Migration of an insulating particle under the action of uniform ambient electric and magnetic fields. Part 2. Boundary formulation and ellipsoidal particles

Abstract

This paper examines the low-Reynolds-number migration of an insulating and rigid particle that is freely suspended in a viscous liquid metal and subject to uniform ambient electric and magnetic fields $\hbox{\bf\itshape E}$ and $\hbox{\bf\itshape B}.$ Under the same physical assumptions as Part 1, a whole boundary formulation of the problem is established. It allows the determination of the particle rigid-body motion without calculating the modified electric field and the flow induced by the Lorentz body force in the fluid domain. The advocated boundary approach, well-adapted for future numerical implementation, makes it possible to obtain an analytical expression for the translational velocity of any ellipsoidal particle (the simplest case of non-spherical orthotropic particles). The behaviour of a spheroid is carefully investigated and discussed both without and with gravity. The migration of this simple non-spherical particle is found to depend on both its nature (prolate or oblate) and the ambient uniform fields $\hbox{\bf\itshape E}$ and $\hbox{\bf\itshape B}.$ The spheroid translates without rotation, and not necessarily parallel to $\hbox{\bf\itshape E}\wedge \hbox{\bf\itshape B}.$ For adequately selected fields $\hbox{\bf\itshape E}$ and $\hbox{\bf\itshape B},$ the spheroid may either migrate parallel or anti-parallel to a sphere and even be motionless.