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N. V. Burmasheva, E. Yu. Prosviryakov


DOI: 10.17804/2410-9908.2020.3.029-046

We study the solvability of the overdetermined system of Navier–Stokes equations, supplemented by the incompressibility equation, which is used to describe isothermal large-scale shear flows of a rotating viscous incompressible fluid. Large–scale flows are studied in a thin-layer approximation (the vertical velocity of the fluid is assumed to be zero). The rotation of a continuous fluid medium is described by three Coriolis parameters. The solution of the reduced system of Navier–Stokes equations is constructed in the Lin–Sidorov–Aristov class. In this case, both nonzero components of the velocity vector, the pressure and temperature fields are assumed to be full linear forms of two Cartesian coordinates, and the dependence on the third Cartesian coordinate has an arbitrary form (including non-polynomial). It is shown that the nonlinear overdetermined system of Navier–Stokes equations and of the incompressibility equation in the framework of the Lin–Sidorov–Aristov class reduces to the equivalent nonlinear overdetermined system of ordinary differential equations, in which the components of the hydrodynamic fields act as unknown functions. The compatibility condition for the equations of the resulting system is derived. It is shown that, if this compatibility condition is fulfilled, the system has a unique solution, and the spatial accelerations in both variables (the linearity with respect to them was postulated when choosing the solution class) prove to be constant functions. These results are a generalization of similar results obtained earlier in the study of solvability in the cases of one and two Coriolis parameters.

Keywords: exact solution, shear flow, Coriolis parameter, solvability, Navier–Stokes equation, overdetermined system, large-scale flow


