A Mathematical Model for Magnetohydrodynamic Convection Flow in a Rotating Horizontal Channel with Inclined Magnetic Field, Magnetic Induction and Hall Current Effects

Swapan Kumar Ghosh, Osman Anwar Bég, Abdul Aziz

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DOI: 10.4236/wjm.2011.13019                

Abstract

Closed-form and asymptotic solutions are derived for the steady, fully-developed hydromagnetic free and forced convection flow in a rotating horizontal parallel-plate channel under the action of an inclined magnetic field and constant pressure gradient along the longitudinal axis of the channel. The magnetic field is strong enough to generate Hall current effects and the magnetic Reynolds number of sufficient magnitude that induced magnetic field effects are also present. Secondary flow is present owing to the Hall current effect. The channel plates are also taken to be electrically-conducting. The conservation equations are formulated in an (x, y, z) coordinate system and non-dimensionalized using appropriate transformations. The resulting non-dimensional coupled ordinary differential equations for primary and secondary velocity components and primary and secondary induced magnetic field components and transformed boundary conditions are shown to be controlled by the dimensionless pressure gradient parameter (p_x), Hartmann number (M²), Grashof number (G), Hall current parameter (m), rotational parameter (K²), magnetic field inclination (Θ), and the electrical conductance ratios of the upper (Θ₁) and lower (Θ₂) plates. Solutions are derived using the method of complex variables. Asymptotic solutions are also presented for very high rotation parameter and Hartmann number of order equal to unity, for which Ekman-Hartmann boundary layers are identified at the plates. A parametric study of the evolution of velocity and induced magnetic field distributions is undertaken. It is shown that generally increasing Hall current effect (m) serves to accentuate the secondary (cross) flow but oppose the primary flow. An increase in rotational parameter (K²) is also found to counteract primary flow intensity. An elevation in the Grashof number i.e. free convection parameter (G) is shown to aid the secondary induced magnetic field component (H_z); however there is a decrease in magnitudes of the primary induced magnetic field component (H_x) with increasing Grashof number. Increasing inclination of the applied magnetic field (Θ, is also found to oppose the primary flow (u₁) but conversely to strongly assist the secondary flow (w₁). Both critical primary (G_cx) and secondary (G_cz)Grashof numbers are shown to be reduced with increasing inclination of the magnetic field (Θ), increasing Hall parameter (m) and rotational parameter (K²). Applications of the study arise in rotating MHD induction power generators and also astrophysical flows.

Keywords

Hydromagnetic Flow, Hall Current, Electrical Conductance, Free And Forced Convection, Critical Grashof Number, Rotation, Complex Variables, Asymptotic Analysis, Ekman-Hartmann Boundary Layers, Astronautics

Conflicts of Interest

The authors declare no conflicts of interest.

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