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.
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See Supplementary Material for a additional graphics concerning the base state and critical conditions
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This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001. Bruno Pelisson Chimetta is grateful to CAPES, and Erick de Moraes Franklin would like to express his gratitude to FAPESP (Grant No. 2018/14981-7), CNPq (Grant No. 400284/2016-2) and FAEPEX/UNICAMP (Grant No. 2112/19) for the financial support they provided.
Conflict of interest
The authors declare that they have no conflict of interest.
This study was funded by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001, Fundação de Amparo à Pesquisa do Estado de São Paulo - FAPESP (Grant No. 2018/14981-7), Conselho Nacional de Desenvolvimento Científico e Tecnológico - CNPq (Grant No. 400284/2016-2) and Fundo de Apoio ao Ensino, Pesquisa e Extensão da Unicamp - FAEPEX/UNICAMP (Grant No. 2112/19).
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The original version of this article was revised: Eq. (23) was incorrectly published and it has been corrected in this version.
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Chimetta, B.P., Franklin, E. An analytical comprehensive solution for the superficial waves appearing in gravity-driven flows of liquid films. Z. Angew. Math. Phys. 71, 122 (2020). https://doi.org/10.1007/s00033-020-01349-x
- Gravity-driven flow
- Generalized Newtonian fluid
- Carreau–Yasuda model
- Temporal stability
- Asymptotic method
Mathematics Subject Classification