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
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
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.
Papers presented to the 12th International Conference on Heat Transfer, Fluid Mechanics and Thermodynamics, Costa de Sol, Spain on 11-13 July 2016.