Stability of the wavy film falling down a vertical plate: The DNS computations and Floquet theory

The paper is devoted to a theoretical analysis of nonlinear two-dimensional waves using the Navier–Stokes equations in their full statement. We computed the steady-state traveling waves and carried out an analysis of their linear stability using the Floquet’s theory. We carried out the linear stability analysis with respect to all possible two-dimensional disturbances on the plane of two parameters (the wavelength L of the nonlinear solution and Re/Ka) for different values of Ka. We obtained that the solutions of the Navier–Stokes equations at relatively small values of the Kapitza number ( Ka ⩽ 2 ) form a wide region of the parameters (L, Re/Ka) where they are stable with respect to all two-dimensional disturbances. We found that the region is splitted into several zones of such stability with the Kapitza number increasing. At the Kapitza number Ka ⩾ 3.5 , we obtained many narrow zones of the stability with respect to arbitrary two-dimensional disturbances at small and moderate values of Re/Ka. In the regions where the nonlinear solutions are stable with respect to arbitrary two-dimensional disturbances, we obtained the “long optimal” wave that demonstrates a minimal value of the averaged film thickness among the nonlinear solutions with different wavelength at constant values of Re/Ka and Ka.