Electrical activity of the Hartmann layers relative to surface viscous shearing in an annular magnetohydrodynamic flow

Abstract

As a first step towards two-phase magnetohydrodynamics (MHD), this paper addresses an original analytical coupling between surface rheology, e.g., a gradually oxidizing liquid metal surface, ruled by the Boussinesq number Bo, and a supporting annular MHD flow, ruled by the Hartmann number Ha, in the general layout of a classical annular deep-channel viscometer, as developed by Mannheimer and Schechter [J. Colloid Interface Sci. 32, 195–211 (1970)]. Using a matched asymptotic expansion based on the small parameter 1/Ha, we can express the surface velocity as a coupling variable in the jump momentum balance at the liquid surface. By solving the latter through the determination of the Green's function, the whole flow can be analytically calculated. A modified Boussinesq number, \documentclass[12pt]{minimal}\begin{document}$\tilde{B_o}$\end{document}Bõ, is produced as a new non-dimensional parameter that provides the balance between surface viscous shearing and the Lorentz force. It is shown that the \documentclass[12pt]{minimal}\begin{document}$\tilde{B_o}$\end{document}Bõ number drives the electrical activation of the Hartmann layers, heavily modifying the MHD flow topology and leading to the emergence of the Lorentz force, for which interaction with the flow is not classical. Finally, the evolution laws given in this study allow the determination of scaling laws for an original experimental protocol, which would make it possible to accurately determine the surface shear viscosity of a liquid metal with respect to the quality of the ambient atmosphere.