Numerical simulation of hydromagnetic Marangoni convection flow in a Darcian porous semiconductor melt enclosure with buoyancy and heat generation effects

We present a mathematical and numerical study of the transient Marangoni thermo-convection flow of an electrically conducting Newtonian fluid in an isotropic Darcy porous rectangular semiconductor melt enclosure with buoyancy and internal heat generation effects, in an (x, y) coordinate system. The governing equations comprising the mass conservation, x-direction momentum, y-direction momentum and energy equation are formulated subject to a quartet of boundary conditions at the four walls of the enclosure. The upper enclosure wall is assumed to be "free" with an appropriate surface tension dynamic boundary condition. A series of transformations are implemented to render the mathematical model dimensionless and into a vorticity form. The governing thermophysical parameters are shown to be the Marangoni number for surface tension (thermo-capillary) effects (Ma), Prandtl number (Pr), Grashof number for buoyancy effects (Gr), enclosure aspect ratio (A), Hartmann hydromagnetic number (Ha), Darcy number for bulk porous resistance (Da), and the internal heat generation parameter (Gamma) the latter being a function of the internal (RaI) and global Rayleigh numbers (Ra). An efficient finite difference numerical method is employed to solve the boundary value problem. Validations with earlier purely fluid solutions (Da -> infinity) are included. A mesh-independence test is included with further validation with other published studies. Isotherms and isovels (streamlines) are computed as are Nusselt numbers at selected boundaries. Solutions for the case of Pr = 0.054 (semiconductor melt) are also compared with earlier studies showing excellent correlation. The model finds applications in the bulk crystal growth of semiconductors, electromagnetic materials processing control and also hybrid fuel cells.