  1. Monin A.S. Theoretical Foundations of Geophysical Hydrodynamics. Leningrad, Gidrometeoizdat Publ., 1988. (In Russian).
  2. Brekhovskikh L., Goncharov V. Mechanics of Continua and Wave Dynamics, transl. from Rusian, Springer-Verlag Berlin Heidelberg, 1985. DOI: 10.1007/978-3-642-96861-7.
  3. Zyryanov V.N. Theory of steady ocean currents, Leningrad, Gidrometeoizdat Publ., 1985. (in Russian).
  4. Pedlosky J. Geophysical fluid dynamics, Berlin, New York, Springer-Verlag, 1987.
  5. Gill A.E. Atmosphere-ocean dynamics, Cambridge, University of Cambridge, 1982.
  6. 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.
  7. Gershuni G.Z., Zhukhovitskii E.M. Convective Stability of Incompressible Fluids, Jerusalem, Keter Publishing House, 1976.
  8. Landau L.D., Lifshitz E.M. Course of Theoretical Physics. Vol. 6. Fluid mechanics, Oxford, Pergamon Press, 1987.
  9. Temam R. Navier–Stokes Equations: Theory and Numerical Analysis, Amsterdam, New York, Oxford, North-Holland Publ., 1977. 
  10. Polyanin A.D., Kutepov A.M., Kazenin D.A., Vyazmin A.V. Hydrodynamics, Mass and Heat Transfer in Chemical Engineering, Boca Raton, CRC Press, 2001.
  11. Couette M. Etudes sur le frottement des liquids. Ann. Chim. Phys., 1890, vol. 21, pp. 433–510.
  12. Guermond, J.L., Migeon C., Pineau G., Quartapel L. Start–up flows in a three-dimensional rectangular driven cavity of aspect ratio 1:1:2 at Re=1000. J. Fluid Mech., 2002, vol. 450, pp. 169–199. DOI: 10.1017/S0022112001006383.
  13. Neto C., Evans D., Bonaccurso E., Butt H.-J., Craig V.S.J. Boundary slip in Newtonian liquids: a review of experimental studies. Reports on Progress in Physics, 2005, vol. 68, pp. 2859–2897. DOI: 10.1088/0034-4885/68/12/R05.
  14. Aristov S.N., Frik P.G. Nonlinear effects of the Ekman layer on the dynamics of large–scale eddies in shallow water. Journal of Applied Mechanics and Technical Physics, 1991, vol. 32, no. 2, pp. 189–194. DOI: 10.1007/BF00858033.
  15. Aristov S.N., Shvarts K.G. Vortical Flows of the Advective Nature in a Rotating Fluid Layer, Perm, Perm State Univ. Publ., 2006. (In Russian).
  16. Aristov S.N., Shvarts K.G. Vikhrevye techeniia v tonkikh sloiakh zhidkosti [Vortical Flows in Thin Fluid Layers]. Kirov, Vyatka State Univ. Publ., 2011. (In Russian).
  17. Gorshkov A.V., Prosviryakov E.Y. 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. Kompaniets L., Pitalskaya O. Exact solutions of Ekmans model for three-dimensional wind-induced flow of homogeneous fluid with geostrophic current. Computer Research and Modeling, 2009, vol. 1, pp. 57–66. DOI: 10.20537/2076-7633-2009-1-1-57-66. (In Russian).
  19. 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.
  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. 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. Constantin A., Johnson R.S. Atmospheric Ekman flows with variable eddy viscosity. Boundary-Layer Meteorol, 2019, vol. 170, pp. 395–414. DOI: 10.1007/s10546-018-0404-0.
  24. 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.
  25. 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).
  26. 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).
  27. Lin C.C. Note on a class of exact solutions in magneto-hydrodynamics. Arch. Rational Mech. Anal., 1958, vol. 1, pp. 391–395.
  28. Aristov S.N., Prosviryakov E.Yu. Large–scale flows of viscous incompressible vortical fluid. Russian Aeronautics, 2015, vol. 58, no. 4, pp. 413–418. DOI: 10.3103/S1068799815040091.
  29. Aristov S.N., Prosviryakov E.Y. Inhomogeneous Couette flow. Russian Journal of Nonlinear Dynamics, 2014, vol. 10, no. 2, pp. 177–182. DOI: 10.20537/nd1402004. (In Russian).
  30. Prosviryakov E.Y. Exact Solutions for three-dimensional potential and vorticity Couette flows of an incompressible viscous fluid. Vestnik Natsional'nogo issledovatel'skogo yadernogo universiteta "MIFI", 2015, vol. 4, no. 6, pp. 501–506 DOI: 10.1134/S2304487X15060127. (In Russian).
  31. Aristov S.N., Prosviryakov E.Y. Unsteady layered vortical fluid flows. Fluid Dynamics, 2016, vol. 51, no. 2, pp. 148–154. DOI: 10.1134/S0015462816020034.
  32. 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.
  33. Burmasheva N.V., Prosviryakov E.Yu. A large-scale layered stationary convection of an incompressible viscous fluid under the action of shear stresses at the upper boundary. Velocity field investigation. VESTNIK SAMARSKOGO GOSUDARSTVENNOGO TEKHNICHESKOGO UNIVERSITETA-SERIYA-FIZIKO-MATEMATICHESKIYE NAUKI, 2017, vol. 21, no. 1, pp. 180–196. DOI: 10.14498/vsgtu1527. (In Russian).
  34. 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. Temperature and presure field investigation. VESTNIK SAMARSKOGO GOSUDARSTVENNOGO TEKHNICHESKOGO UNIVERSITETA-SERIYA-FIZIKO-MATEMATICHESKIYE NAUKI, 2017, vol. 21, no. 4, pp. 736–751 DOI: 10.14498/vsgtu1568. (In Russian).
