Long-waves on thin viscous liquid film derivation of model equations

Abstract

The present paper is devoted to a consistent asymptotic derivation of model equations for two-dimensional nonlinear (2DNL) waves on a viscous dilatable liquid film. Various approximate equations describing the NL evolution in time - space of the thickness of the film are obtained . More precisely, for small amplitudes in the long wave approximation, we consider three cases : high Reynolds number boundary layer regime, moderate Reynolds number of Poiseuille type one, and low Reynolds number second Poiseuille type regime. These various situations arise from the smallness of the thin film aspect ratio parameter E and are distinguished from each other by selecting, as far as order of magnitude are concerned, some dimensionless parameters with respect to E . They are Reynolds, Fronde, Marangoni, Weber, Grashof numbers as well as a Bict one which is assumed, throughout, to be of order unity. For the high Reynolds number case we obtain an equation for the thickness h(t,x) of the film, which is similar to equation previously derived by Oron and Rosenau 1992. From this last evolution equation for h(t,x) a new limiting process, for small amplitude (long) waves, leads to a modified (by Boussinesq effects ) Kuramoto- Sivashinsky (KS) equation. The two other regimes do not lead to a unique consistent equation for h(t,x) unless we focus on small amplitude (long) waves; one finds a KS or a KS with Korteweg-de-Vries (KdV) dispersive term equations. The role of buoyancy is quite different in the various regimes considered; for the low Reynolds number regime the buoyancy coupling effect is “effectively” significant in the KS - KdV model equation.