Rolling of a cylinder in a magnetic field along a surface covered with a layer of an electrically conducting liquid

I. A. Shvarts

Institute of Physics, Latvian Academy of Sciences, Salaspils-1, LV-2169, Latvia

Abstract
This paper considers the magnetohydrodynamic problem of the rolling of a cylinder along a surface covered with a thin layer of an electrically conducting viscous liquid. A homogeneous magnetic field is directed perpendicular to the surface. The ordinary and magnetic Reynolds numbers are assumed to be small. The problem is solved in the two dimensions x and y. The boundary conditions are the same as in the ordinary hydrodynamic problem, whose solution is known. Formulas are obtained for the velocity distribution in the layer of liquid under the cylinder, as well as an integral expression for the pressure. Expressions are given for the pressure in the layer, corresponding to small and large Hartmann numbers. A detailed study is made of the case when the ratio of the thickness of the layer of liquid to the diameter of the cylinder is small. The formulas for the pressure and the velocities are considerably simplified. For the given case it is shown that, with an increase in the Hartmann number, the limits of the region of positive pressure in the layer are narrowed, while the maximal value of the pressure increases. The coefficient of rolling friction is calculated. It is shown that, in a strong magnetic field, the rolling friction coefficient becomes small and does not depend on the viscosity of the liquid.