Axisymmetric MHD viscous flows bounded by a solid plane normal to a uniform ambient magnetic field: fundamental flows and application to a solid sphere translating normal to the wall

ABSTRACT

This work first determines two axisymmetric fundamental Magneto Hydrodynamic (MHD) flows induced, in a conducting Newtonian liquid domain bounded by a plane wall, by distributing either radial or axial points forces on a circular ring located in a plane parallel with the wall and normal to a prescribed uniform ambient magnetic field $\mathbf{B}=B{\mathbf{e}}_z.$ This is achieved, for both a perfectly conducting and an insulating wall, by using the fundamental flow due to a source point analytically obtained elsewhere. Each resulting axisymmetric fundamental MHD flow velocity components (radial and axial ones) and pressure is then analytically expressed in terms of one-dimensional integrals and of the so-called Hartmann layer thickness $d=(\sqrt{\mu /\sigma })/\left|B\right|.$ These quantities are numerically calculated and the wall–ring interactions are then discussed. Such interactions are found to deeply affect the fundamental flows’ streamlines and pressure field prevailing in an unbounded liquid. The derived fundamental flows are then employed to investigate, using a boundary formulation, the drag experienced by a solid sphere immersed in the liquid and translating normal to the wall.

KEYWORDS:

• Magneto hydrodynamics
• axisymmetric fundamental flow
• circular shape source
• bounded flow
• plane wall
• sphere

Disclosure statement

No potential conflict of interest was reported by the authors.