On undercompressive shocks in constrained two-layer flows

We consider the countercurrent flow of two incompressible immiscible viscous fluids in an inclined channel. The lower fluid is denser than the upper fluid, making this configuration relevant to air-water systems. Flow is driven by an imposed pressure gradient and gravity. From a lubrication approximation based on the ratio of the channel height to the downstream disturbance wavelength, we derive a nonlinear system of evolution equations that govern the interfacial shape separating the two fluids and the leading-order pressure. This system includes the physical effects of advection, capillarity, inertia and hydrostatic pressure. Our interest is to compare the dynamics of the solutions under different flow constraints. In particular, we compare the dynamics when the liquid volumetric flow rate and the downstream pressure drop are held fixed to the case when the gas volumetric flow rate and the interfacial height at ends of the channel are held fixed. In both of these systems, Lax shocks, undercompressive shocks and rarefaction waves are investigated. Through a numerical verification, we find that the dynamics of both scenarios are different, resulting in unsteady interfacial profiles for flows driven by fixed liquid flow rate and pressure drop.