Abstract This paper presents an analytical solution for two-layer Couette and Couette–Poiseuille flows of immiscible, incompressible Newtonian fluids in a rectangular channel. Exact series solutions for the velocity field and flow rate are analysed in terms of key dimensionless parameters, including the aspect and viscosity ratios. Particular attention is given to the discontinuity in the velocity gradient at the fluid–fluid interface and to the shift in the location of the maximum velocity between layers. The viscosity contrast produces a viscous coupling between the two fluids that modifies both the velocity field and the hydraulic resistance. This effect is quantified by introducing an intrinsic viscosity for the equivalent single-layer flow and is interpreted through simple asymptotic arguments. The solution provides an exact benchmark for layered Couette–Poiseuille flows in rectangular channels .
Silva et al. (Tue,) studied this question.