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Nonlinear waves in a sheared liquid film on a horizontal plane at small Reynolds numbers

Published online by Cambridge University Press:  11 November 2024

Kai-Xin Hu Open the ORCID record for Kai-Xin Hu [Opens in a new window] ,
Kang Du  and
Qi-Sheng Chen
Kai-Xin Hu*
Affiliation:
Zhejiang Provincial Engineering Research Center for the Safety of Pressure Vessel and Pipeline, Ningbo University, Ningbo, Zhejiang 315211, PR China Key Laboratory of Impact and Safety Engineering (Ningbo University), Ministry of Education, Ningbo, Zhejiang 315211, PR China
Kang Du
Affiliation:
Zhejiang Provincial Engineering Research Center for the Safety of Pressure Vessel and Pipeline, Ningbo University, Ningbo, Zhejiang 315211, PR China Key Laboratory of Impact and Safety Engineering (Ningbo University), Ministry of Education, Ningbo, Zhejiang 315211, PR China
Qi-Sheng Chen
Affiliation:
School of Engineering Science, University of Chinese Academy of Sciences, Beijing 100190, PR China Key Laboratory of Microgravity, Institute of Mechanics, Chinese Academy of Sciences, Beijing 100190, PR China
*
†Email address for correspondence: [email protected]
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Abstract

The nonlinear waves in a sheared liquid film on a horizontal plate at small Reynolds numbers are examined by theoretical and numerical approaches. The analysis employs the long-wave approximation along with finite difference schemes. The results show that the surface tension can suppress disturbances and prevent the occurrence of singularities. While the film flow is driven by the shear stress on the interface, its instability highly depends on the magnitude and direction of gravity. Specifically, when the direction of gravity is opposite to the wall-normal direction, perturbations are stabilized by gravity. In contrast, when these two directions are the same, the gravitational force is destabilizing, and stationary travelling waves can exist if a balance is reached between the effects of gravity and surface tension. For the steady solitary waves, there are quasi-periodic oscillations occurring between two stationary points, indicating the presence of heteroclinic trajectories. For periodic waves, the evolutions are sensitive to several parameters and initial disturbances, while one steady-state wave exhibits a sine function-like behaviour.

JFM classification

Instability: Nonlinear instability Interfacial Flows (free surface): Thin films Low-Reynolds-number flows: Lubrication theory
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 999 , 25 November 2024 , A14
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Copyright
© The Author(s), 2024. Published by Cambridge University Press

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Waves in a sheared liquid film lying on the underside of the horizontal plane
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