Abstract
A wide class of two-dimensional and three-dimensional steady-state and non-steady-state flows of a viscous incompressible fluid is considered. It is assumed that the components of the velocity of a fluid linearly depend on two spatial coordinates. The three-dimensional Navier-Stokes equations in this case are reduced to a closed determining system that consists of six equations with partial derivatives of the third and second orders. A brief review of the known exact solutions of this system and the respective flows of a fluid (Couette-Poiseuille, Ekman, Stokes, Karman, and other flows) is given. The cases of reducing a determining system to one or two equations are described. Many new exact solutions of two-dimensional and three-dimensional nonstationary Navier-Stokes equations containing arbitrary functions and arbitrary parameters are derived. Periodic (both in spatial coordinates and in time) and some other solutions that are expressed in terms of elementary functions are described. The problems of the nonlinear stability of solutions are studied. A number of new hydrodynamic problems are considered. A general interpretation of the solutions as the main terms of the Taylor series expansion in terms of radial coordinates is given.
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References
Kutepov, A.M. Polyanin, A.D., et al., Khimicheskaya gidrodinamika (Chemical Hydrodynamics), Moscow: Byuro Kvantum, 1996.
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.
Loitsyanskii, L.G., Mekhanika zhidkosti i gaza (Fluid Mechanics), Moscow: Nauka, 1973.
Kochin, N.E., Kibel’, I.A., and Roze, N.V., Teoreticheskaya gidromekhanika. (Theoretical Hydromechanics), Fizmatgiz, 1963.
Landau, L.D. and Lifshitz, E.M., Gidrodinamika (Hydrodynamics), Moscow: Nauka, 1986.
Schlichting, H., Boundary Layer Theory, New York: McGraw-Hill, 1974.
Pukhnachev, V.V., Symmetries in the Navier-Stokes Equations, Usp. Mekh., 2006, no. 6, pp. 3–76.
Drazin, P.G. and Riley, N., The Navier-Stokes Equations: A Classification of Flows and Exact Solutions, Cambridge: Cambridge Univ. Press, 2006.
Polyanin, A.D. and Zaitsev, V.F., Handbook of Nonlinear Partial Differential Equations, Boca Raton, FL: Chapman & Hall / CRC, 2004.
Ludlow, D.K., Clarkson, P.A., and Bassom, A.P., Nonclassical Symmetry Reductions of the Two-Dimensional Incompressible Navier-Stokes Equations, Stud. Appl. Math., 1999, vol. 103, pp. 183–240.
Ludlow, D.K., Clarkson, P.A., and Bassom, A.P., Nonclassical Symmetry Reductions of the Three-Dimensional Incompressible Navier-Stokes Equations, J. Phys., A: Math. Gen., 1998, vol. 31, pp. 7965–7980.
Ibragimov, N.H., CRC Handbook of Lie Group to Differential Equations, Boca Raton, FL: CRC, 1995, vol. 2.
Lin, C.C., Note on a Class of Exact Solutions in Magneto-Hydrodynamics, Arch. Rational Mech. Anal., 1958, vol. 1, pp. 391–395.
Sidorov, A.F., On Two Classes of the Solutions of the Equations of Fluid Mechanics and Their Relation to the Theory of Traveling Waves, Prikl. Mekh. Tekh. Fiz., 1989, no. 2, pp. 34–40.
Meleshko, S.V. and Pukhnachev, V.V., On One Class of the Partially Invariant Solutions of the Navier-Stokes Equations, Prikl. Mekh. Tekh. Fiz., 1999, no. 2, pp. 24–33.
Meleshko, S.V., A Particular Class of Partially Invariant Solutions of the Navier-Stokes Equations, Nonlinear Dyn., 2004, vol. 36, no. 1, pp. 47–68.
Hagen, G., Uber die Bewegung des Wasser in engen zylindrischen Rohren, Pogg. Ann., 1839, vol. 46, pp. 423–442.
Poiseuille, J., Recherches experimentelles sur le mouvement des fluides dans les tubes de tres petits diametres, Comptes Rendus, 1841, vol. 12, pp. 112–115.
Couette, M., Etudes sur le frottement des fluides, Ann. Chim. Phys., 1890, vol. 21, pp. 433–510.
