Creeping flow of a couple stress fluid past a semipermeable spherical particle consisting of a solid core: magnetic field effect

The main goal of this research is to explore the influence of an external magnetic field and distinct permeability of the porous media on the flow of a couple stress fluid around a solid sphere featuring a semipermeable layer of porous material. This force acts at a right angle to the fluid’s motion, intending to explore the interaction between magnetic and viscous forces in determining fluid flow patterns. The Brinkman and Darcy equations with distinct permeabilities are utilized to describe porous medium hydrodynamics, incorporating isotropic permeability. To ensure a physically realistic modeling of fluid flow past a sphere, the governing equations are formulated using the spherical polar coordinates according to the proposed model structure and assumptions of this study. The stream functions are employed to transform the governing equations into a sixth-order differential equation for a couple stress fluid flow in a Brinkman porous medium and a second-order differential equation for a couple stress fluid flow in a Darcy porous medium. Analytical solutions for these governing equations, which describe the behavior of the couple stress fluid motion past a semipermeable sphere, are obtained using a variable separable method, expressed in terms of modified Bessel functions. The mathematical expressions of radial velocities, pressures, and drag force are graphically examined by considering different control parameters such as Hartmann number, permeability of the porous structure, separation, and couple stress parameters. The current analysis contrasts its findings with prior studies examining both magnetic influences and their absence. An increase in the Hartmann number results in a heightened viscous drag, as the ratio of electromagnetic to viscous forces rises. Furthermore, the rising value of the couple stress parameter \(\lambda\) $\lambda$ leads to a reduction in the drag growth rate with increasing Hartmann number. The notable finding of this study indicates that if the permeability parameter \(k_1\) $k_1$ of the Brinkman porous medium exceeds that of the Darcy porous medium, the drag force acting on a semipermeable sphere under flow of a couple stress fluid increases compared to situations where the permeability parameter \(k_2\) $k_2$ of the Darcy region dominates over \(k_1\) $k_1$ of the Brinkman porous region. The results of the suggested analysis could assist in assessing the drag experienced by droplets passing through obstacles, the separation of minerals such as uranium and gold, the extraction of proteins and other large molecules in pharmaceutical and biological procedures, and the management of municipal waste.