Stability and bifurcations of the wavy film flow down a vertical plate: the results of integral approaches and full-scale computations

Abstract

This paper is devoted to a theoretical analysis of nonlinear two-dimensional waves using both the Navier–Stokes equations in their full statement and two integral approaches: Shkadov's approach and ‘the regularized integral model’. We found the steady-state travelling waves and carried out an analysis of their linear stability and bifurcations using the Floquet theory. We found that the solutions of the Navier–Stokes equations are qualitatively different from the solutions of Shkadov's integral approach starting from some values of the Kapitza number Ka. It is found that the solutions of all models considered here have an internal vortex at moderate Reynolds numbers Re. A linear stability analysis with respect to the periodic disturbances of the same wavelength L as a period of the nonlinear solution allows us to calculate the bifurcation lines of the nonlinear waves on the plane of two parameters (wavelength L and Re/Ka) for different values of Ka. These lines form a multi-fold and multi-sheet surface where we can compute the different types of solutions at one set of parameters by using the continuation principle and starting the computations with small values of Re/Ka. We found that most of the solutions are unstable.