Analytical Study of Inertialess Instability in Three-Layer Film Flow

Source abstract

This paper presents the analytical study of inertialess instability in a three-layer film flow on an incline at a low capillary number.Using the Maple software enabled us to expand the advantages of the analytical approach to study the instability growth rate as a function of the middle-layer viscosity and thickness. Assuming that the capillary number at the surface is a small parameter, we obtained the asymptotic approximation of the instability growth rate. The comparative study of three-layer films and channel flows was conducted using the asymptotic approximation in both the symmetrical case and the case of weak thickness asymmetry.Our focus was on the so-called dominant wave number, for which the growth rate of the instability is at a maximum. We derived the approximations of the dominant wave number and the corresponding maximum instability growth rate, as well as their partial derivatives with respect to the middle-layer viscosity and thickness. The basic properties of the dominant wave number and the corresponding maximum growth rate were determined using the zeros and singular points of the partial derivatives. The obtained results were compared with the experimental data.