Rheological impact of viscous fluid in the core region of heated curved pipe surrounded by Casson rheological fluid

The immiscible fluids can help design more efficient systems for drug delivery and understanding the blood flow through diseased arteries. It can also have implications for the design of flow control devices or medical implants that involve curved geometries. Therefore, the current article scrutinizes the characteristics of heat transfer on the flow of two immiscible Casson and Newtonian fluids through a curved pipe induced due to a pressure gradient in an axial direction. The pipe is divided into two distinct regions: namely, (i) Region 1 (core) is filled with Newtonian fluid while (ii) Region 2 (peripheral) is filled with non-Newtonian Casson fluid. The complex curvilinear coordinates called toroidal coordinates are used to form a mathematical model for the present problem. Equations governing the flow are considered without ignoring any curvature ratio terms and are solved analytically by considering the perturbation series. The continuity of velocities and shear stresses at the fluid-fluid interface is taken into account along with no-slip, symmetric, and regularity conditions while solving the two-fluid flow problem under consideration. The effect of various fluid parameters such as curvature ratio, Casson fluid parameter, Reynolds number, viscosity ratio, density ratio, Eckert number, and conductivity ratio on the axial velocity and temperature is studied through 2-dimensional graphs and contour plots. In addition, the volumetric flow rate and shear stress are also presented to evaluate the effects of various key parameters. The results show that the axial velocity of immiscible fluids flow increases with an increase in the values of the viscosity ratio parameter and the fluid temperature is decreased by increasing the conductivity ratio parameter.