On the Hartmann hydromagnetic flow

Abstract

The set-up for the Hartmann flow is the simplest geometry studied in the development of magnetohydrodynamic generators. It also illustrates the basic mechanism of a number of technological applications of hydromagnetic flows. The Hartmann flow consists of a conducting body, i.e. a plasma, which is made to move along a rectangular duct of constant cross-section. The enclosing walls are themselves electrically conducting.
The standard fluid equations have been applied to such a system. The results include an analytic form for the velocity profile but only for the stationary regime. In this study, we apply the previously derived modified fluid equations which contain a molecular parameter. The equations have already been applied to classic fluid flows like the Poiseuille flow and the Golub-Swinney flow. This time in order to apply the equation to a conducting fluid, it has an additional term expressing the presence of an extemal magnetic field. Velocity-dependent forces are accounted for in the zero-order kinetic equation where they are included in the Hamiltonian of the external field while the Hamiltonian describing the interactions is treated as a velocity-independent perturbation. The equation is then coupled to the equation of continuity and the Maxwell's equations of electrodynamics. Since the modified fluid equations contain as the correction terms a time integral, it provides a means to investigate the non-stationary behavior only.