We analytically solve the problem of the evolution of small-amplitude waves in a uniform flow of a viscous fluid down an inclined plane. The flow is described in a hydraulic approximation. The flow is supposed to be convectively unstable, and the waves arise as a result of an instantaneous external point disturbance. The solution is presented as a Fourier integral to which the steepest descent method is applied twice. The asymptotics of the growing waves is found analytically as a function of two spatial coordinates and time. We show that the region of growing perturbations is a segment of a circle, that its linear dimensions grow linearly with time, and that it is defined by the characteristics of a system of Saint-Venant differential equations.
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March 2022
Research Article|
March 23 2022
Asymptotic behavior of localized disturbance in a viscous fluid flow down an incline
A. Kulikovskii;
A. Kulikovskii
a)
1
Steklov Mathematical Institute of Russian Academy of Sciences
, Moscow 119991, Russia
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J. Zayko
J. Zayko
b)
2
Institute of Mechanics, Lomonosov Moscow State University
, Moscow 119192, Russia
b)Author to whom correspondence should be addressed: [email protected]
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a)
Electronic mail: [email protected]
b)Author to whom correspondence should be addressed: [email protected]
Physics of Fluids 34, 034119 (2022)
Article history
Received:
December 18 2021
Accepted:
March 02 2022
Citation
A. Kulikovskii, J. Zayko; Asymptotic behavior of localized disturbance in a viscous fluid flow down an incline. Physics of Fluids 1 March 2022; 34 (3): 034119. https://doi.org/10.1063/5.0082782
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