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A new method for constructing exact solutions to three-dimensional Navier-Stokes and Euler equations

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Abstract

Formulas are derived that make it possible to construct new exact solutions for three-dimensional stationary and nonstationary Navier-Stokes and Euler equations using simpler solutions to the respective two-dimensional equations. The formulas contain from two to five additional free parameters, which are not present in the initial solutions to the two-dimensional equations. It is important that these formulas do not contain quadratures in the steady-state case. We consider some examples of constructing new three-dimensional exact solutions to the Navier-Stokes equations using the derived formulas. The results are used to solve some problems of the hydrodynamics of a viscous incompressible liquid. Examples of thenonuniqueness of solutions to steady-state problems are given. Some three-dimensional solutions to the Navier-Stokes equations are constructed using nonviscous solutions to the Euler equations. Two classes of new exact solutions to the Grad-Shafranov equation that contain functional arbitrariness are specified. Note that, in this study, a new method for constructing exact solutions is applied that can be useful for analyzing other nonlinear physical models and phenomena. Several physical and physicochemical systems where this method can be used are considered.

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References

  1. Polyanin, A.D. and Aristov, S.N., Systems of Hydrodynamic Type Equations: Exact Solutions, Transformations, and Nonlinear Stability, Dokl. Phys., 2009, vol. 54, no. 9, p. 429.

    Article  CAS  Google Scholar 

  2. Polyanin, A.D., On the Nonlinear Instability of the Solutions of Hydrodynamic-Type Systems, JETP Lett., 2009, vol. 90, no. 3, p. 217.

    Article  CAS  Google Scholar 

  3. 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, p. 642.

    Article  CAS  Google Scholar 

  4. Aristov, S.N. and Polyanin, A.D., New Classes of Exact Solutions and Some Transformations of the Navier-Stokes Equations, Russ. J. Math. Phys., 2010, vol. 17, no. 1, p. 1.

    Article  Google Scholar 

  5. Pukhnachev, V.V., Symmetries in the Navier-Stokes Equations, Usp. Mekh., 2006, vol. 4, no. 1, p. 6.

    Google Scholar 

  6. Loitsyanskii, L.G., Mekhanika zhidkosti i gaza (Fluid Mechanics), Moscow: Nauka, 1973.

    Google Scholar 

  7. Drazin, P.G. and Riley, N., The Navier-Stokes Equations: A Classification of Flows and Exact Solutions, Cambridge: Cambridge Univ. Press, 2006.

    Book  Google Scholar 

  8. Polyanin, A.D., Handbook of Nonlinear Partial Differential Equations, Boca Raton, Fla.: Chapman & Hall/CRC Press, 2004.

    Google Scholar 

  9. Hocking, L.M., An Example of Boundary Layer Formation, AIAA J., 1963, vol. 1, p. 1222.

    Google Scholar 

  10. Crane, L.J., Flow past a Stretching Plate, Z. Angew. Math. Phys., 1970, vol. 21, p. 645.

    Article  Google Scholar 

  11. Lavrent’ev, M.A. and Shabat, B.V., Metody teorii funktsii kompleksnogo peremennogo (Methods of the Theory of Functions of a Complex Variable), Moscow: Nauka, 1973.

    Google Scholar 

  12. Sedov, L.I., Ploskie zadachi gidrodinamiki i aerodinamiki (Plane Problems of Hydrodynamics and Aerodynamics), Moscow: Nauka, 1980.

    Google Scholar 

  13. Tikhonov, A.N. and Samarskii, A.A., Uravneniya matematicheskoi fiziki (Equations of Mathematical Physics), Moscow: Nauka, 1972.

    Google Scholar 

  14. Polyanin, A.D., Handbook of Linear Partial Differential Equations for Engineers and Scientists, Boca Raton, Fla.: Chapman & Hall/CRC Press, 2002.

    Google Scholar 

  15. Bird, R.B., Stewart, W.E., and Lightfoot, E.N., Transport Phenomena, New York: Wiley, 2001, 2nd ed.

    Google Scholar 

  16. Polyanin, A.D., Kutepov, A.M., Vyazmin, A.V., and Kazenin, D.A., Hydrodynamics, Mass and Heat Transfer in Chemical Engineering, London: Taylor & Francis, 2002.

    Google Scholar 

  17. Kaptsov, O.V., Stationary Vortex Structures in an Inviscid Fluid, J. Exp. Theor. Phys., 1990, vol. 71, no. 2, p. 296.

    Google Scholar 

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Authors and Affiliations

  1. Ishlinsky Institute for Problems in Mechanics, Russian Academy of Sciences, pr. Vernadskogo 101, Moscow, 119526, Russia

    A. D. Polyanin

  2. Institute of Continuous Media Mechanics, Ural Branch, Russian Academy of Sciences, Perm, 614013, Russia

    S. N. Aristov

Authors
  1. A. D. Polyanin
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  2. S. N. Aristov
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Correspondence to A. D. Polyanin.

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Original Russian Text © A.D. Polyanin, S.N. Aristov, 2011, published in Teoreticheskie Osnovy Khimicheskoi Tekhnologii, 2011, Vol. 45, No. 6, pp. 696–701.

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Polyanin, A.D., Aristov, S.N. A new method for constructing exact solutions to three-dimensional Navier-Stokes and Euler equations. Theor Found Chem Eng 45, 885–890 (2011). https://doi.org/10.1134/S0040579511060091

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  • DOI: https://doi.org/10.1134/S0040579511060091

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