MHD duct flows at moderate Hartmann number

A numerical study of steady-state, fully developed, MHD duct flows is presented for the case of an applied transverse magnetic field and a two-dimensional variation of the flow parameters in the plane perpendicular to the flow direction, with no variations in fluid properties. Solutions are found for (i) rectangular ducts with no obstacles and (ii) rectangular ducts containing thin flat plates, long in the flow direction; flow excitation by pressure gradients and moving boundaries is considered. Transverse wakes similar to those found by Hasimoto are studied in several novel circumstances. 1. Introduction The numerical solutions which are described here are concerned with steady-state, fully developed, MHD duct flows. The two-dimensional variations of the flow parameters, in the plane perpendicular to the flow direction, are presented. Ordinary viscous flow hydro-dynamic solutions (OHD) and magnetohydrodynamic solutions (MHD) are found for (i) rectangular ducts with no obstacles and (ii) rectangular ducts with thin flat plates immersed in the flow (the plates are long in the flow direction). Two classes of flow excitation are examined: (i) pressure gradient excitation and (ii) excitation by moving a duct wall or an immersed plate. Hasimoto (1960) investigated (theoretically) the MHD flow excited by moving a long thin plate in a conducting fluid along its own length for the cases when the plate was non-conducting and perfectly conducting and when the plate was semi-infinite in the direction perpendicular to the direction of movement and the magnetic field was perpendicular to the plate. He established the existence of transverse wakes which contained current flow and encompassed the region of changing velocity between the two regions of more uniform velocity (cores). These wakes emanate from the edge of the plate into the fluid along the direction of the magnetic field lines. The case of a long cylinder, being moved along its axis under the influence of a transverse field, has also been investigated by Hasimoto and more fully by Waechter (1966), who produced asymptotic solutions which displayed the character of the flow distribution around the cylinder and the existence of wakes emanating from the edges of the cylinder along the field lines. Hasimoto's solution demonstrated that the velocity of the fluid adjacent to the moving plate should be equal to the plate velocity for a perfectly conducting plate and half the plate velocity for a non-conducting plate. Experiments by Alpher et al. (1960) in an open channel mercury flow experiment …