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


advanced search

IssuesAbout the JournalAuthorContactsNewsRegistration

2019 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






N. V. Burmasheva, E. Yu. Prosviryakov


DOI: 10.17804/2410-9908.2019.5.023-039

The article considers the unidirectional flow of a viscous incompressible fluid in an infinite horizontal layer of a given thickness, which is induced by the thermocapillary effect specified at the upper boundary of the layer and taking into account the condition of fluid slipping at the lower boundary. An exact solution to the Oberbeck-Boussinesq equation system is obtained. A detailed analysis of the velocity field was carried out for various slip length values. It is shown that counterflows may occur in a fluid. Their number is analyzed, the conditions for the appearance of velocity field stratification are written.

Keywords: layered flow, Oberbeck-Boussinesq equation system, exact solution, unidirectional flow, counterflows, Marangoni convection, Navier condition


  1. Marangoni C. Sull espansione delle goccie di un liquido galleggiante sulla superficie di altro liquid, Pavia, Tipografia dei fratelli Fusi, 1865.

  2. Goldstein S.V. Modern Developments in Fluid Mechanics, Oxford, Oxford Univ. Press, 1938.

  3. Prandtl L., Tietjens O. Hydro- und aeromechanic (2 vols.), Berlin, Verlag von Julius Springer, 1931. DOI: 10.1017/S0368393100115366.

  4. Borzenko E.I., Shrager G.R. The structure of viscoplastic fluid flow during filling of a circular pipe/plane channel. Computational continuum mechanics, 2019, vol. 12, no. 2, pp. 129–136. DOI: 10.7242/1999-6691/2019.12.2.11. (In Russian).

  5. Neto C., Evans D., Bonaccurso E. Boundary slip in Newtonian liquids: a review of experimental studies. Reports on Progress in Physics, 2005, vol. 68, no. 12, pp. 2859−2897. DOI: 10.1088/0034-4885/68/12/R05.

  6. Pelenko V.V., Aret V.A., Gusev B.K., Pelenko F.V. The flow of viscoplastic nonlinear media with boundary slip. In: The Bulletin of KrasGAU: Interuniversity Collection of Scientific Papers, Krasnoyarsk, KrasGAU, 2008, no. 2, pp. 54–57. (In Russian).

  7. Navier С.L.M.H. M'emoire sur les lois du mouvement des fluids. M'em. Acad. Sci. Inst. de France, 1823, vol. 2, no. 6. pp. 389–440.

  8. Lauga E., Brenner M., Stone H. Microfluidics: The No-Slip Boundary Condition. In: C. Tropea, A.L. Yarin, J.F. Foss, ed., Springer Handbook of Experimental Fluid Mechanics, Springer, Berlin, Heidelberg, Springer Handbooks, 2007.

  9. Hoffmann J., Johnson C. Computational Turbulent Incompressible Flow, Heidelberg, Berlin, Springer-Verlag, 2007, 397 p.

  10. Borzenko E.I., Diakova O.A., Shrager G.R. Studying the slip phenomenon for a viscous fluid flow in a curved channel. Tomsk State University Journal of Mathematics and Mechanics, 2014, no. 2 (28), pp. 35–44. (In Russian).

  11. Privalova V.V., Prosviryakov E.Yu. Nonlinear isobaric flow of a viscous incompressible fluid in a thin layer with permeable boundaries. Computational continuum mechanics, 2019, vol. 12, no. 2, pp. 230–242. DOI: 10.7242/1999-6691/2019.12.2.20. (In Russian).

  12. Gershuni G.Z., Zhukhovitskii E.M. Convective Stability of Incompressible Fluids, Israel Program for Scientific Translations, Jerusalem, Keter Publishing House, 1976, 330 pp.

  13. Ostroumov G.A. Free convection under the condition of the internal problem, Washington, NACA Technical Memorandum 1407, National Advisory Committee for Aeronautics, 1958.

  14. Birikh R.V. Thermocapillary convection in a horizontal layer of liquid. J. Appl. Mech. Tech. Phys., 1966, vol. 7, no. 3, p. 43.

  15. Sidorov A.F. Two classes of solutions of the fluid and gas mechanics equations and their connection to traveling wave theory. Journal of Applied Mechanics and Technical Physics, 1989, vol. 30, iss. 2, pp 197–203, 1989, no. 2, pp. 34–40. (In Russian).

  16. Andreev V.K., Gaponenko Ya.A., Goncharova O.N., Pukhnachev V.V. Mathematical Models of Convection, Berlin–Boston, De Gryuter Publ., 2012. xv+ + 417 p.

  17. Aristov S.N., Prosviryakov E.Yu. A new class of exact solutions for three-dimensional thermal diffusion equations. Theor. Found. Chem. Eng., 2016, vol. 50, no. 3, pp. 286–293. DOI: 10.1134/S0040579516030027.

  18. Aristov S.N., Frik P.G. Nonlinear effects of the Ekman layer on the dynamics of largescale eddies in shallow water. J. Appl. Mech. Tech. Phys., 1991, vol. 32, no. 2, pp. 189–194. DOI: 10.1007/BF00858033.

  19. 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. In: Tr. In-ta Eksperim. Meteorol., 1996, no. 27 (162), pp. 142–157. (In Russian).

