Electronic Scientific Journal
Diagnostics, Resource and Mechanics 
         of materials and structures


advanced search

IssuesAbout the JournalAuthorContactsNewsRegistration

All Issues

All Issues
2024 Issue 2
2024 Issue 1
2023 Issue 6
2023 Issue 5
2023 Issue 4
2023 Issue 3
2023 Issue 2
2023 Issue 1
2022 Issue 6
2022 Issue 5
2022 Issue 4
2022 Issue 3
2022 Issue 2
2022 Issue 1
2021 Issue 6
2021 Issue 5
2021 Issue 4
2021 Issue 3
2021 Issue 2
2021 Issue 1
2020 Issue 6
2020 Issue 5
2020 Issue 4
2020 Issue 3
2020 Issue 2
2020 Issue 1
2019 Issue 6
2019 Issue 5
2019 Issue 4
2019 Issue 3
2019 Issue 2
2019 Issue 1
2018 Issue 6
2018 Issue 5
2018 Issue 4
2018 Issue 3
2018 Issue 2
2018 Issue 1
2017 Issue 6
2017 Issue 5
2017 Issue 4
2017 Issue 3
2017 Issue 2
2017 Issue 1
2016 Issue 6
2016 Issue 5
2016 Issue 4
2016 Issue 3
2016 Issue 2
2016 Issue 1
2015 Issue 6
2015 Issue 5
2015 Issue 4
2015 Issue 3
2015 Issue 2
2015 Issue 1






A. V. Gorshkov, E. Yu. Prosviryakov


DOI: 10.17804/2410-9908.2021.4.34-49

The paper considers the convective flow of a viscous incompressible fluid over a rotating surface. It studies the angle between the fluid velocity vector in the upper layer and the temperature gradient vector on the free surface. For the study, an analytical solution to the Oberbeck–Boussinesq equations is constructed, which describes the stratified Ekman flow taking into account two components of the Coriolis force. The temperature gradient and the conditions of Marangoni thermocapillary convection are set at the upper (free) boundary, and the condition of fluid adhesion is set on the lower (solid) boundary. The representation of velocities in the form of linear functions of horizontal coordinates is used. It is shown that, when the flow depth tends to infinity, the angle between the upper layer fluid velocity vector and the temperature gradient vector tends to π/2.

