On undercompressive shocks and flooding in countercurrent two-layer flows

We consider the countercurrent flow of two incompressible immiscible viscous fluids in an inclined channel. This configuration may lead to the phenomena of 'flooding', i.e. the transition from a countercurrent regime to a cocurrent regime. This transition is marked by a variety of transient behaviour, such as the development of large-amplitude waves that impede the flow of one of the fluids to the reversal of the flow of the denser fluid. 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, which assumes fluids with disparate density and dynamic viscosity ratios, includes the effects of viscosity stratification, inertia, shear and capillarity. Since the experimental constraints for this effective system are unclear, we consider two ways to drive the flow: either by fixing the volumetric flow rate of the gas phase or by fixing the total pressure drop over a downstream length of the channel. The latter forcing results in a single evolution equation whose dynamics depends non-locally on the interfacial shape. From both of these driven systems, admissible criteria for Lax shocks, undercompressive shocks and rarefaction waves are investigated. These criteria, through a numerical verification, do not depend significantly on the inertial effects within the more dense layer. The choice of the local/non-local constraints appears to play a role in the transient growth of undercompressive shocks, and may relate to the phenomena observed near the onset of flooding.