Free surface flows down a slope occur in various real-life scenarios, such as civil engineering, industry, and natural hazards. Unstable waves can develop at the free surface when inertia is sufficiently strong, indicated by the Reynolds number exceeding a critical value. Although this instability has been investigated for specific fluids with different rheologies, a common framework is still lacking to facilitate comparison among the various models. In this study, we investigate the linear stability of a generalized Newtonian fluid, where the viscosity eta ( gamma (center dot) ) remains unspecified. We meticulously construct new dimensionless quantities to minimize dependence on the rheology, and subsequently derive the Orr-Sommerfeld equation of stability for any generalized Newtonian fluid, which has never been done before. We conduct a long-wave expansion and generate a novel analytical expression for the wave celerity, along with the critical Reynolds number. The originality in this study is that the analytical expressions obtained are valid for any rheology, and are easy to compute from a rheological measurement or from a base flow profile measurement. These results are subsequently scrutinized using various shear-thinning, shear-thickening, and viscoplastic rheology models. They exhibit excellent agreement with experimental or numerical data as well as theoretical findings from existing literature. Furthermore, the novel analytical expressions enable a much more comprehensive investigation into the impact of rheology on stability. While our approach does not encompass singular or non-monotonous rheology, the analytical expressions derived from the long-wave expansion exhibit remarkable resilience and they continue to accurately predict both the wave speed and the instability threshold in such cases.
