Integral Model of Wavy Flow Regimes of Thin Viscous Liquid Layer Taking into Account Second Order Terms and Influence of the Gas Flow

Abstract
In this paper, a model is obtained which takes into account the influence of second-order terms of smallness in the long-wavelength parameter, as well as tangential and normal stresses from the gas flow. The linear stability of the obtained system of equations was studied and the results were compared with the exact solution of the Orr-Sommerfeld equation. It is shown that integral models differ from the exact solution even at low Reynolds numbers, which is due to the fact that the longitudinal velocity profile differs from the semi-parabolic one. The evolution of nonlinear waves was simulated for both a free falling and a liquid film entrained in a gas stream. The profiles of stationary-traveling waves for different Reynolds numbers are obtained. A comparison of the profiles of waves obtained by different models has been carried out. It is shown that the differences in the results obtained by different models are insignificant, and the interaction with the gas flow leads to a decrease in the wavelength of the disturbances.

Keywords
thin liquid layer, longwave approximation, linear analysis, nonlinear waves
Acknowledgements
The work was financially supported by the grant of the RFBR No. 18-31-00269 mol_a  
УДК 532.516

Integral Model of Wavy Flow Regimes of Thin Viscous Liquid Layer Taking into Account Second Order Terms and Influence of the Gas Flow