Hydrodynamics of wave flow



In this paper a method for obtaining periodical solutions of the nonlinear equations of motion is proposed. The method is applied to the case of a falling liquid film in wave motion. It consists in a triple series expansion; the first is a Taylor expansion, with respect to the distance y1 from the free surface; a second one, representing the periodicity condition, is a Fourier expansion with respect to the variable zk(xct), and a third one is a Taylor expansion with respect to the amplitude 2iϕ1. The calculation of the different coefficients is made easy by the fact that the algebraic equations obtained are linear and do not simulataneously contain all the unknowns. This allows the performance of the computation step by step in increasing order of the powers of ϕ1. The periodicity conditions allows the determination of all physical quantities as functions of one of them. The amplitude |2iϕ11| was selected as the parameter.

The existence of a dimensionless quantity for the wave flow is outlined. Arguments are adduced in support of the fact that the amplitude| 2iϕ1| depends only on ψ and a universal curve |2iϕ1| vs. ψ is plotted on the basis of experimental data. Theoretical equations for the wave length, the wave velocity and the film thickness as a function of ψ are established. There is good agreement between the theoretical equations and experiment.