Fluid Dynamics

, Volume 51, Issue 5, pp 581–587 | Cite as

Nonuniform convective Couette flow

  • S. N. AristovEmail author
  • E. Yu. Prosviryakov


An exact solution describing the convective flow of a vortical viscous incompressible fluid is derived. The solution of the Oberbeck–Boussinesq equation possesses a characteristic feature in describing a fluid in motion, namely, it holds true when not only viscous but also inertia forces are taken into account. Taking the inertia forces into account leads to the appearance of stagnation points in a fluid layer and counterflows, as well as the existence of layer thicknesses at which the tangent stresses vanish on the lower boundary. It is shown that the vortices in the fluid are generated due to the nonlinear effects leading to the occurrence of counterflows and flow velocity amplification, compared with those given by the boundary conditions. The solution of the spectral problem for the polynomials describing the tangent stress distribution makes it possible to explain the absence of the skin friction on the solid surface and in an arbitrary section of an infinite layer.


vortical viscous incompressible fluid Couette flow convection exact solutions layered flows counterflows 


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  1. 1.
    O. A. Ladyzhenskaya, “Sixth Problem of the Thousand Years: Navier–Stokes Equations, Existence and Smoothness,” Usp. Mat. Nauk 58 (2), 45 (2003).MathSciNetCrossRefGoogle Scholar
  2. 2.
    L. D. Landau and E. M. Lifshitz, Fluid Dynamics, Pergamon, London (1987).zbMATHGoogle Scholar
  3. 3.
    S. N. Aristov, D. E. Knyazev, and A. D. Polyanin, “Exact Solutions of the Navier–Stokes Equations with the Linear Dependence of the Velocity Components on Two Spatial Variables,” Teoret. Osnovy Khim. Tekhnol. 43, 547 (2006).Google Scholar
  4. 4.
    G. I. Broman and O. V. Rudenko, “Submergent Landau Jet: Exact Solutions, Their Meaning and Applications,” Usp. Fiz. Nauk 180 (1), 97 (2010).CrossRefGoogle Scholar
  5. 5.
    V. V. Pukhnachev, “Symmetry in the Navier–Stokes Equations,” Usp. Mekh. No. 6, 3 (2006).Google Scholar
  6. 6.
    M. Couette, “Etudes sur le frottement des liquides,” Ann. Chim. Phys. Sér. 6 21, 433 (1890).zbMATHGoogle Scholar
  7. 7.
    L. Kh. Ingel’ and M. V. Kalashnik, “Nontrivial Features of the Thermohydrodynamics of Sea Water and Other Stratified Solutions,” Usp. Fiz. Nauk 182, 379 (2012).CrossRefGoogle Scholar
  8. 8.
    S. N. Aristov and K. G. Shvartz, Advective Vortical Flows in a Rotating Fluid Layer [in Russian], Perm (2006).zbMATHGoogle Scholar
  9. 9.
    S. N. Aristov and K. G. Shvartz, Vortical Flows in Thin Fluid Layers [in Russian], Kirov (2011).zbMATHGoogle Scholar
  10. 10.
    O. I. Skul’skii and S. N. Aristov, Mechanics of Anomalously Viscous Fluids [in Russian], Moscow & Izhevsk (2003).Google Scholar
  11. 11.
    G. A. Ostroumov, Free Convection under the Conditions of an Internal Problem [in Russian], GITTL, Moscow (1952).Google Scholar
  12. 12.
    R. V. Birikh, “Thermocapillary Convection in a Horizontal Fluid Layer,” Zh. Prikl. Mekh. Tekhn. Fiz. No. 3, 69 (1966).Google Scholar
  13. 13.
    L. G. Napolitano, “Plane Marangoni–Poiseuille Flow of Two Immiscible Fluids,” Acta Astronaut. No. 7, 461 (1980).ADSCrossRefzbMATHGoogle Scholar
  14. 14.
    O. Goncharova and O. Kabov, “Gas Flow and Thermocapillary Effects of Fluid Flow Dynamics in a Horizontal Layer,” Microgravity Sci. Technol. 21, Suppl. 1, 129 (2009).CrossRefzbMATHGoogle Scholar
  15. 15.
    V. K. Andreev, Birikh’s Solutions of the Convection Equations and Their Certain Generalizations [in Russian], Krasnoyarsk (2010).Google Scholar
  16. 16.
    S. N. Aristov and E. Yu. Prosviryakov, “Layered Flows in Plane Free Convection,” Nelin. Dinam. 9, 651 (2013).CrossRefGoogle Scholar
  17. 17.
    C. C. Lin, “Note on a Class of Exact Solutions in Magnetohydrodynamics,”Arch. Ration. Mech. Anal. 1 (1), 391 (1958).Google Scholar
  18. 18.
    A. F. Sidorov, “On Two Classes of Solutions of Fluid Mechanics Equations and Their Relation with Theory of TravelingWaves,” Zh. Prikl. Mekh. Tekhn. Fiz. No. 2, 34 (1989).MathSciNetGoogle Scholar
  19. 19.
    S. K. Betyaev, “Fluid Dynamics: Problems and Paradoxes,” Usp. Fiz. Nauk 165, 299 (1995).CrossRefGoogle Scholar
  20. 20.
    S. N. Aristov and E. Yu. Prosviryakov, “Nonuniform Couette Flows,” Nelin. Dinam. 10, 177 (2014).CrossRefzbMATHGoogle Scholar
  21. 21.
    G. K. Korotaev, E. N. Mikhailova, and N. B. Shapiro, Theory of Equatorial Counterflows in the World Ocean [in Russian], Naukova Dumka, Kiev (1986).Google Scholar
  22. 22.
    E. E. Tyrtyshnikov, Matrix Analysis and Linear Algebra [in Russian], Fizmatlit, Moscow (2007).zbMATHGoogle Scholar
  23. 23.
    V. I. Arnold and B. A. Khesin, Topological Methods in Fluid Dynamics [in Russian], Moscow (2007).zbMATHGoogle Scholar

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© Pleiades Publishing, Ltd. 2016

Authors and Affiliations

  1. 1.Institute of Continuous Media MechanicsUral Branch of Russian Academy of SciencesPermRussia
  2. 2.Kazan National Research Technical University named after A.N. TupolevKazanRussia
  3. 3.Institute of Engineering ScienceUral Branch of Russian Academy of SciencesEkaterinburgRussia

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