A two-layer model for buoyant displacement flows in a channel with wall slip

We study theoretically buoyant displacement flows of two generalized Newtonian fluids in a two-dimensional (2-D) channel with wall slip. We assume that a pseudo-interface separates two miscible (immiscible) fluids at the limit of negligible molecular diffusion (negligible surface tension). A heavy fluid displaces a light fluid at near-horizontal channel inclinations, implying that a stratified flow assumption is relevant. We develop a classical lubrication approximation model as a semi-analytical framework that includes a number of dimensionless parameters, such as a buoyancy number, the viscosity ratio, the non-Newtonian properties and the upper and lower wall slip coefficients. For specified interface heights and slopes, the reduced model can furnish the flux and velocity functions in displacing and displaced phases. We numerically solve the interface kinematic condition for four different wall slip cases: no slip (Case I), slip at the lower wall (Case II), slip at the upper wall (Case III) and slip at both walls (Case IV). The solutions for these cases deliver the interface propagation in time, for which leading and trailing displacement front heights, shapes and speeds and several key displacement features, such as front characteristic spreading lengths and short time behaviours, can be directly predicted by simplified analyses. The results reveal in detail how the presence of a channel wall slip may significantly affect the overall displacement flow and the interface evolution characteristics, for both Newtonian and non-Newtonian fluids. Regarding the latter, our analysis quantifies in particular the appearance and removal of static residual wall layers of the displaced phase, versus the wall slip cases.