Tuning a magnetic field to generate spinning ferrofluid droplets with controllable speed via nonlinear periodic interfacial waves

Two-dimensional free surface flows in Hele-Shaw configurations are a fertile ground for exploring nonlinear physics. Since Saffman and Taylor's work on linear instability of fluid-fluid interfaces, significant effort has been expended to determining the physics and forcing that set the linear growth rate. However, linear stability does not always imply nonlinear stability. We demonstrate how the combination of a radial and an azimuthal external magnetic field can manipulate the interfacial shape of a linearly unstable ferrofluid droplet in a Hele-Shaw configuration. We show that weakly nonlinear theory can be used to tune the initial unstable growth. Then, nonlinearity arrests the instability and leads to a permanent deformed droplet shape. Specifically, we show that the deformed droplet can be set into motion with a predictable rotation speed, demonstrating nonlinear traveling waves on the fluid-fluid interface. The most linearly unstable wave number and the combined strength of the applied external magnetic fields determine the traveling wave shape, which can be asymmetric.

Figure 1

Schematic illustration of a Hele-Shaw cell confining a ferrofluid droplet, initially circular with radius $R$. The azimuthal magnetic field ${\mathrm{H}}_a$ is produced by a long wire conveying an electric current $I$. The radial magnetic field ${\mathrm{H}}_r$ is produced by a pair of anti-Helmholtz coils with equal currents $I_{AH}$ in opposite directions. The combined external magnetic field $\mathrm{H}$ deforms the droplet, and its interface is given by $h(\theta ,t)$. In comparison, the fluid exterior to the droplet (e.g., air) is assumed to have negligible viscosity and velocity.

Figure 2

(a) The evolution of the first five harmonic modes from fully nonlinear simulation (solid) and weakly nonlinear approximation (dashed), for ${\mathrm{N}}_{\mathrm{Ba}}=1.0, {\mathrm{N}}_{\mathrm{Br}}=37, R=1$, and $\chi =1$ [same parameters for (b), (c) and (d)]. (b) The fully nonlinear evolution of the interface from a small perturbation of the flat base state into a permanent traveling wave (rotating droplet). (c) Comparison between the final shape from fully nonlinear simulation (solid) and weakly nonlinear approximation (dashed). (d) Stability diagram based on the first two harmonic modes of the final shape (marked with $▴$) shown in (b); $\circ$ (resp. $\times$) denotes the stable (resp. unstable) initial conditions, solid (resp. dashed) curves track the stable (resp. unstable) evolution trajectories. The unstable region is shaded, and the “$f$” superscript represents the final harmonic mode amplitude.

Figure 3

(a) Comparison of the propagation velocity predicted by linear theory $v_f^L$ (dashed), weakly nonlinear theory $v_f^W$ (empty symbols), and fully nonlinear simulation $v_f^F$ (filled symbols). The circles represent results for case (i) $R=1$ fixed and ${\mathrm{N}}_{\mathrm{Ba}}\in [0,{10}^{-3},{10}^{-2},{10}^{-1},1,3,5]$, the triangles represent results for case (ii) ${\mathrm{N}}_{\mathrm{Ba}}=1$ fixed with ${\mathrm{N}}_{\mathrm{Br}}$ varying according to $R\in [0.8,0.9,1.1,1.2]$, and the squares represent case (iii) $k_m\in [5,6,7,8,9], R=1$ and ${\mathrm{N}}_{\mathrm{Ba}},{\mathrm{N}}_{\mathrm{Br}}$ determined so that $v_f^L=85.16$. (b) The skewness $Sk$ of the fully nonlinear profile. (c) The permanent wave shape (only one wavelength shown).

Figure 4

(a) Time evolution of the wave profile asymmetry for different combinations of ${\mathrm{N}}_{\mathrm{Ba}}\in [0,0.1,1,3,5,6,7,8,9]$ and ${\mathrm{N}}_{\mathrm{Br}}$ varying so that $k_m=7$ for $R=1$. Solid curves represent stable cases yielding a propagating profile; dashed curves represent unstable cases in which the profile distorts and grows without bound. (b) Permanent wave profiles that emerge and propagating in a stable manner. (c) Stable (top, with ${\mathrm{N}}_{\mathrm{Ba}}=1,{\mathrm{N}}_{\mathrm{Br}}=37$) and unstable (bottom, ${\mathrm{N}}_{\mathrm{Ba}}=8,{\mathrm{N}}_{\mathrm{Br}}=41$) evolution of the profile. The instants of time [at which the shapes in (c) are shown] are marked with white dots in (a), superimposed on the asymmetry profiles.

Figure 5

The unstable evolution of the first five harmonic modes ($k=4,8,12,16,20$) from fully nonlinear simulation (solid) and their stable evolution from weakly nonlinear approximation (dashed) for (a) nonrotating ($k_m=4, {\mathrm{N}}_{\mathrm{Ba}}=0$) and (b) rotating ($k_m=4, {\mathrm{N}}_{\mathrm{Ba}}=1$) shapes.

Figure 6

Flow chart of the vortex-sheet algorithm using the Crank-Nicolson method for time advancement and fixed-point iteration for resolving the implicit nonlinear terms.

Figure 7

Grid convergence study for fundamental modes $k_m=7$ (black) and $k_m=9$ (blue) with ${\mathrm{N}}_{\mathrm{Ba}}=1$ and $N=256$ (dotted), $N=512$ (dot-dashed), $N=1024$ (dashed), and $N=2048$ (solid). (a) Spectral energy of harmonic modes ($k, 2k, 3k, \cdots$). (b) The root-mean-square error taking the $N=2048$ solution $\widehat{z}$ as “exact.” (c) Grid convergence of the evolution of the skewness $Sk\left(t\right)$. (d) Grid convergence of the evolution of the asymmetry $As\left(t\right)$.