Buoyant displacement flow of immiscible fluids in inclined ducts: A theoretical approach

We study the buoyant displacement flow of two immiscible Newtonian fluids in an inclined duct (two-dimensional channel) theoretically. The fluids may have different viscosities. The displacing fluid is denser than the displaced one, i.e., a density-unstable configuration. For simplicity, the fluids are assumed to behave as neutrally wetting in the vicinity of duct walls. The small diameter-to-length ratio of the duct considered (⁠$δ≪1$⁠) has been used as the perturbation parameter in developing a lubrication model (negligible inertia). Appropriate Navier-slip conditions have been applied at the walls to overcome contact-line problem singularity. The lubrication model developed has then been numerically solved using a robust total variation diminishing finite difference scheme. Completely different flow patterns have been observed compared to the miscible limit. Fluids immiscibility is found to cause a capillary ridge in the vicinity of the displacing front, which diminishes as the surface tension is increased. For small values of surface tension parameter, the fluids immiscibility is found to decelerate the advancement of interpenetrating heavy and light layers. More efficient displacement (less fingering within the displacing layer) has been observed at small density differences and when the displacing fluid is more viscous than the displaced one. The limit of zero imposed velocity corresponding to the exchange flow has further been considered in the lubrication model. An interesting jump in the interface height occurs close to the vicinity of the gate region due to the immiscibility, which has been similarly reported in other recent computational works. Detailed mathematical notes on the similarity solution of the flow at long times are moreover provided. Investigating the short-time dynamics of the flow reveals the dominance of diffusive surface tension effects over buoyancy.