ИСТИНА

We consider a thin film of viscous fluid flow down a plane with sinusoidal corrugations. By use of integral boundary layer approach evolution equations are derived. The main goal of the work is to analyse the stability of steady-state flow. For the stability analysis, we have an fourth-order ordinary differential equation with periodic coefficients. Floquet theory is a main tool to check existence of a growing disturbance. It was shown that one eigenvalue corresponds to a growing solution for any parameters, but this solution has no physical meaning. This result leads to the idea that we have to analyses how the eigenvalues move at a complex plane while parameters of the system, mainly amplitude and wavenumber of corrugation change. For small ratio a between the corrugation amplitude and mean film thickness, we use an expansion onto series of a and find an approximate analytical solution for corrections of the growth rate and phase velocity. These corrections are proportional to the second power of amplitude and the approximate theory coincides with numerical simulations within few percents for amplitude value up to 20 The corrugations mostly have stabilizing effect. There is a resonance between cor- rugations and natural instability of the film which leads to the stongest stabilisation. This effect takes place for relatively small values of corrugation period: disturbances with such values of wavelength are damping in a film at a flat wall. By the developed approach, we obtained corrections for critical Reynolds number at inclined corrugated plane and proved the existence of the ”islands of instability” 1 . We compare our results of analytical and numerical solutions with experiments 23 . We also checked the results of linear stability analysis by simulations of nonlinear waves evolution, considering nonstationary problem on a large space interval. Some disturbance indeed become damping. If small disturbances are unstable, nonlinear waves are developed and the corrugations have a minor effect. If the wavelength of the disturbance and the corrugation are close long-wave structures like beatings appear. The work is supported by Russian Foundation for Basic Research (18-01-00762) and Russian Federation President Grants Council (MK-1798.2017.1).