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.

本文专门研究当Carreau-Yasuda模型给出流体流变性时，由重力作用在倾斜平面上驱动的液膜的基本流动和时间稳定性的解析解，该一般描述适用于不同类型的流体。为了获得不稳定性开始的基本状态和临界条件，提出了两组渐近展开式，从中可以找到四个新的方程式，它们描述了参考流以及不稳定性的相速度和增长率。这些结果导致了临界雷诺数的方程式，该方程式决定了降膜不稳定开始的条件。与以前的工作不同，本文提出了在不假设的情况下获得的不稳定性的增长速度，波长和速度的渐近解。因此，先验的是精确的流体流变性，因此对不同种类的流体都有效。我们的发现代表了迈向理解非牛顿流体重力流稳定性的重要一步。