MHD flow in an insulating rectangular duct under a non-uniform magnetic field

Followed by a review of previous studies of magnetohydrodynamic (MHD) duct flows in a non-uniform magnetic field at the entry into a magnet (fringing magnetic field), the associated MHD problem is revisited for a particular case of a nonconducting rectangular duct of a small aspect ratio ε; = b/a (here, b is the duct half-width in the magnetic field direction, and a is the half-height). The suggested model includes a realistic three-component div- and curl-free fringing magnetic field as well as inertia terms and takes into account the mechanism of electric current exchange between the core of the flow and the Hartmann layers. The original three-dimensional flow equations are reduced to a quasi-two-dimensional (Q2D) form for three basic scalar quantities: the vorticity, the streamfunction and the electric potential. This Q2 D formulation implies that the velocity field in the core region between the two Hartmann layers does not change in the magnetic field direction and thus is two-dimensional, while the induced electric current forms both cross-sectional and axial circuits and is essentially three-dimensional. A new parameter R = ε;²Re/Ha has been identified to characterize the role of inertia in duct flows with insulating walls (Re and Ha stand for the Reynolds and Hartmann numbers). Computations and analytical studies are performed for inertialess (R << 1) and inertial (R >> 1) flows at ε; = 0.2 for Re up to 300,000 resulting in new scaling laws for typical lengths, velocities, electric current densities and pressure drops, which provide a new theoretical basis for potential applications.