An analytical comprehensive solution for the superficial waves appearing in gravity-driven flows of liquid films

This paper is devoted to analytical solutions for the base flow and temporal stability of a liquid film driven by gravity over an inclined plane when the fluid rheology is given by the Carreau-Yasuda model, a general description that applies to different types of fluids. In order to obtain the base state and critical conditions for the onset of instabilities, two sets of asymptotic expansions are proposed, from which it is possible to find four new equations describing the reference flow and the phase speed and growth rate of instabilities. These results lead to an equation for the critical Reynolds number, which dictates the conditions for the onset of the instabilities of a falling film. Different from previous works, this paper presents asymptotic solutions for the growth rate, wavelength and celerity of instabilities obtained without supposinga priorithe exact fluid rheology, being, therefore, valid for different kinds of fluids. Our findings represent a significant step toward understanding the stability of gravitational flows of non-Newtonian fluids.