Stability of wavy downflow of films calculated by the Navier–Stokes equations

The process of downflow of viscous films on a smooth surface is analyzed theoretically with the use of the full Navier–Stokes equations. The limits of applicability of the asymptotic and integral approaches to the description of waves on falling films are determined. Various nonlinear wavy downflow regimes are calculated in a wide range of the Reynolds and Kapitsa numbers, and stability of these regimes is studied. For low values of the Kapitsa number, the results of the asymptotic approach are demonstrated to be inapplicable almost for all Reynolds numbers. For high values of the Kapitsa number, the solution obtained by the asymptotic method starts to differ significantly from the result obtained by solving the Navier–Stokes equations beginning from moderate Reynolds numbers. For high Reynolds numbers, the wavelength of neutral disturbances is independent of the flow rate of the liquid, and the phase velocity of neutral disturbances is close to the velocity of the free surface. Calculations of nonlinear wavy regimes with moderate Reynolds numbers predict the existence of internal vortex zones. It is shown that there are only a few families of steady traveling solutions (a countable set of different families of such solutions was obtained in calculations by the integral model).