Analytical study on flow of two non-miscible laminar layers of Newtonian fluids in a curved channel with wall slippage

The velocity slippage with combined effects of buoyancy force plays an important role in the extraction of crude oil since it reduces the pressure drop along the pipelines and well-bore. Due to this, it becomes difficult to push the oil out of the reservoir. So, the main goal is to analyse the velocity slip and no-slip on two immiscible viscous fluids flowing in a curved channel with viscous dissipation. To the best of the author's knowledge, there is no study given on two non-miscible laminar Newtonian layers of viscous-viscous fluids. The curved channel is divided into two regions and both regions are occupied with viscous fluids. The flow is caused by the constant pressure gradient and buoyancy force. The analytical solutions of highly nonlinear mathematical equations are obtained using perturbation series by taking $\epsilon (=\mathrm{Pr}Ec)\ll 1$. The impact of the curvature ratio parameter, velocity slips, Reynolds number, buoyancy force, viscosity ratio parameter, and height ratio parameter on heat and mass transfer are examined and presented through graphs. Moreover, the behaviour of shear stresses, skin drag and Nusselt number on both walls of the channel is analyzed and presented through bar charts. It is concluded that the velocities of the immiscible fluids augment by increasing the curvature parameter whereas the temperatures and shear stresses fall. It is also observed that the shear stress at the lower wall is more significant with a rise in curvature parameter in the case when velocity slippage is not considered as compared to velocity slippage.