Liquid-metal flow in a rectangular duct with thin metal walls and with a non-uniform magnetic field

Abstract

This paper treats the steady flow of an electrically conducting, incompressible liquid in a duct with a constant rectangular cross section, with thin, electrically conducting walls and with a non-uniform transverse magnetic field which is parallel to two duct walls. With the x axis along the centerline of the duct, the dimensionless magnetic field B = B x(x, y) x ̂ + B x(x, y) y ̂ , where x̂ and ŷ are Cartesian unit vectors, while B x and B y are odd and even functions of y, respectively. For a small magnetic Reynolds number, B x and B y satisfy the Cauchy-Riemann equations and boundary conditions at the pole faces of the external magnet. Previous treatments have used a simplified magnetic field B = B y(x) y ̂ , even though this field does not satisfy the Cauchy-Riemann equations. We consider a magnetic field in which B y at the plane of symmetry varies from 0.98 to 0.54 over a short axial distance. For this field the maximum values of B x and of the y variation of B y are 0.25 and 0.125 in the liquid-metal region. These values are certainly significant, but they are neglected in the simplified magnetic field model. Nevertheless the results for the simplified and complete magnetic fields are virtually identical, so that the simplified field gives excellent results. Previous treatments have also assumed that the Hartmann number M is large. A possible error arises from the implicit assumption that α = cM 1 2 ⪢1 , where c is the wall conductance ratio, while realistic values of a are generally not particularly large. Comparison of the three-dimensional results for α⪢1 and for α = O(1) reveals that the results for α⪢1 can be corrected with simple scaling factors derived from much simpler solutions for fully developed flow.