Analysis and Finite Element Simulation of Mhd Flows, with an Application to Seawater Drag Reduction

Much research effort has recently been devoted to the electromagnetic control of saltwater flows, exploiting the macroscopic interaction of saltwater with electric currents and magnetic fields. This interaction is governed by the equations of viscous incompressible MHD, essentially, the Navier-Stokes equations coupled to Maxwell’s equations. A major problem in the analysis and numerical solution of these equations is the fact that while the Navier-Stokes equations are posed in the fluid domain, Maxwell’s equations are generally posed on all of space. Consequently, electric and magnetic fields do not satisfy standard boundary conditions, but jump or continuity relations on the surface of the fluid domain (and other interfaces). Frequently the resulting difficulties are circumvented by prescribing more or less artificial boundary conditions. In this paper we present a novel formulation of the MHD equations that avoids some inherent difficulties of more traditional approaches by employing the electric current density rather than the magnetic field as the primary electromagnetic variable. This formulation leads to initialboundary value problems for a system of integro-differential equations in the fluid domain and lends itself naturally to the use of finite-element based discretization techniques. As a first application we describe a mixed finite-element method for the numerical solution of a class of stationary MHD flow problems and report on the computational simulation of a simple drag reduction experiment.