Hydromagnetic Flow of Two Immiscible Couple Stress Fluids through Porous Medium in a Cylindrical Pipe with Slip Effect

Abstract

In this study, the steady hydromagnetic flow of two immiscible couple stress fluids through a uniform porous medium in a cylindrical pipe with slip effect is investigated analytically. Essentially, the flow system is divided into two regions, region I and region II, which occupy the core and periphery of the system, respectively. The flow is driven by a constant pressure gradient applied in a direction parallel to the cylinder’s axis, and an external uniform magnetic field is applied in the direction perpendicular to the direction of fluid motion. Instead of the classical no-slip condition, the slip velocity along with vanishing couple stress boundary conditions is taken on the surface of the rigid cylinder, and continuity conditions of velocity, vorticity, shear stress, and couple stress are imposed at the fluid-fluid interface. The governing equations are modeled using the fully developed flow conditions. The resulting differential equations governing the flow in the two regions are converted to nondimensional forms using appropriate dimensionless variables. The nondimensional equations are solved analytically, and closed-form expressions for the flow velocity, flow rate, and stresses are derived in terms of the Bessel functions. The impacts of several parameters pertaining to the flow such as the magnetic number, couple stress parameters, Darcy number, viscosity ratio, Reynolds number, and slip parameter on the velocities in respective regions are examined and illustrated through graphs. The flow rate’s numerical values are also calculated for different fluid parameters and displayed in tabular form. It is found that increasing the magnetic number, viscosity ratio, Reynolds number, and slip parameters decreases the velocities of the fluids whereas increasing the couple stress parameter, Darcy number, and pressure gradient increases fluid velocities. The results obtained in this paper show an excellent agreement with the already existing results in the literature as limiting cases.