Numerical Modeling of Two-Phase Hydromagnetic Flow and Heat Transfer in a Particle-Suspension through a non-Darcian Porous Channel

A mathematical model is presented for the steady, two-dimensional magneto-convection heat transfer of a two-phase, electrically-conducting, particle-suspension in a channel containing a non-Darcian porous medium intercalated between two parallel plates, in the presence of a transverse magnetic field. The channel walls are assumed to be isothermal but at different temperatures. The governing equations for the one-dimensional steady flow are formulated following Marble (1970) and extended to include the influence of Darcian porous drag, Forcheimmer quadratic drag, buoyancy effects, Lorentz body force (hydromagnetic retardation force) and particle-phase viscous stresses. Special boundary conditions for the particle-phase wall conditions are implemented. The governing coupled, non-linear differential equations are reduced from an (x,y) coordinate system to a one-dimensional (y) coordinate system. A series of transformations is then employed to non-dimensionalize the model in terms of a single independent variable, eta, yielding a quartet of coupled ordinary differential equations which are solved numerically using the finite element method, under appropriate transformed boundary conditions. The influence of for example Grashof free convection number (Gr), Haffmann hydromagnetic number (Ha), inverse Stokes number (Sk(m)), Darcy number (Da), Forcheimmer number (Fs),particle loading parameter (PO, buoyancy parameter (B) on the fluid-phase velocity and particle-phase velocity are presented graphically. A number of special cases of the transformed model are also studied. The mathematical model finds applications in solar collector devices, electronic fabrication, jet nozzle flows, industrial materials processing transport phenomena, MHD energy generator systems etc.