  Gravity-driven flows of liquid films are frequent in nature and industry,
such as in landslides, lava flow, cooling of nuclear reactors, and coating
processes. In many of these cases, the liquid is non-Newtonian and has
particular characteristics. In this paper, we analyze numerically the temporal
stability of films of non-Newtonian liquids falling by gravity, on the onset of
instability. The liquid flows over an incline, where surface waves appear under
certain conditions, and we do not fix a priori its rheological behavior. For
that, we made used of the Carreau-Yasuda model without assigning specific
values to its constants, and we compute general stability solutions. The
numerical strategy is based on expansions of Chebyshev polynomials for
discretizing the Orr-Sommerfeld equation and boundary conditions, and a
Galerkin method for solving the generalized eigenvalue problem. In addition, an
Inverse Iteration method was implemented to increase accuracy and improve
computational time. The result is a robust and light numerical tool capable of
finding the critical conditions for different types of fluids, which we use to
analyze some key fluids. We show that the outputs of the general code match
previous solutions obtained for specific computations. Besides increasing our
knowledge on surface-wave instabilities in non-Newtonian liquids, our findings
provide a new tool for obtaining comprehensive solutions on the onset of
instability.
