Two-Layer Flow with One Viscous Layerin Inclined Channels

Abstract

We study pressure-driven, two-layer flow in inclined channels with high density andviscosity contrasts. We use a combination of asymptotic reduction, boundary-layer theory and theKarman-Polhausen approximation to derive evolution equations that describe the interfacial dynamics.Two distinguished limits are considered: where the viscosity ratio is small with densityratios of order unity, and where both density and viscosity ratios are small. The evolution equationsaccount for the presence of inertia, gravity, capillarity and viscous retardation; attention isrestricted to situations in which the flow is laminar. The results of our linear stability analysis andnumerical simulations indicate that the flow is destabilised by positive channel inclination in thestably stratified case. The dependence of the nonlinear wave dynamics on system parameters isalso explored.