Long-wave linear stability theory for two-fluid channel flow including compressibility effects

We present the linear stability of the laminar flow of an immiscible system of a compressible gas and incompressible liquid separated by an interface with large surface tension in a thin inclined channel. The flow is driven by an applied pressure drop and gravity. Following the air-water case, which is found in a variety of engineering systems, the ratio of the characteristic values of the gas and liquid densities and viscosities are assumed to be disparate. Under the lubrication approximation, and assuming ideal gas behaviour and isothermal conditions, this approach leads to a coupled non-linear system of partial differential equations describing the evolution of the interface between the gas and the liquid and the streamwise density distribution of the gas. This system also includes the effects of viscosity stratification, inertia, shear and capillarity. A linear stability analysis that allows for physically relevant non-zero pressure-drop base state is then performed. In contrast to the zero-pressure drop case which is amenable to the classical normal-mode approach, this configuration requires numerically solving a boundary-value problem for the gas density and interfacial deviations from the base state in the streamwise coordinate. We find that the effect of the gas compressibility on the interfacial stability in the limit of vanishingly small wavenumber is destabilizing, even for Stokes flow in the liquid. However, for finite wavenumber disturbances, compressibility may have stabilizing effects. In this regime, sufficient shear is required to destabilize the flow.