Nelineinaya Dinamika [Russian Journal of Nonlinear Dynamics]
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Nelin. Dinam., 2015, Volume 11, Number 1, Pages 89–97 (Mi nd466)  

This article is cited in 4 scientific papers (total in 4 papers)

The exact solutions of the problem of a viscous fluid flow in a cylindrical domain with varying radius

Denis V. Knyazeva, Ilia Yu. Kolpakovb

a Institute of Continuous Media Mechanics UB RAS, Korolev St. 1, Perm, 614013, Russia
b Perm National Research Polytechnic University, Komsomolsky Ave. 29, Perm, 614990 Russia

Abstract: In the frameworks of a class of exact solutions of the Navier–Stokes equations with linear dependence of part the speed components on one spatial variable the axisymmetrical nonselfsimilar flows of viscous fluid in the cylindrical area which radius changes over the time under some law calculated during the solution are considered. The problem is reduced to two-parametrical dynamic system. The qualitative and numerical analysis of the system allowed to allocate three areas on the phase plane corresponding to various limit sizes of a pipe radius: radius of a pipe and stream velocity tend to infinity for finite time, the area of a cross section of the cylinder tend to zero during a finite time span, radius of the tube infinitely long time approaches to a constant value, and the flow tend to the state of rest. For a case of ideal fluid flow the solution of the problem is obtained in the closed form and satisfying the slip condition.

Keywords: Navier–Stokes equations, exact solutions, pipe flow.

Funding Agency Grant Number
Far Eastern Branch of the Russian Academy of Sciences 12-С-1-1006
Ural Branch of the Russian Academy of Sciences 12-С-1-1006
Siberian Branch of Russian Academy of Sciences 12-С-1-1006
Russian Foundation for Basic Research 12-01-00023

Full text: PDF file (305 kB)
References: PDF file   HTML file
UDC: 532.516
MSC: 76D05
Received: 15.04.2014
Revised: 16.12.2014

Citation: Denis V. Knyazev, Ilia Yu. Kolpakov, “The exact solutions of the problem of a viscous fluid flow in a cylindrical domain with varying radius”, Nelin. Dinam., 11:1 (2015), 89–97

Citation in format AMSBIB
\by Denis~V.~Knyazev, Ilia~Yu.~Kolpakov
\paper The exact solutions of the problem of a viscous fluid flow in a cylindrical domain with varying radius
\jour Nelin. Dinam.
\yr 2015
\vol 11
\issue 1
\pages 89--97

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    This publication is cited in the following articles:
    1. M. S. Deryabina, S. I. Martynov, “Periodicheskoe techenie vyazkoi zhidkosti s zadannym gradientom davleniya i temperatury”, Nelineinaya dinam., 14:1 (2018), 81–97  mathnet  crossref  elib
    2. A. S. Mozokhina, S. I. Mukhin, “Nekotorye tochnye resheniya zadachi o techenii zhidkosti v sokraschayuschemsya elastichnom sosude”, Matem. modelirovanie, 31:3 (2019), 124–140  mathnet  crossref  elib
    3. V. V. Privalova, E. Yu. Prosviryakov, M. A. Simonov, “Nonlinear Gradient Flow of a Vertical Vortex Fluid in a Thin Layer”, Rus. J. Nonlin. Dyn., 15:3 (2019), 271–283  mathnet  crossref  mathscinet
    4. V. A. Galkin, A. O. Dubovik, “O modelirovanii sloistogo techeniya vyazkoi provodyaschei zhidkosti v oblasti, izmenyayuscheisya vo vremeni”, Matem. modelirovanie, 32:4 (2020), 31–42  mathnet  crossref
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