Asymptotic expansion of the solution of the equation of a slow axisymmetric electrovortex flow between two planes

Electrovortex flows are of great interest both from the viewpoint of theoretical magnetohydrodynamics and for applications. They arise when the electric current of variable density passes through a conducting medium (such as a liquid metal). The interaction between the current and self magnetic field of the current induce a nonpotential electromagnetic force that causes a vortex flow of the liquid. As a model problem, we consider a stationary flow between two parallel planes. The flow is described by the stream function for which we obtain a nonlinear fourth-order partial differential equation. For moderate values of the electrovortex flow parameter, we investigate the problem by the asymptotic series expansion in the powers of this parameter. We describe the process of successive approximations of various degrees and present the flow pattern that is given by two first terms. This asymptotic solution is shown to be rather close to the numerical solution of the problem.