Shape transformations in rotating ferrofluid drops

Abstract

Floating drops of magnetic fluid can be brought into rotation by applying a rotating magnetic field. We report theoretical and experimental results on the transition from a spheroid equilibrium shape to non-axissymmetrical three-axes ellipsoids at certain values of the external field strength. The transitions are continuous for small values of the magnetic susceptibility and show hysteresis for larger ones. In the non-axissymmetric shape the rotational motion of the drop consists of a vortical flow inside the drop combined with a slow rotation of the shape. Nonlinear magnetization laws are crucial to obtain quantitative agreement between theory and experiment.PACS numbers: 47.20.Hw, 47.55.Dz, 75.50.Mm The equilibrium shapes of rotating fluid bodies are of importance in various fields of physics as, e.g., astrophysics [1], nuclear fission [2], plasma [3], and biological physics [4]. The famous controversy between Newton and Cassini on whether the Earth has the form of an oblate or prolate rotational ellipsoid started a series of ingenious investigations of the equilibrium shapes of heavenly bodies including work by Maupertius, MacLaurin, Jacobi, Riemann, Poincaré and others. Among the suprising results discovered are the possibility of threeaxes ellipsoids as shown by Jacobi and the emergence of pear-shaped configurations found by Poincaré. In rotating non-neutral plasmas, laser cooled in a Penning trap, a novel equilibrium state with non-axissymmetric surface has recently been observed [5]. Tank-treading elliptical membranes in a shear flow have been used as a theoretical model for the motion of human red blood cells [6].Experimental investigations of the stationary shapes of rotating bodies in the laboratory usually start with a static drop of fluid floating in another, immiscible fluid of the same density. The rotational motion of the drop is then set up by, e.g., using a rotating shaft [7] or applying an acoustic torque [8]. An elegant way to spin up drops made from polarizable fluids is to use rotating electric [5] or magnetic fields [9,10].In the present letter we report theoretical and experimental investigations of rotating ferrofluid drops and in particular perform the first quantitative study of a transition from a spheroidal to a non-axissymmetric equilibrium shape in this system. Moreover, providing an approximate solution of the hydrodynamic flow problem inside and outside the drop we are able to analyze the rotational motion of the drop shape and to separate it from the internal hydrodynamic flow.Ferrofluids are suspensions of ferromagnetic nanoparticles in suitable carrier liquids combining the hydrodynamic behaviour of a Newtonian fluid with the magnetic properties of a super-paramagnet [11]. A rotating external field induces a rotational motion of the nanoparticles which due to their viscous coupling to the surrounding liquid transfer the angular momentum to the whole drop. The system has been studied previously by Bacri et al. using microdrops with a typical radius of 10 µm, very small surfac...

Rotating Ferrofluid Drops

“…Part of our results was published in the short communication [9]. In the present manuscript we give a substantially enlarged and comprehensive analysis of the problem.…”

Design Considerations for Magnetically Actuated Biomimetic Cilia

“…In the magnetic energy term (Morozov et al, 2002), G is the applied magnetic field (in A/m), M' is the magnetization of the cilium, which in general is a function of H, the internal field (in A/m). The angle between the cilium and the applied field is given by , and N x and N y are the demagnetization factors parallel and perpendicular to the axis of the cilium, respectively.…”

“…In addition, a few authors have published on the energy of rotating magnetic droplet (Cebers, 2002;Morozov, Engel, & Lebedev, 2002). While many of these models take into account dynamics such as fluid-structure interactions and polymer viscosity, we will confine ourselves here to a quasi-static model.…”

Ferrofluid drops in rotating magnetic fields

“…Rotating shafts [6] or acoustic torques [7] may then be used to spin up these drops. For drops made from either electrically or magnetically polarizable fluids rotating electric [3] or magnetic fields [8]- [10] are a convenient tool to set up a rotation drop.…”

“…Part of our results have already been published in the short communication [10] where emphasis was placed on the influence of saturation effects on the shape and rotational frequency of the drop. Accordingly, in this paper saturation effects will be dealt with only in brief.…”