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New Class of Exact Solutions of Navier–Stokes Equations with Exponential Dependence of Velocity on Two Spatial Coordinates

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Abstract

A new class of exact solutions of nonlinear and linearized Navier–Stokes equations has been proposed, which generalize the well-known family of exact solutions in which the velocity is linear in some coordinates. The case of the quadratic dependence of the velocities on two horizontal (longitudinal) coordinates with coefficients that are the functions of the vertical (transverse) coordinate and time was considered in detail. The solutions were generalized for rotating liquids. Equations for constructing exact solutions with an arbitrary dependence of velocities on the horizontal coordinates were derived.

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REFERENCES

  1. Aristov, S.N., Knyazev, D.V., and Polyanin, A.D., Exact solutions of the Navier–Stokes equations with the linear dependence of velocity components on two space variables, Theor. Found. Chem. Eng., 2009, vol. 43, no. 5, pp. 642–662.https://doi.org/10.1134/S0040579509050066

    Article  CAS  Google Scholar 

  2. Lin, C.C., Note on a class of exact solutions in magneto-hydrodynamics, Arch. Ration. Mech. Anal., 1957, vol. 1, no. 1, pp. 391–395. https://doi.org/10.1007/BF00298016

    Article  Google Scholar 

  3. Sidorov, A.F., Two classes of solutions of the fluid and gas mechanics equations and their connection to traveling wave theory, J. Appl. Mech. Tech. Phys., 1989, vol. 30, no. 2, pp. 197–203. https://doi.org/10.1007/BF00852164

    Article  Google Scholar 

  4. Aristov, S.N., Vortex flows in thin liquid layers, Extended Abstract of Doctoral (Phys.–Math.) Dissertation, Perm: Inst. of Continuous Media Mechanics, Ural Branch, Academy of Sciences of the USSR, 1990.

  5. Aristov, S.N. and 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. https://doi.org/10.1134/S0040579516030027

    Article  CAS  Google Scholar 

  6. Pukhnachev, V.V., Mathematical model of an incompressible viscoelastic Maxwell medium, J. Appl. Mech. Tech. Phys., 2010, vol. 51, no. 4, pp. 546–554. https://doi.org/10.1007/s10808-010-0071-5

    Article  Google Scholar 

  7. Pukhnachev, V.V. and Pukhnacheva, T.P., The Couette problem for a Kelvin–Voigt medium, J. Math. Sci., 2012, vol. 186, no. 3, pp. 495–510. https://doi.org/10.1007/s10958-012-1003-0

    Article  Google Scholar 

  8. Pukhnachev, V.V., Exact solutions of the equations of motion for an incompressible viscoelastic Maxwell medium, J. Appl. Mech. Tech. Phys., 2009, vol. 50, no. 2, pp. 181–187. https://doi.org/10.1007/s10808-009-0025-y

    Article  Google Scholar 

  9. Polyanin, A.D. and Vyazmin, A.V., Decomposition of three-dimensional linearized equations for Maxwell and Oldroyd viscoelastic fluids and their generalizations, Theor. Found. Chem. Eng., 2013, vol. 47, no. 4, pp. 321–329.https://doi.org/10.1134/S004057951304026X

    Article  CAS  Google Scholar 

  10. Polyanin, A.D., Exact solutions to new classes of reaction-diffusion equations containing delay and arbitrary functions, Theor. Found. Chem. Eng., 2015, vol. 49, no. 2, pp. 169–175. https://doi.org/10.1134/S0040579515020104

    Article  CAS  Google Scholar 

  11. Polyanin, A.D., Sorokin, V.G., and Vyazmin, A.V., Exact solutions and qualitative features of nonlinear hyperbolic reaction–diffusion equations with delay, Theor. Found. Chem. Eng., 2015, vol. 49, no. 5, pp. 622–635. https://doi.org/10.1134/S0040579515050243

    Article  CAS  Google Scholar 

  12. Aristov, S.N. and Prosviryakov, E.Yu., Large-scale flows of viscous incompressible vortical fluid, Russ. Aeronaut., 2015, vol. 58, no. 4, pp. 413–418. https://doi.org/10.3103/S1068799815040091

    Article  Google Scholar 

  13. Aristov, S.N. and Prosviryakov, E.Yu., Unsteady layered vortical fluid flows, Fluid Dyn., 2016, vol. 51, no. 2, pp. 148–154. https://doi.org/10.1134/S0015462816020034

    Article  Google Scholar 

  14. Aristov, S.N., Prosviryakov, E.Yu., and 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. https://doi.org/10.1134/S0040579516020019

    Article  CAS  Google Scholar 

  15. Knyazev, D.V., Two-dimensional flows of a viscous binary fluid between moving solid boundaries, J. Appl. Mech. Tech. Phys., 2011, vol. 52, no. 2, pp. 212–217. https://doi.org/10.1134/S0021894411020088

    Article  CAS  Google Scholar 

  16. Aristov, S.N. and Prosviryakov, E.Yu., Nonuniform convective Couette flow, Fluid Dyn., 2016, vol. 51, no. 5, pp. 581–587. https://doi.org/10.1134/S001546281605001X

    Article  Google Scholar 

  17. Aristov, S.N. and Shvarts, K.G., Convective heat transfer in a locally heated plane incompressible fluid layer, Fluid Dyn., 2013, vol. 48, no. 3, pp. 330–335. https://doi.org/10.1134/S001546281303006X

    Article  CAS  Google Scholar 

  18. Andreev, V.K. and Bekezhanova, V.B., Stability of non-isothermal fluids (Review), J. Appl. Mech. Tech. Phys., 2013, vol. 54, no. 2, pp. 171–184. https://doi.org/10.1134/S0021894413020016

    Article  Google Scholar 

  19. Goncharova, O.N. and Rezanova, E.V., Modeling of two-layer fluid flows with evaporation at the interface in the presence of the anomalous thermocapillary effect, J. Sib. Fed. Univ., Math. Phys., 2016, vol. 9, no. 1, pp. 48–59. https://doi.org/10.17516/1997-1397-2016-9-1-48-59

    Article  Google Scholar 

  20. Efimova, M.V., On one two-dimensional stationary flow of a binary mixture and viscous fluid in a plane layer, J. Sib. Fed. Univ., Math. Phys., 2016, vol. 9, no. 1, pp. 30–36.https://doi.org/10.17516/1997-1397-2016-9-1-30-36

    Article  Google Scholar 

  21. Goncharova, O.N., Kabov, O.A., and Pukhnachov, V.V., Solutions of special type describing the three dimensional thermocapillary flows with an interface, Int. J. Heat Mass Transfer, 2012, vol. 55, no. 4, pp. 715–725. https://doi.org/10.1016/j.ijheatmasstransfer.2011.10.038

    Article  Google Scholar 

  22. Goncharova, O. and Rezanova, E., Mathematical modelling of the evaporating liquid films on the basis of the generalized interface conditions, MATEC Web Conf., 2016, vol. 84, article no. 00013. https://doi.org/10.1051/matecconf/2016840001310.1051/matecconf/20168400013

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  1. Institute of Engineering Science, Ural Branch, Russian Academy of Sciences, 620049, Yekaterinburg, Russia

    E. Yu. Prosviryakov

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  1. E. Yu. Prosviryakov
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Translated by L. Smolina

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Prosviryakov, E.Y. New Class of Exact Solutions of Navier–Stokes Equations with Exponential Dependence of Velocity on Two Spatial Coordinates. Theor Found Chem Eng 53, 107–114 (2019). https://doi.org/10.1134/S0040579518060088

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