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

TL;DR

This paper provides analytical solutions for the stability of gravity-driven liquid films with non-Newtonian rheology, offering new equations for critical conditions and instability characteristics applicable to various fluid types.

Contribution

It introduces asymptotic solutions for instability parameters without assuming specific fluid rheology, advancing understanding of non-Newtonian film flow stability.

Findings

Derived equations for critical Reynolds number and instability onset.

Provided asymptotic expressions for growth rate, wavelength, and phase speed.

Applicable to a wide range of non-Newtonian fluids.

Abstract

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 supposing a priori the 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.