J. Fluid Mech. (1983), vol. 126, pp. 413430 Printed in Great Britain 413 Azimuthal rotation in the axisymmetric meridional flow due to an electric-current source By V. BOJAREVICS AND E. V. SHCHERBININ Institute of Physics, Latvian S.S.R. Academy of Sciences, Salaspils, Riga 229021, U.S.S.R. [Received 15 January 1982 and in revised form 20 July 1982) The steady laminar flow driven by the meridional electromagnetic force due to an electric-current point source on a plane is considered. The previous studies of the problem (Shercliff 1970; Shilova & Shcherbinin 1971) lead to a self-similar solution of the full Navier-Stokes equations analogous to the classic Landau jet. The solution breaks down when a critical electric-current magnitude is exceeded (Sozou 1971). In the present paper the converging meridional flow is shown to be unstable to an axisymmetric azimuthal perturbation when the corresponding critical Reynolds number is exceeded. The flow solution breakdown is eliminated for the coupled converging and rotating flow. The physical process is suggested by the draining-vortex formation. The fluid-flow equations are solved by the Galerkin method, using expansions in Gegenbauer functions. The mechanism sustaining the rotation is examined; the increased angular momentum in the fluid region is maintained by t)he balance of viscous diffusion upstream and convection to the axis of symmetry. The experimental evidence for vortex formation is considered. 1. Introduction In the last decade some authors have studied the problem of an electrically conducting fluid flow due to an electric-current point source since this is a convenient model for the investigation of some high-current industrial processes (electrical arcs, electro-slag welding, etc.)and natural phenomena (lightning discharge in a conducting fluid, tornado). The fluid motion in these processes is driven by the rotational Lorentz force set up by a diverging current and its associated magnetic field. The problem appeared in the papers of Zhigulev (1960), Lundquist (1969), Shercliff (1970), Sozou (1971) and Shilova & Shcherbinin (1971). Further interest in the problem can be motivated by the fact that the fluid flow due to an electric-current source is described by a class of exact solutions of the Navier-Stokes equations. The class was introduced by Landau (1944), Yatseyev (1950) and Squire (1951). These solutions correspond to the axisymmetric meridional flows with velocity fields inversely proportional to a spherical radius. Goldshtik (1960) added an azimuthal velocity and considered the potential vortex viscous flow above a rigid plane. Wu (1961) extended the class of exact solutions to magnetohydro- dynamics with the meridional magnetic field inversely proportional to a spherical radius. A full formulation of the class of exact solutions in magnetohydrodynamics was given by Shcherbinin (1969). The solution obtained by Lundquist (1969) describes in the Stokes approximation the slow fluid flow due to an electric-current source on a plane. The flow converges along the plane and ascends along the axis of symmetry. A similar meridional flow is induced by a vortex line normal to a rigid plane (Goldshtik 1960). Goldshtik’s 14-2