Rapidly convergent series for solving the electrovortex flow problem in a hemispherical vessel

A linear boundary value problem describing the axially symmetric steady viscous electrovortex flow in a hemispherical container is considered. The electrovortex flow is generated due to the interaction of an electric current flowing through the medium with the magnetic field produced by this current. In earlier works, formal double series in terms of the eigenfunctions of the Dirichlet problem for the Laplacian in a hemispherical layer were obtained for the solution of this problem. The Fourier coefficients were expressed in terms of hypergeometric functions, and they contained the eigenvalues of the hemispherical layer. In this paper, the classical solution of the boundary value problem under study is represented in the form of single series in terms of associated Legendre functions. The expansion coefficients are elementary functions of the radial variable. The first few terms are sufficient for the correct representation of the solution. The rate of decay of the terms is estimated. The smoothness of the solution is proved using Weyl’s lemma. The results can be useful in the study of other boundary value problems involving a vector Laplacian.