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Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2023, Volume 322, Pages 146–156
DOI: https://doi.org/10.4213/tm4346
(Mi tm4346)
 

On Waves on the Surface of an Unstable Layer of a Viscous Fluid Flowing Down a Curved Surface

A. G. Kulikovskiia, J. S. Zaykob

a Steklov Mathematical Institute of Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991 Russia
b Institute of Mechanics, Lomonosov Moscow State University, Michurinskii pr. 1, Moscow, 119192 Russia
References:
Abstract: We consider the evolution of linear waves of small perturbations of an unstable flow of a viscous fluid layer over a curved surface. The source of perturbations is assumed to be given by initial conditions defined in a small domain (in the limit, in the form of a $\delta $-function) or by an instantaneous localized external impact. The behavior of perturbations is described by hydrodynamic equations averaged over the thickness of the layer, with the gravity force and bottom friction taken into account (Saint-Venant equations). We study the asymptotic behavior of one-dimensional perturbations for large times. The inclination of the surface to the horizon is defined by a slowly varying function of the spatial variable. We focus on the perturbation amplitude as a function of time and the spatial variable. To study the asymptotics of perturbations, we use a simple generalization of the well-known method, based on the saddle-point technique, for finding the asymptotics of perturbations developing against a uniform background. We show that this method is equivalent to the one based on the application of the approximate WKB method for constructing solutions of differential equations. When constructing the asymptotics, it is convenient to assume that $x$ is a real variable and to allow time $t$ to take complex values.
Keywords: linear waves, fluid layer, flow over a surface, instability, asymptotics, saddle-point method, WKB method.
Funding agency Grant number
Russian Science Foundation 20-11-20141
Ministry of Science and Higher Education of the Russian Federation МК-4090.2022.4
The work of A. G. Kulikovskii (statement of the problem and description of general approaches to the solution of similar problems; Sections 1–4) was supported by the Russian Science Foundation under grant no. 20-11-20141, https://rscf.ru/en/project/20-11-20141/, and performed at the Steklov Mathematical Institute of Russian Academy of Sciences. The work of J. S. Zayko (specification of the problem as applied to the study of perturbations in a layer of flowing fluid, and calculation of the flow; Sections 4 and 5) was supported by a grant of the President of the Russian Federation for young scientists (project no. MK-4090.2022.4) and performed at the Institute of Mechanics of the Lomonosov Moscow State University.
Received: February 25, 2023
Revised: March 20, 2023
Accepted: April 18, 2023
English version:
Proceedings of the Steklov Institute of Mathematics, 2023, Volume 322, Pages 140–150
DOI: https://doi.org/10.1134/S0081543823040120
Bibliographic databases:
Document Type: Article
UDC: 532.5
Language: Russian
Citation: A. G. Kulikovskii, J. S. Zayko, “On Waves on the Surface of an Unstable Layer of a Viscous Fluid Flowing Down a Curved Surface”, Modern Methods of Mechanics, Collected papers. On the occasion of the 90th birthday of Academician Andrei Gennad'evich Kulikovskii, Trudy Mat. Inst. Steklova, 322, Steklov Math. Inst., Moscow, 2023, 146–156; Proc. Steklov Inst. Math., 322 (2023), 140–150
Citation in format AMSBIB
\Bibitem{KulZay23}
\by A.~G.~Kulikovskii, J.~S.~Zayko
\paper On Waves on the Surface of an Unstable Layer of a Viscous Fluid Flowing Down a Curved Surface
\inbook Modern Methods of Mechanics
\bookinfo Collected papers. On the occasion of the 90th birthday of Academician Andrei Gennad'evich Kulikovskii
\serial Trudy Mat. Inst. Steklova
\yr 2023
\vol 322
\pages 146--156
\publ Steklov Math. Inst.
\publaddr Moscow
\mathnet{http://mi.mathnet.ru/tm4346}
\crossref{https://doi.org/10.4213/tm4346}
\transl
\jour Proc. Steklov Inst. Math.
\yr 2023
\vol 322
\pages 140--150
\crossref{https://doi.org/10.1134/S0081543823040120}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85180263161}
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    Труды Математического института имени В. А. Стеклова Proceedings of the Steklov Institute of Mathematics
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