A three-equation model for thin films down an inclined plane

Abstract

We derive a new model for thin viscous liquid films down an inclined plane. With an asymptotic expansion in the long-wave limit, the Navier–Stokes equations and the work–energy theorem are averaged over the fluid depth. This gives three equations for the mass, momentum and energy balance which have the mathematical structure of the Euler equations of compressible fluids with relaxation source terms, diffusive and capillary terms. The three variables of the model are the fluid depth, the average velocity and a third variable called enstrophy, related to the variance of the velocity. The equations are numerically solved by classical schemes which are known to be reliable and robust. The model gives satisfactory results both for the neutral stability curves and for the depth profiles of wavy films produced by a periodical forcing or by a random noise perturbation. The numerical calculations agree fairly well with experimental measurements of Liu & Gollub (Phys. Fluids, vol. 6, 1994, pp. 1702–1712). The calculation of the wall shear stress below the waves indicates a flow reversal at the first depth minimum downstream of the main hump, in agreement with experiments of Tihon et al. (Exp. Fluids, vol. 41, 2006, pp. 79–89).