Ferrofluid annulus in crossed magnetic fields

We study the dynamics and pattern formation of a ferrofluid annulus enveloped by two nonmagnetic fluids in a Hele-Shaw cell, subjected to an in-plane crossed magnetic field configuration involving the combination of radial and azimuthal magnetic fields. A perturbative, second-order mode-coupling analysis is employed to investigate how the ferrofluid annulus responds to variations in the relative strength of the radial and azimuthal magnetic field components, as well as in the thickness of magnetic fluid ring. By tuning the magnetic field components and the annulus' thickness, we have found the development of several stationary annular shapes, presenting polygon-shaped structures typically having skewed, peaked fingers. Such fingered structures may vary their skewness, sharpness, and number and arise on the inner, outer, or even both boundaries of the annulus. In addition to controlling the morphologies of the ferrofluid annuli, the external field can be used to put the annulus into a rotational motion, with an angular velocity having prescribed magnitude, and direction. Our second-order theory is utilized to obtain a correction to the linear stability analysis prediction of such angular velocity, usually resulting in a decreased weakly nonlinear value as compared with the magnitude predicted by purely linear theory. These theoretical results suggest the use of magnetic-field-controlled ferrofluid annuli in Hele-Shaw cells as a potential laboratory for microscale applications related to the manipulation of shape-programmable magnetic fluid objects and tunable fluidic-mixing devices in confined environments.

Figure 1

Schematic of the ferrofluid annulus confined flow problem under the action of crossed magnetic fields (radial and azimuthal). Initially, the ferrofluid annular structure has circular boundaries of radii $R_1$ and $R_2$ (dashed circles). The ferrofluid annulus has viscosity ${\eta }_2$, while the inner and outer fluids are nonmagnetic, having viscosities ${\eta }_1$, and ${\eta }_3$, respectively. The combined magnetic field $\mathbf{H}$ may deform the inner and outer interfaces of the ferrofluid annulus (solid curves). The interfacial perturbation amplitudes in the distorted ferrofluid ring structure are denoted by $\zeta =\zeta (\theta ,t)$ and $\varepsilon =\varepsilon (\theta ,t)$, where $\theta$ is the azimuthal angle.

Figure 2

Typical weakly nonlinear, annular ferrofluid patterns formed for a fixed radial magnetic Bond number $N_{\mathrm{Br}}=56$, and three increasing values of the azimuthal magnetic Bond number: $N_{\mathrm{Ba}}=0$ [(a) and (d)], $N_{\mathrm{Ba}}=4$ [(b) and (e)], and $N_{\mathrm{Ba}}=20$ [(c) and (f)]. In the top (bottom) panels we take $R=0.70$ ($R=0.94$). Moreover, we set $\chi =1.5$, $R_2=1$, and final time $t_f=0.03$. These patterns, and all others presented in this work, are plotted by considering the coupling of $N=40(n,2n,3n,\cdots ,40n)$ participating sine and cosine modes. The fundamental mode $n$ is given by the closest integer to the fastest growing mode $n_{\max }$ evaluated at $t=t_f$. Here, we have that $n=n_{\max }=8$ [(a)–(d)], $n=n_{\max }=7$ (e), and $n=n_{\max }=6$ (f). In this figure, and throughout this study, we take $A_{12}=\mathcal{A}=1$, $A_{23}=-1$, $\mathrm{sgn}\left(I\right)=1$, and $\sigma =1$.

Figure 3

Time evolution of the perturbation amplitudes of the inner $\left[\right|{\zeta }_n\left(t\right)|=\sqrt{a_n^2\left(t\right)+b_n^2\left(t\right)}/2]$ (dashed curves) and outer $\left[\right|{\varepsilon }_n\left(t\right)|=\sqrt{{\overline{a}}_n^2\left(t\right)+{\overline{b}}_n^2\left(t\right)}/2]$ (solid curves) interfaces, corresponding to the situations leading to the formation of the nonlinear annular ferrofluid structures displayed in Fig. 2. Amplitudes for modes $n$, $2n$, $3n$, and $4n$ are shown.

Figure 4

Representative weakly nonlinear, annular ferrofluid patterns formed for a constant azimuthal magnetic Bond number $N_{\mathrm{Ba}}=100$, and three increasing values of the radial magnetic Bond number: $N_{\mathrm{Br}}=0$ [(a) and (d)], $N_{\mathrm{Br}}=14$ [(b) and (e)], and $N_{\mathrm{Br}}=50$ [(c) and (f)]. In the top (bottom) panels we take $R=0.70$ ($R=0.86$). In addition, we set $\chi =0.9$, $R_1=0.7$, and final time $t_f=0.04$. Here we have that $n=n_{\max }=6$ [(a)–(d)], $n=n_{\max }=5$ (e), and $n=n_{\max }=4$ (f). The remaining physical parameters, initial conditions, and the number of participating modes are the same as those used in Fig. 2.

Figure 5

Time evolution of the perturbation amplitudes of the inner $\left[\right|{\zeta }_n\left(t\right)|=\sqrt{a_n^2\left(t\right)+b_n^2\left(t\right)}/2]$ (dashed curves) and outer $\left[\right|{\varepsilon }_n\left(t\right)|=\sqrt{{\overline{a}}_n^2\left(t\right)+{\overline{b}}_n^2\left(t\right)}/2]$ (solid curves) boundaries of the ferrofluid ring, corresponding to the situations resulting in the nonlinear annular patterns depicted in Fig. 4. Amplitudes for modes $n$, $2n$, $3n$, and $4n$ are shown.

Figure 6

(a) Overlaid snapshots of the pattern investigated in Fig. 2, taken at times $t=3.02\times {10}^{-2}$, $t=3.04\times {10}^{-2}$, and $t=3.05\times {10}^{-2}$, illustrating a counterclockwise rotational motion of the ferrofluid annulus. Likewise, (b) displays superposed snapshots of the pattern examined in Fig. 4, shown at times $t=2.29\times {10}^{-2}$, $t=2.31\times {10}^{-2}$, and $t=2.33\times {10}^{-2}$, exhibiting a clockwise spin of the ferrofluid ring. The curly black arrows indicate the directions of rotation.

Figure 7

Linear (dashed) and weakly nonlinear (solid) phase velocities $v_i(n,t)$ with $i=1,2$, and $n=n_{\max }$, as a function of time $t$ for the time evolution of the annular ferrofluid patterns portrayed in (a) Fig. 6 and (b) Fig. 6.