The flow of two immiscible viscous fluids in a thin inclined channel is considered, in either a cocurrent or countercurrent regime. Following the air-water case, which is found in a variety of engineering systems, we allow the upper fluid to be either compressible or incompressible. The disparity of the length scales and the density and viscosity ratios of the two fluids is exploited through a lubrication approximation of the conservation of mass and the Navier-Stokes equations. As a result of this long-wave theory, a coupled nonlinear system of partial differential equations is obtained that describes the evolution of the interfacial thickness and the leading-order pressure. This system includes the effects of viscosity stratification, inertia, shear, and capillarity, and reduces to the single-phase falling film Benney equation for sufficiently thin liquid films and constant gas density.
The case of two incompressible fluids is investigated first. Since the experimental conditions for this effective system are unclear, we consider several 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, or by fixing liquid flow rate and gas pressure drop. The forcing with prescribed pressure drop results in a single evolution equation whose dynamics depends nonlocally on the interfacial shape. From weakly nonlinear analysis in this case, we obtain the modified Kuramoto-Sivashinsky equation with an additional integral term, influencing the speed of propagation but not the shape of the interfacial wave. For the strongly nonlinear case, admissible criteria for Lax shocks, undercompressive shocks and rarefaction waves are investigated. Through a numerical verification we find that these criteria do not depend significantly on the inertial effects within the more dense layer. The choice of the local/nonlocal boundary conditions appears to play a role in the transient growth of undercompressive shocks, and may relate to the phenomena observed near the onset of flooding.
We then perform a linear stability analysis when the gas phase is compressible. The base-state profile for the density is spatially dependent when a pressure drop over the length of the channel is prescribed. The case when zero pressure drop is prescribed is amenable to a normal-mode analysis. When the liquid film thickness is sufficiently thin, the stability matches that of the single-phase falling film case with the exception that the compressible quiescient gas is stabilizing. When the liquid film thickness is sufficently thick, the density mode within the thin gas layer is destabilizing. In the general case, over a finite domain, a general stability diagram of film thickness and pressure drop is found. For sufficently large countercurrent pressure drops, the interfacial mode becomes unstable, with the location of the largest deformation found near the liquid inlet.