Fundamental MHD Creeping Flow and Electric Potential for a Conducting Fluid Bounded by Two Parallel PlaneWalls

Abstract

This work determines the MHD fundamental viscous flow and electric potential induced by a concentrated force, with arbitrary strength s, immersed in a conducting Newtonian liquid bounded by two motionless, parallel and plane no-slip walls. The walls are perfectly conducting or insulating surfaces normal to the imposed uniform ambient magnetic field B. Each fundamental quantity (velocity component, pressure, electric potential) is the analytical one prevailing in the absence of walls plus another ‘confinement’ quantity due to the walls. By performing direct and inverse two-dimensional Fourier transforms, each such confinement quantity is obtained in closed form solely in terms of one-dimensional Bessel-type integrals. The resulting fundamental flow and electric potential are found to depend upon the concentrated force location, the wall-wall gap, the properties of the walls and the problem Hartmann layer thickness d = (μ∕σ)∕|B| where μ and σ > denote the liquid uniform viscosity and conductivity, respectively. For a force normal to the walls there the electric potential vanishes and the fundamental velocity components and pressure are independent of the nature of the walls and also receive tractable closed forms. These properties remain true for the fundamental pressure and velocity component normal to the walls in case of a force tangent to the walls. In contrast, the electric potential and the velocity component tangent to the walls admit quite involved closed forms and deeply depend upon the nature of the walls when the concentrated force is parallel with the walls.