The flow around an electromagnetically self-propelled sphere

We present a numerical analysis of the flow behind an electromagnetically self-propelled sphere. Two cases are considered. In the first part the sphere contains the electromagnetic coil system which creates a traveling magnetic field on the sphere surface. The control parameter of the problem is the interaction parameter N of the traveling field. For a given value of N the sphere moves in an electrically conducting fluid with some velocity, which defines the corresponding Reynolds number. It is found that in the self-propelled regime when the hydrodynamic drag force and the electromagnetic thrust force balance each other, the flow behind the sphere has a much smaller separation bubble than in the absence of the Lorentz force. The size of the separation bubble depends from the distribution of the electromagnetic field near the sphere surface and the value of the interaction parameter N. Vortex separation can be fully suppressed. We study the linear stability of the axisymmetric flow. It is found that the 3D instability occurs at much larger Reynolds numbers than in the absence of the Lorentz force. For the unstable 2D flow case we calculated a full 3D flow behind the sphere. It is found that even in this unstable case the drag of the sphere can be smaller than the drag of the sphere without Lorentz force.
In the second part we studied a so-called conductive system. The sphere contains a system of sectioned electrodes on the surface and a coil system inside. The Lorentz force is three-dimensional now and the flow behind the sphere is three-dimensional too, due to the 3D Lorentz forcing. We calculated the 3D flow behind a sphere in the self-propelled regime and found that the drag can be significantly reduced. We consider the energy consumption due to the drag reduction as well.