  20. Pukhnachev V.V. Non-stationary analogues of the Birikh solution. Izvestiya AltGU, 2011, no. 1–2, pp. 62–69. (In Russian).

  21. Aristov S.N., Prosviryakov E.Yu. On laminar flows of planar free convection. Russian Journal of Nonlinear Dynamics, 2013, vol. 9, no. 4, pp. 651–657. DOI: 10.20537/nd1304004. (In Russian).

  22. Andreev V.K. Resheniya Birikha uravneniy konvektsii i nekotorye ego obobshcheniya: preprint [Birikh Solutions of Convection Equations and Some of its Generalizations: preprint]. Krasnoyarsk, 2010, № 1–10. (In Russian).

  23. Pukhnachev V.V. Group-theoretical methods in the convection problems. In: Application of Mathematics in Technical and Natural Sciences, M.D. Todorov and C.I. Christov, eds., American Institute of Physics, CP 1404, Melwille, NY, 2011, pp. 31–42.

  24. Burmasheva N.V., Prosviryakov E.Yu. Convective layered flows of a vertically whirling viscous incompressible fluid. Velocity field investigation. Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2019, vol. 23, no. 2, pp. 341–360. DOI: 10.14498/vsgtu1670.

  25. Aristov S.N., Prosviryakov E.Yu., Spevak L.F. Unsteady-state Bénard–Marangoni convection in layered viscous incompressible flows. Theor. Found. Chem. Eng., 2016, vol. 50, no. 2, pp. 132–141. DOI: 10.1134/S0040579516020019.

  26. Burmasheva N.V., Prosviryakov E.Yu. Temperature field investigation in layered flows of a vertically swirling viscous incompressible fluid under two thermocapillar forces at a free boundary. Diagnostics, Resource and Mechanics of materials and structures, 2019, iss. 1, pp. 6–42. DOI: 10.17804/2410-9908.2019.1.006-042. Available at: http://dream-journal.org/DREAM_Issue_1_2019_Burmasheva_N.V._et_al._006_042.pdf

  27. Gorshkov A.V., Prosviryakov E.Yu. Layered Benard–Marangoni convection during heat transfer according to the Newton’s law of cooling. Comp. Research and Modeling, 2016, vol. 8, no. 6, pp. 927–940. (In Russian).

  28. Goncharova O.N., Kabov O.A. Gravitational-thermocapillary convection of fluid in the horizontal layer in co-current gas flow. Dokl. Phys., 2009, vol. 54, no. 5, pp. 242–247. DOI: 10.1134/S1028335809050061.

  29. Bekezhanova V.B. Convective instability of Marangoni-Poiseuille flow under a longitudinal temperature gradient. Journal of Applied Mechanics and Technical Physics, 2011, vol. 52, no. 1, pp. 74–81. DOI: 10.1134/S0021894411010111.

  30. Gordeeva V.Y., Lyushnin A.V. Influence of the thermocapillary effect on the dynamics and stability of motion of a thin evaporating film. Technical Physics, 2013, vol. 58, no. 3, pp. 351–357. DOI: 10.1134/S1063784213030092.

  31. Aktershev S.P. Thermocapillary effect and periodic structures on the surface of a heated viscous liquid film. In: Proceedings of the Institute of Mechanics of Ural Branch of RAS, 2007, no. 5, pp. 79–84. DOI: 10.21662/uim2007.1.005. (In Russian).

  32. Schlunder E.U. Heat Exchanger Design Handbook, Hemisphere Publishing Corporation, 1983.

  33. Aristov S.N., Prosviryakov E.Yu. Nonuniform convective Couette flow. Fluid Dynamics, 2016, vol. 51, no. 5, pp. 581–587. DOI: 10.1134/S001546281605001X.

  34. Prosviryakov E.Yu. A new class of exact solutions of the Navier – Stokes equations with a power-law dependence of velocities on two spatial coordinates. Theoretical Foundations of Chemical Engineering, 2019, vol. 53, no. 1, pp. 112–120. DOI: 10.1134/S0040357118060118. (In Russian).

  35. Burmasheva N.V., Prosviryakov E.Yu. A large-scale layered stationary convection of a incompressible viscous fluid under the action of shear stresses at the upper boundary. Velocity field investigation. Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.]. 2017, vol. 21, no. 1, pp. 180–196. DOI: 10.14498/vsgtu1527. (In Russian).

  36. Knutova N.S., Shvarts K.G. A study of behavior and stability of an advective thermocapillary flow in a weakly rotating liquid layer under microgravity. Fluid Dyn., 2015, vol. 50, no. 3, pp. 340–350. DOI: 10.1134/S0015462815030047.

  37. Couette M. Études sur le frottement des liquids. Ann. Chim. Phys. Ser. 6, 1890, vol. 21, pp. 433–510.


Article reference

Burmasheva N. V., Prosviryakov E. Yu. An Exact Solution to the Description of a Unidirectional Marangoni Flow of a Viscous Incompressible Fluid with the Navier Boundary Condition. Velocity Field Investigation // Diagnostics, Resource and Mechanics of materials and structures. - 2019. - Iss. 5. - P. 23-39. -
DOI: 10.17804/2410-9908.2019.5.023-039. -
URL: http://eng.dream-journal.org/issues/2019-5/2019-5_259.html
(accessed: 09/29/2023).


impact factor
RSCI 0.42


MRDMS 2022
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) 2014-2023, www.imach.uran.ru