Waves induced by instability in falling films of finite thickness

Abstract

Nonlinear gravity-capillary waves in falling viscous films of finite thickness are here investigated. Mathematical modelling of this problem is based on an evolution system with a single similarity parameter δ, which depends on the mean film thickness, gravity and liquid properties. To investigate spatially periodic structures, we apply the Galerkin procedure to obtain a dissipative dynamical system which contains disturbance wavenumber in addition to the parameter δ. Two-parametric manifold of its limit cycles is numerically constructed. Attracting invariant solutions, limit cycles and invariant torii, developing from different initial data are revealed. Parallel with the analysis of solutions to the dynamical system, the corresponding periodic waves in physical space are analysed and their mechanical meaning is discussed. The dominating waves that correspond to the attracting limit cycles have successfully been compared with experimental data.