Unsteady magnetohydrodynamic Hartmann–Couette flow and heat transfer in a Darcian channel with Hall current, ionslip, viscous and Joule heating effects: network numerical solutions

A theoretical study of unsteady magnetohydrodynamic viscous Hartmann–Couette laminar flow and heat transfer in a Darcian porous medium intercalated between parallel plates, under a constant pressure gradient is presented. Viscous dissipation, Joule heating, Hall current and ionslip current effects are included as is lateral mass flux at both plates. The dimensionless conservation equations for the primary (x*-direction), secondary (z*-direction) momentum and also energy conservation equation are derived and solved using a computational technique known as Network Simulation Methodology (NSM). Velocity distributions (u*, w*) and temperature distribution (T*) at the channel centre (y* = 0) over time (t*) are studied graphically for the effects of Darcy number (Da), Hartmann number (Ha), transpiration (Nt), Hall current parameter (Be), ionslip parameter (Bi), pressure gradient parameter (dP/dx*) with Prandtl number prescribed at 7.0 (electrically conducting water), Eckert number held constant at 0.25 (heat convection from the plates to the fluid) and Reynolds number (Re) fixed at 5.0 (for Re < 10, Darcian model is generally valid). Increasing Darcy number causes an increase in temperature, T*; values are however significantly reduced for the higher Hartmann number case (Ha = 10). For the case of low transpiration (i.e. Nt = 1 which corresponds to weak suction at the upper plate and weak injection at the lower plate), both primary velocity (u*) and secondary velocity (w*) are increased with a rise in Darcy number (owing to a simultaneous decrease in Darcian porous drag); temperature T* is also increased considerably with increasing Da. However, for stronger transpiration (Nt = 10), magnitudes of u*, w* and T* are significantly reduced and also significant overshoots are detected prior to the establishment of steady state flow. With increasing Hall current parameter, Be, (for the purely fluid regime i.e. Da → ∞), primary velocity is considerably increased, whereas secondary velocity is reduced; temperatures are decreased in the early stages of flow but effectively increased in the steady state with increasing Be. With strong Darcian drag present (Da = 0.01 i.e. very low permeability), magnitudes of u*, w* and T* are considerably reduced and temperatures are found to be reduced for all t*, with increasing Hall current effect (Be). Increasing ionslip current parameter (Bi) increases primary velocity (u*), decreases secondary velocity (w*) and also temperature (T*) for all time (t*), in the infinite permeability case (Da → ∞). For weakly Darcian flow, ionslip parameter (Bi) has a much reduced effect on the velocity distributions. Temperature, T* is strongly increased with a rise in pressure gradient parameter, dP*/dx*, as is primary velocity (u*); however, secondary velocity (w*) is reduced. The present study has applications in hybrid magnetohydrodynamic (MHD) energy generators, materials processing, geophysical hydromagnetics, etc