Dynamics of a vertically falling film in the presence of a first-order chemical reaction

The evolution of a vertically falling film in the presence of a simple first-order (exothermic or endothermic) chemical reaction is considered. The heat of reaction sets up surface tension gradients that induce thermocapillary stresses on the free-surface, thus affecting the evolution of the film. By using a long-wave expansion of the equations of motion and associated boundary conditions, we derive a nonlinear partial differential equation of the evolution type for the local film thickness. We demonstrate that, when the surface tension is an increasing function of temperature an exothermic reaction has a stabilizing effect on the free surface while an endothermic reaction is destabilizing. We construct bifurcation diagrams for permanent solitary waves and show that, in all cases the solution branches exhibit limit points and multiplicity with two branches, a lower branch and an upper branch. Time-dependent computations of the free-surface evolution equation show that the system always approaches a train of coherent structures that resemble the lower branch solitary waves. We also examine the absorption characteristics through the interface and we demonstrate that an endothermic reaction enhances absorption and mass transport. The opposite is true for an exothermic reaction.