Spatial and time evolution of non linear waves in falling liquid films by the harmonic expansion method with predictor-corrector integration

Abstract

Falling film flows in vertical or inclined planes, and pipes, are present in the energy and chemical industry (Chemical reactors, evaporators, condensers…). The occurrence of waves in these falling films is of relevance because it enhances the heat and mass transfer in comparison with a flat film. Perturbation theory can be applied to the Navier-Stokes (NS) equations expressing the velocity and the pressure in terms of an order formal parameter representing the smallness of the stream wise spatial derivative. Normally good results are obtained for this kind of problems solving the first order NS equations. In the present work we use the integral approach method and we expand the velocity profile of the falling liquid in a complete orthogonal set of harmonic functions satisfying the boundary conditions of the NS problem in first order approximation of the formal expansion. The present model does not assume self-similar profile of the velocity and its convergence to the solution is good with few harmonics. The problem is discretized by means of a uniform grid. Then the partial differential equations are integrated over the length of an arbitrary node. Proceeding in this way we have obtained a set of coupled ordinary differential equation system (ODES) for the harmonics of the flow rate and the film thickness at each grid node The resulting coupled ODES is integrated by a semi-implicit predictor-corrector method of the Adams-Moulton type that converges, with one iteration, at each time step. The method predicts well the experimental data on the evolution of the waves with time, the height of the waves, the wave separation, and the wave profiles for different experimental conditions. Providing a physical understanding of the non-linear wave phenomena produced in falling films.