Thermocapillary long waves in a liquid film flow. Part 1. Low-dimensional formulation

We consider the dynamics of a thin liquid film falling down a uniformly heated wall. The heating sets up surface tension gradients that induce thermocapillary stresses on the free surface, thus affecting the evolution of the film. We model this thermocapillary flow by using a gradient expansion combined with a Galerkin projection with polynomial test functions for both velocity and temperature fields. We obtain equations for the evolution of the velocity and temperature amplitudes at first- and second-order in the expansion parameter. These equations are fully compatible close to criticality with the Benney long-wave expansion. Models of reduced dimensionality for the evolution of the local film thickness, flow rate and interfacial temperature only, are proposed.