Mathematical Notes

, Volume 103, Issue 1–2, pp 145–154 | Cite as

A Priori Estimates of the Solution of the Problem of the Unidirectional Thermogravitational Motion of a Viscous Liquid in the Plane Channel

  • E. N. Cheremnykh


We consider an initial boundary-value problem describing the unidirectional motion of a liquid in the Oberbeck–Boussinesq model in a plane channel with rigid immovable walls on which the temperature distribution is given (or the upper wall is heat-insulated). For this problem, we obtain a priori estimates, find an exact stationary solution, and determine conditions under which the solution converges to its stationary regime.


initial boundary-value problem inverse problem a priori estimate 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    R. B. Birikh, “Thermocapillary convection in a horizontal layer of liquid,” J. Appl. Mech. Tech. Phys. 7 (3), 43–44 (1966).CrossRefGoogle Scholar
  2. 2.
    V. L. Katkov, “Exact solutions of certain convection problems,” Prikl. Mat. Mekh. 32 (3), 482–487 (1968) [J. Appl. Math. Mech. 32, 489–495 (1969)].MATHGoogle Scholar
  3. 3.
    V. V. Pukhnachev, “Nonstationary analogs of the Birikh solution,” Izv. Altaisk. Gos. Univ., No. 1-2, 62–69 (2011).Google Scholar
  4. 4.
    V. K. Andreev, Yu. A. Gaponenko, O. N. Goncharova, and V. V. Pukhnachev, Mathematical Models of Convection, in De Gruyter Stud. Math. Phys. (Walter de Gruyter, Berlin, 2012), Vol. 5.CrossRefGoogle Scholar
  5. 5.
    V. K. Andreev and E. N. Cheremnykh, “A joint creeping motion of three fluids in a flat layer: a priori estimates and convergence to a stationary regime,” Sib. Zh. Ind. Mat. 19 (1), 3–17 (2016).MathSciNetMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Institute of Numerical SimulationRussian Academy of SciencesKrasnoyarskRussia
  2. 2.Siberian Federal UniversityKrasnoyarskRussia

Personalised recommendations