Single-mode instability of a ferrofluid-mercury interface under a nonuniform magnetic field

This work reports an experimental and a numerical study of the interfacial instability in a mercury-ferrofluid system caused by a spatially nonuniform magnetic field against the action of gravity and interfacial tension. The interface evolution is observed to be continuous till its movement is hindered by a physical boundary. In contrast to the behavior of the ferrofluid interface under uniform field, we noted the instability growth to be monotonic under a field gradient. A steepness in the growth curve is noticed during the later stages of the instability, indicating a high magnitude of the growth velocities. Some unique phenomena, such as similarity of the growth at the initial stage, a slope transition in the growth curve at a later stage, and wrapping and pinning of the interface are observed, both in experiments and simulations.

Figure 1

(a) Schematic of the classical gravity driven Rayleigh-Taylor instability showing Taylor unstable interface. (b) Hele-Shaw cell used in the current study and (c) the scheme of placing the cell onto the magnets.

Figure 2

(a)–(e) The image sequence showing the unstable finger growth, (f)–(j) the corresponding binary image sequence, and (k)–(o) the interface tracked from the binary images with (k) showing the terminology used in the text.

Figure 3

The splash after the spike touches the lower wall of the cell. The phenomenon is noted to be so rapid that the physical time difference calculated by correlating the image (a) and (e) is only $0.018\mathrm{s}$.

Figure 4

(a) Normalized finger-tip position with time for the magnet field strengths of $2.03,2.64$, and $3.27\text{kG}$. (b) Relative interface length with time for magnet field strengths of $2.03$. The simulation curve is also shown in both (a) and (b).

Figure 5

(a) Schematic of the problem domain showing field and flow boundary conditions (not to scale). (b) Measured field values due to the neodymium magnets (NM) and (c) the field distribution for a single magnet fitted with a Gaussian curve.

Figure 6

Case studies for validation of numerical scheme. (a) The displacement of RTI spike and bubble tip under the gravity compared with He et al. [17] and (b) the FF droplet shapes and their axis length ratio compared with Flament et al. [30] and Shi et al. [31].

Figure 7

(a) The initial magnetic field strength distribution normalized with the maximum magnet surface strength $\left(2.03\mathrm{kG}\right)$. (b) the same at horizontal center of the domain, decreasing as cube of the distance from bottom wall. The polynomial for this particular case is $H/H_o=-56.87{(y/W)}^3-8.812{(y/W)}^2-1.623(y/W)+0.264$.

Figure 8

Phase patterns obtained from simulation (right) vis-à-vis experimental image sequence (left) for field strength $3.27\mathrm{kG}$. The normalized time instants are $(a,f)=(0.0134,0.013),(b,g)=(0.271,0.262),(c,h)=(0.417,0.393),(d,i)=(0.478,0.459)$, and $(e,j)=(0.503,0.485)$.

Figure 9

The simulations depicting the monomode nature of the instability under the influence of single column of magnets. The growth of central spike dominates over the wavy protuberances on both sides. (a)–(f) depicts the growth after initial interface perturbation given using Eq. (14) while (g)–(l) use Eq. (15).

Figure 10

(Top) Velocity vector field, (inset) the velocity field in the close proximity of the interface (wrapping the interface) and (bottom) maximum-value normalized vorticity contours as the finger grows. The three normalized time instants are $0.262,0.393$, and $0.459$ under field strength case of $3.27\mathrm{kG}$.

Figure 11

The pinning of the interface at the vortex centers (left) from experiment and (right) simulation.

Figure 12

The vortex dynamics when the initial interface is perturbed in a wavy fashion. The vortex pair surrounding the dominantly growing mode survives while the pairs around the attenuating modes die out. (a), (b), and (c), respectively, correspond to Figs. 9, 9, and 9.