Stability of a Two-Layer Film Flow of Viscous Fluids down an Inclined Plane

The stability of a two-layer film flow of viscous fluids down an inclined plane is studied by the use of the derivative expansion method. Multiple scales are introduced, and two nonlinear asymptotic equations are derived for the elevation of the free and internal surfaces. The stability of the two-layer flow is governed by four nondimensional parameters of ratios of density, kinematic viscosity, surface tension and layer thickness. For specific case of water and benzene layers, it is shown that, when the heavier fluid (water) is in the lower layer, the two-layer flow is similar to a single fluid flow. On the other hand, when the heavier fluid is in the upper, layer reversion is expected to occur under a certain condition of the film thickness ratio.