  35. Gorshkov A.V., Prosviryakov E.Yu. Analytic solutions of stationary complex convection describing a shear stress field of different signs. Trudy Inst. Mat. i Mekh. UrO RAN, 2017, vol. 23, no. 2, pp. 32–41. DOI: 10.21538/0134-4889-2017-23-2-32-41. (In Russian).
  36. Vereshchagin V.P., Subbotin Yu.N., Chernykh N.I. Description of a helical motion of an incompressible nonviscous fluid. Proc. Steklov Inst. Math., 2015, vol. 288, pp. 202–210. DOI: 10.1134/S0081543815020212.
  37. Vereshchagin V.P., Subbotin Yu.N., Chernykh N.I. Some solutions of continuum equations for an incompressible viscous medium. Proc. Steklov Inst. Math., 2014, vol. 287, pp. 208–223. DOI: 10.1134/S008154381409020X.
  38. Vereshchagin V.P., Subbotin Yu.N., Chernykh N.I. On the mechanics of helical flows in an ideal incompressible nonviscous continuous medium. Proc. Steklov Inst. Math., 2014, vol. 284, pp. 159–174. DOI: 10.1134/S008154381402014X.
  39. Zubarev N.M., Prosviryakov E.Yu. Exact solutions for the layered three-dimensional nonstationary isobaric flows of viscous incompressible fluid. Journal of Applied Mechanics and Technical Physics, 2019, vol. 60, no. 6 (358), pp. 65–71. DOI: 10.15372/PMTF201. (In Russian).
  40. Pukhnachev V.V. Point vortex in a viscous incompressible fluid. Journal of Applied Mechanics and Technical Physics, 2014, vol. 55, no. 2, pp. 345–351. DOI: 10.1134/S0021894414020175.
  41. Birikh R.V., Pukhnachev V.V., Frolovskaya O.A. Convective flow in a horizontal channel with non-Newtonian surface rheology under time-dependent longitudinal temperature gradient. Fluid Dynamics, 2015, vol. 50, no. 1, pp. 173–179. DOI: 10.1134/S0015462815010172.
  42. Golubkin V.N., Sizykh G.B. Maximum principle for the Bernoulli function. TsAGI Science Journal, 2015, vol. 46, no. 5, pp. 485–490. DOI: 10.1615/TsAGISciJ.v46.i5.50.
  43. Sizykh G.B. Axisymmetric helical flows of viscous fluid. Russian Mathematics, 2019, vol. 63, no. 2, pp. 44–50. DOI: 10.3103/S1066369X19020063.
  44. Sizykh G.B.  The splitting of Navier–Stokes equations for a class of axisymmetric flows. VESTNIK SAMARSKOGO GOSUDARSTVENNOGO TEKHNICHESKOGO UNIVERSITETA-SERIYA-FIZIKO-MATEMATICHESKIYE NAUKI, 2020, vol. 24, no. 1, pp.  163–173. DOI: 10.14498/vsgtu1740. (In Russian).
  45. Markov V.V. , Sizykh G.B. Exact solutions of the Euler equations for some two-dimensional incompressible flows. Proc. Steklov Inst. Math., 2016, vol. 294, no. 1, pp. 283–290. DOI: 10.1134/S0081543816060195.
  46. Ershkov S.V., Shamin R.V., Giniyatullina A.R. On a new type of non-stationary helical flows for incompressible 3D Navier-Stokes equations. Journal of King Saud University – Science, 2020, vol. 32, no. 1, pp. 459–467. DOI: 10.1016/j.jksus.2018.07.006.
  47. Kovalev V.P., Prosviryakov E.Yu., Sizykh G.B. Obtaining examples of exact solutions of the Navier–Stokes equations for helical flows by the method of summation of velocities. Trudy MFTI, 2017, vol. 9, no. 1, pp. 71–88. (In Russian).
  48. Burmasheva N.V., Prosviryakov E.Yu. Thermocapillary convection of a vertical swirling liquid. Theoretical Foundations of Chemical Engineering, 2020, vol. 54, no. 1, pp. 230–239. DOI: 10.1134/S0040579519060034.
  49. Ul'yanov O. N. A class of viscous fluid flows. Proc. Steklov Inst. Math. (Suppl.), 2003, suppl. 2, S173–S181.
  50. Pukhnachev V.V.  Nonlinear diffusion and exact solutions to the Navier-Stokes equations. The Bulletin of Irkutsk State University. Series Mathematics, 2010, vol. 3, no. 1, pp. 61–69.
  51. Burmasheva N.V., Prosviryakov E.Yu. Convective layered flows of a vertically whirling viscous incompressible fluid. Velocity field investigation. VESTNIK SAMARSKOGO GOSUDARSTVENNOGO TEKHNICHESKOGO UNIVERSITETA-SERIYA-FIZIKO-MATEMATICHESKIYE NAUKI, 2019, vol. 23, no. 2, pp. 341–360. DOI: 10.14498/vsgtu1670.
  52. Privalova V.V., Prosviryakov E.Yu., Simonov M.A. Nonlinear gradient flow of a vertical vortex fluid in a thin layer. Russian Journal of Nonlinear Dynamics, 2019, vol. 15, no. 3, pp. 271–283. DOI: 10.20537/nd190306.
  53. Aristov S.N., Privalova V.V., Prosviryakov E.Yu. Stationary nonisothermal Couette flow. Quadratic heating of the upper boundary of the fluid layer. Nelin. Dinam., 2016, vol. 12, no. 2, pp. 167–178. (In Russian).
  54. Polyanin A.D., Zaitsev V.F. Handbook of exact solutions for ordinary differential equations, 2nd еd., Boca Raton: Chapman& Hall/CRC, 2003.


Article reference

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. - P. 29-46. -
DOI: 10.17804/2410-9908.2020.3.029-046. -
URL: http://eng.dream-journal.org/issues/2020-3/2020-3_291.html
(accessed: 05/22/2024).


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