  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.
