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Math. Model. Nat. Phenom. Vol. 3, No. 1, 2008, pp. 126-148
DOI: 10.1051/mmnp:2008037
Two-Layer Flow with One Viscous Layer in Inclined Channels
O.K. Matar1, G.M. Sisoev2 and C.J. Lawrence31 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
o.matar@imperial.ac.uk
Published online: 17 July 2008
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.
Mathematics Subject Classification. 35Q30, 35Q35, 76D05, 76D08, 76D33, 76D45, 76T10
Key words: slug flows -- interfacial instability -- two-layer flow -- channel flow -- modelling