Short-Wave Instability and Formation of Nonlinear Waves in a Thin Layer of Flowing Viscous Liquid

Abstract

The flow of a viscous liquid film over a solid surface deviating slightly from the horizon is considered. Linear and nonlinear waves excited by hydrodynamic instability are investigated. A new system of differential equations for the average thickness, local fluid flow rate, and friction on a solid surface is derived. The stability of a uniform flow with respect to small periodic wave disturbances in the longitudinal coordinate is investigated. It is shown that there is a finite instability interval in the long-wave range and an unlimited instability interval in the wave number in the short-wave range. It is established that with a spectral representation of solutions by a finite number of terms of the Fourier series, the existence of a regular nonlinear wave regime is possible over a large time interval. The regular solution agrees with experimental observations on the boundaries of linear instability and on the parameters of the nonlinear wave regime.