Analysis of magnetohydrodynamic natural convection in closed cavities through integral transforms

- A hybrid numerical-analytical solution has been obtained, based on the Generalized Integral Transform Technique (GITT), for the coupled nonlinear partial differential equations governing the 2-D MHD flow with heat transfer inside a square cavity.
- The convergence behavior for both the streamfunction and temperature fields at prescribed positions, as well as for the average Nusselt number at the hot wall, was investigated. The GITT results are fully converged to at least three significant digits in all cases.
- A verification and validation analysis was performed by comparing the results for fluid velocities and temperature, as well as for Nusselt number, against results (numerical and experimental) reported in the literature. good agreement was achieved, and relative deviations in general increased for higher Grashof and Hartmann numbers.
- The approach can be extended to more involved problems to investigate the effects of various parameters, including cavity inclination angle and aspect ratio, internal volumetric heat generation rate, spatial variation of temperature and/or heat flux at the boundaries, spatial variation of thermophysical properties, and irregular geometries.

A hybrid numerical-analytical solution is proposed to analyze MHD (magnetohydrodynamic) natural convection of an electrically-conducting fluid within a square cavity, differentially heated at the sidewalls and subjected to an inclined external magnetic field. The first goal is to expand the spectrum of application of the so called Generalized Integral Transform Technique (GITT), dealing with a multiphysics formulation, while further demonstrating the relative merits of the proposed eigenfunction expansion approach in handling highly nonlinear and coupled systems of partial differential equations. The second goal is to provide a set of benchmark results in this important application for quantities of practical interest in determining the heat transfer rates, such as the average Nusselt number. The two-dimensional steady state equations are written in dimensionless form using the streamfunction-only formulation and are subsequently solved with the GITT approach, under automatic relative error control. Critical comparisons are performed against previous work reported in the literature, both computational and experimental, together with the corresponding physical interpretations, for different values of the governing parameters, such as Grashof number, Hartmann number, Prandtl number, and magnetic field inclination angle.