In this paper, we study the flow of two immiscible fluids namely, couple stress fluid and Jeffrey fluid in a porous channel. Instead of the classical no-slip conditions on the boundaries, we used slip boundary conditions, which are more realistic and meaningful. In addition, we used inclined magnetic field effects on the fluid flow. The couple stress fluid and Jeffrey fluid are flowing adjacent to each other in the region I and in the region II, respectively, of the horizontal porous channel. The nondimensionalized governing equations are solved analytically by using slip conditions at the lower and upper boundaries and interface conditions at the fluid-fluid interface. The analytical expressions for the velocity components in both regions are obtained in closed form. The effects of slip parameter, Hartmann number, couple stress parameter, Jeffrey parameter, angle of inclination, and Darcy number on velocity components in both regions are investigated. In the absence of slip, couple stress parameter, and Jeffrey parameters, limiting cases are obtained and discussed.

In this study, we investigated the effects of slip boundary and magnetic field on the flow of two immiscible couple stress fluids in an inclined porous media channel. The Stokes model was used to define the two couple stress fluids. The fluids in both the regions are assumed to be incompressible and immiscible and the transport properties in the two regions are assumed to be constant. The governing coupled linear ordinary differential equations are made dimensionless by using appropriate fundamental quantities. The expressions for the velocity components in both the regions are obtained in closed form using the undetermined coefficient method; by applying slip boundary condition at the boundary of the channel and at fluid-fluid interface, the continuity of velocities, vorticities, shear stresses, and couple stresses are applied. The effects of the nondimensional parameters including the couple stress parameters, slip parameter, Hartmann number, Darcy number, and the ratio of basic fluid properties on the velocity components are presented graphically; the flow rate and shear stress are presented in table and discussed briefly. It is reported that the presence of couple stresses and magnetic field have a declining impact while slip boundary has an enhancement impact on the fluid velocity and the volumetric flow rate. In the absence of couple stress, the results for the liming case as s ⟶ ∞ η ⟶ 0 for Newtonian flow are achieved and further limiting cases are established and briefly addressed.