How a rotating magnetic field causes ferrofluid to rotate

The article explores the various mechanisms by which a rotating magnetic field causes to rotate a magnetic fluid contained in a cylinder (the so-called spin-up of ferrofluids). It is shown first that the generally accepted theory of spin diffusion cannot explain this phenomenon due to the extremely low spin viscosity ${\eta }^{\prime }$: Our calculations give ${\eta }^{\prime }\sim {10}^{-14}\mathrm{g}\mathrm{cm}/\mathrm{s}$ which is 11 orders of magnitude less than the most optimistic estimates. Then we develop a theory linking the rotation of a free surface layer to the shape of the meniscus. The latter, equally as the strength and frequency of the rotating field, determines both the direction and the magnitude of the fluid velocity. However, the magnetic torque created on the curved surface of the liquid captures only a thin subsurface layer and does not cause rotation in the volume. As for the volumetric flow, we associate it with the release of heat in microvortices that arise around rotating magnetic particles. In a vertical cylinder, such heating forms a parabolic temperature profile, which makes the magnetization of the fluid spatially inhomogeneous. As shown, this is sufficient to induce an observable bulk flow of the ferrofluid. The influence of natural convection on this spin-up flow is also investigated. In conclusion, it is shown that the stress tensor of a magnetic fluid can always be brought to a symmetric form.

Figure 1

Rotation of particles in a ferrofluid about their own axes in a rotating magnetic field. Each nanoparticle rotates with the angular velocity $\mathit{\omega }$, however, the macroscopic fluid velocity vanishes by symmetry. The sketch depicts the configuration of streamlines. The circulation caused by the rotation of the particle is precisely compensated by the oppositely directed circulation of its neighbors. Observe the existence of stagnation points because of the closure of all streamlines upon themselves.

Figure 2

Ferrofluid particle model: magnetic core of diameter $d_m$ covered with a surfactant layer of the thickness $(d_h-d_m)/2$.

Figure 3

The (a) concave and (b) convex profiles of the free surface of the ferrofluid. The contact angles are $\psi ={45}^{\circ }$ and $\psi ={135}^{\circ }$, correspondingly. (a) The concave shape produces counter-rotation, (b) convex—corotation.

Figure 4

The azimuthal velocity of the ferrofluid surface in a cylinder of $R=2.33\mathrm{cm},\gamma /2\pi =60\mathrm{Hz}$. The symbols mark the experimental data: $◯-H=100\mathrm{Oe},△-H=200\mathrm{Oe}$ [15], and the solid line is described by the expression (31). The maximum point on this curve has coordinates $r^{*}/R=0.865$ and $u\left(r^{*}\right)/u_0=0.822$.

Figure 5

Nondimensional azimuthal velocity $G$ in the Kármán problem as a function of $\xi =z\sqrt{\mathrm{\Omega }/\nu }$. The blue curve taken from Fig. 7 in Ref. [31] is the exact solution; the red curve—our approximation by the function $G=exp(-\xi /1.3)$.

Figure 6

Dependence on the Langevin parameter of the temperature difference $\mathrm{\Theta }$ between the axis and the cylinder wall, and the maximum speed $u_{\mathrm{m}}$ of fluid rotation; dashed line, relative magnetization $\mathcal{L}\left(\alpha \right)=M\left(\alpha \right)/M_{\mathrm{sat}}$.

Figure 7

Convective flow structure caused by internal heat sources in a long cylinder closed at the top and bottom. The dashed line—the interface between ascending and descending flows.