Abstract:

We show that the full set of magnetohydrodynamic (MHD) equations, for resistive and viscous incompressible fluids, allows in cylindrical coordinates an exact reduction to a pair of coupled, nonlinear, ordinary differential equations (ODE) for two scalar potentials. The ODEs represent self-similar MHD flows in channels, bounded by non-parallel plane walls, intersecting on the z-axis, akin to the Jeffery-Hamel flows of non-conducting fluids. We consider the case in which an external current, flows along the z-axis, and exerts a body force that controls the flow. Only one nonlinear ODE governs the solution in this case. Besides the Reynolds number (Re) and the angle of the walls, two other non dimensional parameters determine the solutions, the magnetic Reynolds number (Rm), and the Hartmann number (Ha). We give a preliminary study of the properties of the high resistivity regime, in which the Hartmann number becomes important. The main result is that for Ha larger than 2 the flow reversal near the walls, which is a typical feature of channels with Ha=0 and Re∼O(1) or larger, tends to disappear. Moderate values of Ha∼6 are sufficient to suppress flow reversal up to Re∼25 for angular widths of the channel as large as 240°. The configuration may be of interest for applications to flow control of liquid metals, and industrial treatment of molten metals.