Effect of pressure gradient on MHD over a flat plate

Summary. The magnetohydrodynamic (MHD) boundary layer flow over a fiat plate is examined here for two cases, viz. a uniform free-stream velocity and a uniform hydrostatic pressure. The nonlinear boundary layer equations are solved using a reliable finite-difference method, The boundary layer physical parameters such as skin-friction coefficient, displacement, momentum and energy thicknesses of the boundary layer are determined. It is found that the normal surface velocity gradient decreases with the local magnetic interaction parameter for the cases of a uniform hydrostatic pressure, whereas in the case of a uniform free-stream velocity it increases with the interaction parameter. Within the boundary layer, the velocity increases from zero at the surface to the free-stream velocity at the edge of the boundary layer and, therefore, velocity gradients may be appreciable, even if the viscosity is small. Determination of the wall-shearing stress is one of the important objectives in the solution of the boundary layer equations. The equations governing the boundary layer flow in general become nonsimilar due to the presence of a magnetic field or variable fluid properties. Wu [1] has studied the effects of suction or injection on a steady two dimensional magnetohydrodynamic (MHD) boundary layer flow on a flat plate. He assumed that both the free-stream velocity and the hydrostatic pressure were constant as in the case of boundary layers wherein the magnetic force term or Lorentz force term is absent in the equation of motion. Chuang [2] has pointed out the shortcomings of Wu's model and suggested to assume either free-stream velocity, or the hydrostatic pressure as constant in the solution of the boundary layer equations. The pressure gradient across the boundary layer is of the order of the boundary layer thickness and the pressure can be assumed constant across this thin layer. The pressure gradient along the flow direction may, in certain specific cases, be small or even zero; but, in general, it is determined by the external flow. Since the effects of viscosity are confined to a thin layer of fluid adjacent to the boundary, the pressure may be calculated on the basis of potential flow past the surface. This approach yields a reasonably accurate prediction of the pressure gradient when the boundary layer is not near to separation. If the free-stream velocity is constant, then the hydrostatic and magnetic pressure gradients are counter balancing with each other. For the case of zero hydrostatic pressure gradient, the free-stream velocity decreases along the flat plate due to the presence of a magnetic force. Motivated by the work of the above-mentioned authors, the effect of the pressure gradient on the MHD boundary layer over a flat plate is examined here. The nonlinear boundary layer equations were solved numerically by the finite-difference method and obtained the boundary layer physical parameters.