Advertisement

Journal of Mathematical Sciences

, Volume 230, Issue 5, pp 813–817 | Cite as

Exact Solution of the Navier–Stokes Equation Describing Nonisothermal Large-Scale Flows in a Rotating Layer of Liquid with Free Upper Surface

  • K. G. Shvarts
Article
  • 8 Downloads

Abstract

We present an analytic representation of an exact solution of the Navier–Stokes equations that describe flows of a rotating horizontal layer of a liquid with rigid and thermally isolated bottom and a free upper surface. On the upper surface, a constant tangential stress of an external force is given, and heat emission governed by the Newton law occurs. The temperature of the medium over the surface of the liquid is a linear function of horizontal coordinates. We find the solution of the boundary-value problem for ordinary differential equations for the velocity and temperature. and examine its form depending on the Taylor, Grashof, Reynolds, and Biot numbers. In rotating liquid, the motion is helical; account of the inhomogeneity of the temperature makes the helical motion more complicated.

Keywords and phrases

horizontal convection exact solution nonisothermal flow 

AMS Subject Classification

76U05 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    S. N. Aristov and P. G. Frik, “Nonlinear effects of the action of Ekman layers on the dynamics of large-scale vortices in shallow water,” Prikl. Mekh. Tekhn. Fiz., 2, 49–54 (1991).Google Scholar
  2. 2.
    S. N. Aristov and K. G. Schwarz, “New two-dimensional model of large-scale oceanic circulation,” in: Proc. 2nd Int. Conf. “Computer Modelling in Ocean Engineering’91,” Barcelona, Sept. 31–Oct. 4, 1991, Balkema, Rotterdam (1991), pp. 49–54.Google Scholar
  3. 3.
    S. N. Aristov and K. G. Shvarts, “Evolution of wind circulation in a nonisothermal ocean,” Okeanologiya, 30, No. 4, 562–566 (1990).Google Scholar
  4. 4.
    S. N. Aristov and K. G. Shvarts, “On the influence of salinity exchange on the circulation of a fluid in an enclosed basin,” Sov. J. Phys. Oceanogr., 2, No. 4, 293–298 (1991).CrossRefGoogle Scholar
  5. 5.
    S. N. Aristov and K. G. Shvarts, Vortex Flows in Thin Layers of Liquids [in Russian], Kirov (2011).Google Scholar
  6. 6.
    G. Z. Gershuni and E. M. Zhukhovitsky, Convective Stability of Incompressible Liquids [in Russian], Nauka, Moscow (1972).Google Scholar
  7. 7.
    T. M. Haeusser and S. Leibovich, “Pattern formation in the marginally unstable Ekman layer,” J. Fluid Mech., 479, 125–144 (2003).MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    V. F. Kozlov, “A model of two-dimensional vortex motion of a liquid with an entrainment mechanism,” Izv. Ross. Akad. Nauk. Ser. Mekh. Zhidk. Gaza, 6, 49–56 (1992).Google Scholar
  9. 9.
    J. Pedlosky, Geophysical Fluid Dynamics, Springer-Verlag (1987).Google Scholar
  10. 10.
    D. G. Seidov, Modeling of Synoptic and Climatic Variability of the Ocean [in Russian], Gidrometeoizdat, Leningrad (1985).Google Scholar
  11. 11.
    K. G. Shvarts, “On the stability of flows appearing under the action of tangential stresses on the upper surface of a rotating layer of a liquid,” in: Proc. 15th Winter School on the Continuum Mechanics, Vol. 3, Yekaterinburg (2007), pp. 266–269.Google Scholar
  12. 12.
    K. G. Shvarts, Models of Geophysical Fluid Dynamics [in Russian], Perm (2006).Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Perm State UniversityPermRussia

Personalised recommendations