An analytical comprehensive solution for the superficial waves appearing in gravity-driven flows of liquid films
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Zeitschrift f¨ ur angewandte Mathematik und Physik ZAMP
An analytical comprehensive solution for the superficial waves appearing in gravitydriven flows of liquid films Bruno Pelisson Chimetta and Erick Franklin Abstract. 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. Mathematics Subject Classification. 76E17. Keywords. Gravity-driven flow, Generalized Newtonian fluid, Carreau–Yasuda model, Temporal stability, Asymptotic method.
1. Introduction The study of gravity-driven flows of liquids and suspensions has been the object of interest of many authors through the last decades, not only because of practical engineering applications, such as frictionreducing effects, reactor cooling or coating processes, but also because of different geophysical flows such as mud, glaciers and lava flows. In the case of Newtonian fluids, this class of problems has been extensively studied for almost a century. Among the first studies, Kapitza and Kapitza [1,2] carried out experimental and analytical investigations of Newtonian films flowing in the presence of a vertical wall. Over the last decades, several authors studied falling films by using asymptotic approaches for long and short waves [3–6], while others carried out numerical investigations in order to increase accuracy and obtain new results [7,8]. These works established the foundations of the stability analysis of gravity-driven flows of Newtonian films. However, the rheological behavior of a wide class of fluids cannot be properly described by the Newtonian constitutive equation. To overcome this difficulty, it is possible to use a generalized Newtonian fluid, for which the viscosity η is a function of the shear rate γ. ˙ This class of fluids satisfies the following constitutive equation [9,10]: ˙ γ˙ (1) τ = η(γ) where η(γ) ˙ is a scalar function and γ˙ = |γ|. ˙ To describe the viscosity behavior, there are different models according to the considered fluid, some examples being the power-law [11], Bingham [12] and Cross (Carreau) [13] fluids. For the modeling of a liquid layer in the presence of free su
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