Two-Layer Flow with One Viscous Layer in Inclined Channels

¹ Department of Chemical Engineering, Imperial College London South Kensington Campus, SW7 2AZ, UK
² School of Mathematics, University of Birmingham, Edgbaston Birmingham, B15 2TT, UK
³ Institutt for Energiteknikk, P.O. Box 40, 2027 Kjeller, Norway

Corresponding author: o.matar@imperial.ac.uk

Abstract

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