

Figure 4

Representative weakly nonlinear, ferrofluid patterns (top panels) considering the coupling of three Fourier modes $n$, $2n$, and $3n$. The values of the magnetic Bond number and participating modes are (a) $N_B=50$, $n=7$, $2n=14$, and $3n=21$; (b) $N_B=105$, $n=10$, $2n=20$, and $3n=30$; and (c) $N_B=175$, $n=13$, $2n=26$, and $3n=39$. The corresponding time evolution of the cosine amplitudes $a_n\left(t\right)$ for modes $n$, $2n$, and $3n$ is depicted in the bottom panels. In addition, we set $\chi =1$, $S_0=S=0$. The values of the droplet initial radius in panels (a)–(c) are equal to those used in Fig. 3. The times at which the patterns are plotted are (a) $t=0.03$, (b) $t=0.0036$, and (c) $t=0.015$.

Figure 5

Representative weakly nonlinear, ferrofluid patterns (top panels) considering the coupling of four Fourier modes $n$, $2n$, $3n$, and $4n$. The values of the magnetic Bond number and participating modes are (a) $N_B=50$, $n=7$, $2n=14$, $3n=21$, and $4n=28$; (b) $N_B=105$, $n=10$, $2n=20$, $3n=30$, and $4n=40$; and (c) $N_B=175$, $n=13$, $2n=26$, $3n=39$, and $4n=52$. The corresponding time evolutions of the cosine amplitudes $a_n\left(t\right)$ for modes $n$, $2n$, $3n$, and $4n$ are depicted in the bottom panels. In addition, we set $\chi =1$, $S_0=S=0$. The values of the droplet initial radius in panels (a)–(c) are equal to those used in Fig. 3. The times at which the patterns are plotted are (a) $t=0.02$, (b) $t=0.006$, and (c) $t=0.003$.

Figure 6

Perturbative and exact stationary solutions related to the ferrofluid situation studied in Fig. 5. Top panels: comparison between exact (solid curves) and four mode weakly nonlinear (WNL) solutions (dashed curves) for the ferrofluid interface shape when (a) $N_B=50$, (b) $N_B=105$, and (c) $N_B=175$. Bottom panels: corresponding plot of the absolute value of the cosine Fourier amplitudes $|a_n|$ as a function of the azimuthal mode number $n$. The values for the constant of integration in the exact solutions are (a) $a=-62.8$, (b) $a=-172.741226$, and (c) $a=-313.318659$.

Figure 7

Representative weakly nonlinear, magnetorheological fluid patterns (top panels) considering the coupling of two Fourier modes $n$, and $2n$. The values of the magnetic-field-dependent yield-stress parameter $S$ and participating modes are (a) $S=102.40$, $n=3$, and $2n=6$; (b) $S=93.13$, $n=4$, and $2n=8$; and (c) $S=84.13$, $n=5$, and $2n=10$. The corresponding time evolutions of the cosine amplitudes $a_n\left(t\right)$ for modes $n$ and $2n$ are depicted in the bottom panels. In addition, we set $\chi =0.5$, $S_0=57.6$, and $N_B=256$. The values of the magnetorheological fluid droplet unperturbed radius are (a) $R=0.845625$, (b) $R=0.908623$, and (c) $R=0.923848$. The times used to plot the patterns are (a) $t=1.5$, (b) $t=0.15$, and (c) $t=0.08$.

Figure 8

Representative weakly nonlinear, magnetorheological fluid patterns (top panels) considering the coupling of three Fourier modes $n$, $2n$, and $3n$. The values of the magnetic-field-dependent yield-stress parameter $S$ and participating modes are (a) $S=102.40$, $n=3$, $2n=6$, and $3n=9$; (b) $S=93.13$, $n=4$, $2n=8$, and $3n=12$; and (c) $S=84.13$, $n=5$, $2n=10$, and $3n=15$. The corresponding time evolutions of the cosine amplitudes $a_n\left(t\right)$ for modes $n$, $2n$, and $3n$ are depicted in the bottom panels. In addition, we set $\chi =0.5$, $S_0=57.6$, and $N_B=256$. The values of the droplet initial radius in panels (a)–(c) are equal to those used in Fig. 7. The times at which the patterns are plotted are (a) $t=0.064$, (b) $t=0.065$, and (c) $t=0.06$.

Figure 9

Representative weakly nonlinear, magnetorheological fluid patterns (top panels) considering the coupling of four Fourier modes $n$, $2n$, $3n$, and $4n$. The values of the magnetic-field-dependent yield-stress parameter $S$ and participating modes are (a) $S=102.40$, $n=3$, $2n=6$, $3n=9$, and $4n=12$; (b) $S=93.13$, $n=4$, $2n=8$, $3n=12$, and $4n=16$; and (c) $S=84.13$, $n=5$, $2n=10$, $3n=15$, and $4n=20$. The corresponding time evolution of the cosine amplitudes $a_n\left(t\right)$ for modes $n$, $2n$, $3n$, and $4n$ is depicted in the bottom panels. In addition, we set $\chi =0.5$, $S_0=57.6$, and $N_B=256$. The values of the droplet initial radius in panels (a)–(c) are equal to those used in Fig. 7. The times at which the patterns are plotted are (a) $t=0.12$, (b) $t=0.1$, and (c) $t=0.08$.

Figure 10

Perturbative and exact stationary solutions related to the magnetorheological fluid situation studied in Fig. 9. Top panels: comparison between exact (solid curves) and four-mode weakly nonlinear (WNL) solutions (dashed curves) for the magnetorheological interface shape when (a) $S=102.40$, (b) $S=93.13$, and (c) $S=84.13$. Bottom panels: corresponding plot of the absolute value of the cosine Fourier amplitudes $|a_n|$ as a function of the azimuthal mode number $n$. The values for the constant of integration in the exact solutions are (a) $a=-23.910252$, (b) $a=-32.436107$, and (c) $a=-40.324116$.