Abstract
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 e = 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 = e2Re/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 e = 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.
Highlights
Magnetohydrodynamic (MHD) flows in a non-uniform magnetic field, or in ducts of a varying cross-sectional area, have been extensively studied since the seminal paper of Hartmann [1], where the formation of the axial current loops and the origin of the pressure losses associated with the fringing magnetic field either at the entry or exit of a magnet are qualitatively described
6 Discussion and conclusion In this paper, the problem of the flow in a non-uniform magnetic field is revisited by paying special attention to accurate representation of the applied fringing magnetic field, electric current exchange between the core and the Hartmann layers, and the role of inertia
A second important point is the explicit consideration of the Hartmann layers and the electric current exchange mechanism between them and the core flow
Summary
Magnetohydrodynamic (MHD) flows in a non-uniform magnetic field, or in ducts of a varying cross-sectional area, have been extensively studied since the seminal paper of Hartmann [1], where the formation of the axial current loops and the origin of the pressure losses associated with the fringing magnetic field either at the entry or exit of a magnet are qualitatively described. Unlike previous theories, where this mechanism is not introduced or included in an approximate way, we derive the three current components directly based on the properties of the Hartmann layers, the vorticity equation and by satisfying the charge conservation equation This is different, for example, from the approach in Ref. Such an approach seems not to be valid in the flow region where the M-shaped velocity profile is formed and the local current density can differ significantly from its averaged value Despite this limitation, Alboussière’s analysis has the merit to renew the basic scaling law in this problem, introducing a new important length scale of the order of Ha-1/4 in the case of a pipe flow.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