Keywords: analytical solution, thermocapillary convection, Ekman flow


  1. Pedlosky J. Geophysical fluid dynamics, Berlin, New York, Springer, 1987.
  2. Gill A.E. Atmosphere-Ocean Dynamics, New York, Academic Press, 1982.
  3. Ekman V.W. On the influence of the Earth’s rotation on ocean-currents. Ark. Mat. Astron. Fys., 1905, vol. 2, No. 11, pp. 1–52.
  4. Aristov S.N., Knyazev D.V., Polyanin A.D. Exact solutions of the Navier-Stokes Equations with the linear dependence of velocity components on two space variables. Theoretical Foundations of Chemical Engineering, 2009, vol. 43, No. 5, pp. 642–662. DOI: 10.1134/S0040579509050066.
  5. Burmasheva N.V., Prosviryakov E.Yu. Exact solution of Navier–Stokes equations describing spatially inhomogeneous flows of a rotating fluid. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2020, vol. 26, No. 2, pp. 79–87. (In Russian).
  6. Burmasheva N.V., Prosviryakov E.Yu. A class of exact solutions for two–dimensional equations of geophysical hydrodynamics with two Coriolis parameters. The Bulletin of Irkutsk State University. Series Mathematics, 2020, vol. 32, pp. 33–48. DOI: 10.26516/1997-7670.2020.32.33. (In Russian).
  7. Felzenbaum A.I. Teoreticheskie osnovy i metody rascheta ustanovivshikhsya morskikh techeniy [Theoretical Foundations and Methods for Calculating Steady Sea Currents]. AN SSSR Publ., 1960, 127 p. (In Russian).
  8. Dolgansky F.V. Lektsii po geophizicheckoy gidrodinamike [Lectures on Geophysical Hidrodynamics]. Мoscow, IVM RAN Publ, 2006, 378 p. (In Russian).
  9. Korotaev G.K., Mikhaylova E.N., Shapiro N.B. Teoriya ekvatorialnykh protivotecheniy v Mirovom okeane [Theory of Equatorial Countercurrents in the World Ocean]. Kiev, Nauk. Dumka Publ., 1986, 208 p. (In Russian).
  10. Zyryanov V.N. Teoriya ustanovivshchikhsya okeanicheskikh techeniy [Theory of Steady Ocean Currents]. Leningrad, Gidrometeoizdat Publ., 1985. (In Russian).
  11. Aristov S.N., Prosviryakov E.Yu. Aristov S.N., Prosviryakov E.Y. On laminar flows of planar free convection. Rus. J. Nonlin. Dyn., 2013, vol. 9, No. 4, pp. 651–657. DOI: 10.20537/nd1304004. (In Russian).
  12. Aristov S.N., Shvarts K.G. Vikhrevye techeniya advektivnoy prirody vo vrashchayushchemsya sloe zhidkosti [Vortical Flows of Advective Nature in a Rotating Fluid Layer]. Perm, Perm. Gos. Univ. Publ., 2006, 154 p. (In Russian).
  13. Aristov S.N., Shvarts K.G. Vikhrevye techeniya v tonkikh sloyakh zhidkosti [Vortical Flows in Thin Fluid Layers]. Kirov, VyatGU Publ., 2011. (In Russian).
  14. Aristov S.N., Shvarts K.G. Advective flow in a rotating liquid film. Journal of Applied Mechanics and Technical Physics, 2016, vol. 57, No. 1, pp. 188–194. DOI: 10.1134/S0021894416010211.
  15. Ingel L.Kh., Aristov S.N. The class of exact solutions of nonlinear problems on thermal circulation associated with volumetric heat release in the atmosphere. Tr. In-ta Eksperim. Meteorol., 1996, No. 27 (162), pp. 142–157. (In Russian).
  16. Gorshkov A.V., Prosviryakov E.Yu. Convective flow in the solid rotation of a viscous incompressible fluid. AIP Conference Proceedings, 2017, vol. 1915, 040020. DOI: 10.1063/1.5017368.
  17. Gorshkov A.V., Prosviryakov E.Yu. Ekman Convective Layer Flow of a Viscous Incompressible Fluid. Izvestiya, Atmospheric and Oceanic Physics, 2018, vol. 54, No. 2, pp. 189–195. DOI: 10.1134/S0001433818020081.
  18. Gorshkov A.V. Prosviryakov E.Yu. Analytical Study of the Ekman Angle for the Isothermal Flow of a Viscous Incompressible Fluid in View of the Navier Boundary Condition. AIP Conference Proceedings, 2020, vol. 2315, 020018. DOI: 10.1063/5.0036889.
  19. Constantin A., Johnson R.S. Atmospheric Ekman flows with variable eddy viscosity. Boundary-Layer Meteorology, 2019, vol. 170, pp. 395–414. DOI: 10.1007/s10546-018-0404-0.
  20. Shrira V.I., Almelah R.B. Upper-ocean Ekman current dynamics: a new perspective. Journal of Fluid Mechanics, 2020, vol. 887, A24. DOI: 10.1017/jfm.2019.1059.
  21. Fečkan M., Guan Y., O’Regan D., Wang J.R. Existence and uniqueness and first order approximation of solutions to atmospheric Ekman flows. Monatshefte für Mathematik, 2020, 193, pp. 623–636. DOI: 10.1007/s00605-020-01414-7.
  22. Ortiz-Tarin J.L., Lee S., Flores O., Sarkar S. Global modes and large-scale structures in an Ekman boundary layer. Journal of Physics: Conference Series, 2020, vol. 1522, 012011. DOI: 10.1088/1742-6596/1522/1/012011.
  23. Prosviryakov E.Y. New class of exact solutions of Navier–Stokes equations with exponential dependence of velocity on two spatial coordinates. Theoretical Foundations of Chemical Engineering, 2019, vol. 53, No. 1, pp. 107–114. DOI: 10.1134/S0040579518060088.
  24. Aristov S.N., Prosviryakov E.Y. A new class of exact solutions for three-dimensional thermal diffusion equations. Theoretical Foundations of Chemical Engineering, 2016, vol. 50, No. 3, pp. 286–293. DOI: 10.1134/S0040579516030027.
  25. Burmasheva N.V., Prosviryakov E.Yu. Isothermal layered flows of a viscous incompressible fluid with spatial acceleration in the case of three Coriolis parameters. Diagnostics, Resource and Mechanics of materials and structures, 2020, iss. 3, pp. 29–46. DOI: 10.17804/2410-9908.2020.3.029-046. Available at: http://dream-journal.org/issues/2020-3/2020-3_291.html
  26. Burmasheva N.V., Privalova V.V., Prosviryakov E.Yu. Layered Marangoni convection with the Navier slip condition. SADHANA-ACADEMY PROCEEDINGS IN ENGINEERING SCIENCES, 2021, vol. 46, iss. 1, No. 55. DOI: 10.1007/s12046-021-01585-5.
  27. Ershkov Sergey V., Prosviryakov Evgeniy Yu., Burmasheva Natalya V., and Christianto Victor. Towards understanding the algorithms for solving the Navier–Stokes equations. Fluid Dynamics Research, 2021, vol. 53, No. 4, pp. 044501. DOI:10.1088/1873-7005/ac10f0.


Article reference

Gorshkov A. V., Prosviryakov E. Yu. Analytical Study of the Ekman Angle for the Benard–marangoni Convective Flow of Viscous Incompressible Fluid // Diagnostics, Resource and Mechanics of materials and structures. - 2021. - Iss. 4. - P. 34-48. -
DOI: 10.17804/2410-9908.2021.4.34-49. -
URL: http://eng.dream-journal.org/issues/content/article_340.html
(accessed: 06/22/2024).


impact factor
RSCI 0.42


MRDMS 2024
Google Scholar



Founder:  Institute of Engineering Science, Russian Academy of Sciences (Ural Branch)
Chief Editor:  S.V. Smirnov
When citing, it is obligatory that you refer to the Journal. Reproduction in electronic or other periodicals without permission of the Editorial Board is prohibited. The materials published in the Journal may be used only for non-profit purposes.
Home E-mail 0+

ISSN 2410-9908 Registration SMI Эл № ФС77-57355 dated March 24, 2014 © IMACH of RAS (UB) 2014-2024, www.imach.uran.ru