# A Study of Unsteady Rotating Hydromagnetic Free and Forced Convection in a Channel Subject to Forced Oscillation under an Oblique Magnetic Field

Authors

1 Magnetohydrodynamics Research, Department of Mathematics, Narajole Raj College, P.O.- Narajole, Dist. – Midnapore (West), Pin code – 721 211, West Bengal, India.

2 Biomechanics and Astronautics Research, Aerospace Engineering Program, Mechanical Engineering Subject Group, Department of Engineering and Mathematics, Sheffield Hallam University, Sheffield, S1 1WB, UK.

3 Fluid Dynamics Research, Fundamental and Applied Sciences Department, Universiti Teknologi Petronas, 31750 Tronoh, Bandar Seri Iskandar, Perak Darul Ridzuan, Malaysia.

Abstract

A theoretical analysis is presented for transient, fully-developed magnetohydrodynamic free and forced convection flow of a viscous, incompressible, Newtonian fluid in a rotating horizontal parallel-plate channel subjected to a uniform strength, static, oblique magnetic field acting at an angle  to the positive direction of the axis of rotation. A constant pressure gradient is imposed along the longitudinal axis of the channel. Magnetic Reynolds number is sufficiently small to negate the effects of magnetic induction. The channel plates are electrically non-conducting. The conservation equations are formulated in an (x,y,z) coordinate system and normalized using appropriate transformations. The resulting non-dimensional coupled ordinary differential equations for primary and secondary velocity components and transformed boundary conditions are found to be reciprocal of the Ekman number ( 2 K =1/Ek), non-dimensional pressure gradient parameter (Px), Hartmann number ( 2 M ), Grashof number (Gr), magnetic field inclination () and oscillation frequency (). Complex variables are employed to solve the two-point boundary value problem. A steady state resonance of the velocity field is identified for  4 4 4 1/ 2 16 2 1   K M Sin  . Furthermore the solutions indicate that the condition  1/2 1 4 4 4 cos 16 2   T K M Sin  signifies an oscillatory turbulent dynamo mechanism. A critical Grashof number (Grcx) is also evaluated for which primary flow reversal arises at the upper channel plate. A similar criterion for Grashof number (Grcy) is established for the onset of secondary flow reversal at the upper plate. A detailed assessment of the influence of the control parameters on primary and secondary velocity evolution in the channel is also conducted. The model finds applications in MHD (Magneto Hydro Dynamic) energy generators and also magnetic materials processing systems.

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