Magnetohydrodynamic Flow of Immiscible Couple Stress and Newtonian Fluids in a Porous Pipe With Navier Slip Effect

Abstract

This work deals with the magnetohydrodnamic (MHD) flow of non-miscible couple stress and Newtonian fluids within a horizontally-oriented porous cylinder. The overall flow domain is divided into two separate regions. In the core area, identified as region I, the flow of the couple stress fluid takes place, while in region II, which forms the outer part of the flow area, the flow of Newtonian fluid occurs. The linear Navier slip condition on the cylinder's surface and continuity conditions for velocities and shear stresses, along with vanishing couple stress at the fluid-fluid surface, have been taken as boundary and interface conditions, respectively. The nonlinear partial differential equations describing the flow situation along with the boundary conditions are first mathematically formulated and, then cast in a dimensionless form. Closed-form solutions for velocities, wall shear stress, and total flow rate have been obtained by solving the non-dimensionalized governing equations through the direct method. The influences of different flow parameters on the velocity in both flow areas are depicted graphically. The numerical values of the total flow rate and the wall stress for various flow parameters are also tabulated. The examination of the obtained results indicates that the fluid velocities diminish with increases in the Hartmann number, viscosity ratio, and porosity parameter. Conversely, they escalate with higher Reynolds numbers, pressure gradients, and slip parameters. Furthermore, the increase in the couple stress parameter increases the velocity of the couple stress fluid (core region), while the velocity of the Newtonian fluid (peripheral region) remains nearly constant. The findings of this research align very well with the results documented in the existing literature. This study is novel as it examines, for the first time, the effects of slip and magnetic fields on the flow of two immiscible fluids (couple stress and Newtonian) through a porous medium in cylindrical coordinates.