On Marangoni drying: Nonlinear kinematic waves in a thin film

In the field of industrial drying, a recent innovation has exploited the occurrence of Marangoni effects in such a way that the resultant free-surface flow enhances the drying process. To this end, alcohol vapour, soluble in water, is introduced above a drying film and as a result of diffusion through the air and water phases a favourable concentration gradient gives rise to the required shear flow. We consider here a simple process driven by this mechanism, and by means of asymptotic simplification and the concepts of singular perturbation theory a leading-order approximation is obtained in which the alcohol concentration in the water is a specified function of space and time. The evolution of the free surface thus reduces to a single nonlinear partial differential equation of a similar form to the Korteweg-de Vries and Burgers equations, higher-derivative terms corresponding to surface tension and gravity effects. Numerical solutions of this equation are obtained and are compared to the application of first order nonlinear kinematic wave theory with corresponding shock solutions.