Finite difference simulation of unsteady MHD flow through a non-darcian porous medium past a vertical cone in geothermal systems

Abstract This study has important applications in geothermal energy extraction, where heat and fluid flow through underground porous rocks must be properly understood for efficient energy production. It is also useful in petroleum engineering for enhanced oil recovery, in environmental engineering for groundwater studies, and in industrial processes involving electrically conducting fluids under magnetic fields. Additionally, such simulations help in improving the design of heat transfer systems and controlling fluid flow in engineering and energy-related technologies. The study deals with the numerical analysis of time-dependent flow of an electrically conducting fluid under the influence of a magnetic field. The flow occurs through a porous medium where Darcy’s law alone is not sufficient to describe the fluid behavior (Non-Darcian effects are considered). The fluid moves past a vertical cone, and the governing equations are solved using an implicit Crank-Nicolsion Finite Difference Method (FDM), which converts complex differential equations into simpler algebraic forms for computation (Thomas algorithm). This type of analysis is especially important in understanding heat and fluid flow behavior in geothermal systems. The velocity, temperature, and concentration distributions have been examined to study the effects of the electrical conductivity parameter and the Prandtl number. The local skin-friction coefficient, Nusselt number, and Sherwood number are also presented and analyzed graphically. It is observed that the velocity decreases with increasing non-Darcian parameter, while the temperature distribution increases. An increase in the velocity slip parameter leads to a reduction in velocity as well as boundary layer thickness, whereas the temperature distribution and thermal boundary layer thickness increase. The present results are compared with existing results available in the literature and are found to be in good agreement. A stability analysis is performed to examine the convergence and reliability of the finite difference scheme applied to the current problem. The results confirm the numerical stability and accuracy of the method under the chosen physical and computational parameters.