Driven Magnetic Particles on a Fluid Surface: Pattern Assisted Surface Flows

Magnetic microparticles suspended on the liquid-air interface and subjected to an alternating magnetic field exhibit spontaneous formation of dynamic localized snake patterns. These patterns are accompanied by four large-scale hydrodynamic vortices located at the opposite ends of the snake patterns. We report detailed studies of these large-scale vortices and their relationship to the collective response of magnetic particles in the presence of an alternating magnetic field. We present a model based on the amplitude equation for surface waves coupled to the large-scale hydrodynamic mean flow equation. The model describes both the formation of the dynamic snake patterns and the induced structure of the experimentally observed hydrodynamic flows.

Figure 1

Dynamic self-assembled snake structure generated by a vertical alternating magnetic field (100 Oe, 60 Hz) and induced surface flows. White round spots in the figure are $90\mu \mathrm{m}$ nickel particles. One sees individual particles as well as linear chains formed by particles. Arrows designate the velocity field of the surface flows obtained by means of particle image velocimetry. Background colors reflect the amplitude of the flow velocities. The flow velocity at the tails of the snake pattern reaches $1.2\mathrm{cm}/\mathrm{s}$.

Figure 2

Effective vortex strength $\Omega$ vs external magnetic field frequency $f$. Amplitude of the driving field was fixed to 100 Oe during the experiments. Inset: frequency behavior of the vortex strength obtained from the simulations based on the proposed phenomenological model. Bottom panel: onset of the snake instability (experiment: 70 Hz, 100 Oe driving).

Figure 3

Experimental critical driving field (squared) as a function of magnetic particles density. The data were taken for 80 Hz excitations. Dashed line is a fit to a $\propto 1/\rho$ dependence.

Figure 4

Patterns and hydrodynamic flows obtained by numerical modeling: (a) density of particles at the surface of liquid. Dark color corresponds to the low density of particles; (b) Surface wave amplitude induced by the collective response of particle to the external driving. (c) velocity field generated in vicinity of the snake pattern which is located horizontally in the center of Figure. Parameters in Eqs. (1, 2, 3) are $\epsilon =1$, $b=2$, $\gamma =2.5$, $D=1$, $\beta =5$, $\omega =3$, $\nu =200$ in domain of $200\times 200$ dimensionless units.

Figure 5

Evolution of two interacting snake patterns. (a)–(c) results of simulations. Color reflects density of particles: brighter colors correspond to high densities of particles; (d) experimental realization of the round snake pattern obtained at 100 Oe, 40 Hz driving.