Stokes, G.G., On the Effect of the Internal Friction of Fluid on the Motion of Pendulums, Trans. Cambridge Philos. Soc., 1851, vol. 9, pp. 8–106.
Ekman, V.W., On the Influence of the Earth’s Rotation on Ocean Currents, Ark. Mat. Astron. Fys., 1905, vol. 2, pp. 1–5.
Berker, R., A New Solution of the Navier-Stokes Equations for the Motion of a Fluid Contained between Parallel Plates Rotating about the Same Axis, Arch. Mech. Stosow, 1979, vol. 31, no. 2, pp. 265–280.
Berker, R., An Exact Solution of the Navier-Stokes Equation: the Vortex with Curvilinear Axis, Int. J. Eng. Sci, 1981, vol. 20, no. 2, pp. 217–230.
Lai, C.-Y., Rajagopal, K.R., and Szeri, A.Z., Asymmetric Flow between Parallel Rotating Disks, J. Fluid Mech., 1984, vol. 446, pp. 203–225.
Polyanin, A.D., Exact Solutions of the Navier-Stokes Equations with the Generalized Separation of Variables, Dokl. Akad. Nauk, 2001, vol. 380, no. 4, pp. 491–496 [Dokl. Phys. (Engl. Transl.), vol. 46, no. 10, pp. 726–731].
Polyanin, A.D. and Zaitsev, V.F., Equations of the Nonstationary Boundary Layer: General Transformations and Exact Solutions, Teor. Osn. Khim. Technol., 2001, vol. 35, no. 6, pp. 563–573.
Hiemenz, K., Die Grenzschicht An Einem in Den Gleichformigen Flussigkeitsstrom Eingetauchten Geraden Kreiszylinder, Dinglers Polytech. J, 1911, vol. 326, pp. 321–324.
Stuart, J.T., The Viscous Flow near a Stagnation Point When the External Flow Has Uniform Vorticity, J. Aerosp. Sci., 1959, vol. 26, pp. 124–125.
Tamada, K., Two-Dimensional Stagnation Point Flow Impinging Obliquely on a Plane Wall, J. Phys. Soc. Jpn., 1979, vol. 46, pp. 310–311.
Dorrepaal, J.M., An Exact Solution of the Navier-Stokes Equation Which Describes Non-Orthogonal Stagnation Point Flow in Two Dimensions, J. Fluid Mech., 1986, vol. 163, pp. 141–147.
Wang, C.Y., Exact Solutions of the Unsteady Navier-Stokes Equations, Appl. Mech. Rev., 1989, vol. 42, no. 11, pp. 269–282.
Wang, C.Y., Exact Solutions of the Steady-State Navier-Stokes Equations, Annu. Rev. Fluid Mech., 1991, vol. 23, pp. 159–177.
Wang, C.Y., Exact Solutions of the Navier-Stokes Equations — the Generalized Beltrami Flows, Review Extension, Acta Mech., 1990, vol. 81, pp. 69–74.
Crane, L.J., Flow past a Stretching Plate, Z. Angew. Math. Phys., 1970, vol. 21, pp. 645–647.
Brady, J.F. and Acrivos, A., Steady Flow in a Channel or Tube with an Accelerating Surface Velocity. An Exact Solution to the Navier-Stokes Equations with Reverse Flow, J. Fluid Mech., 1981, vol. 112, pp. 127–150.
Berman, A.S., Laminar Flow in Channels with Porous Walls, J. Appl. Phys., 1953, vol. 24, pp. 1232–1235.
Terrill, R.M. and Shrestha, G.M., Laminar Flow through Parallel and Uniformly Porous Walls of Different Permeability, Z. Angew. Math. Phys., 1965, vol. 16, pp. 470–482.
Novikov, P.A. and Lyubin, L.Ya., Gidrodinamika shchelevykh sistem (Hydrodynamics of Slot Systems), Minsk: Nauka i Tekhnika, 1988.
von Karman, T., Uber Laminare und Turbulente Reibung, Z. Angew. Math. Mech., 1921, vol. 1, pp. 233–252.
Bodewadt, U.T., Die Drehstromung uber Festem Grunde, Z. Angew. Math. Mech., 1940, vol. 20, pp. 241–253.
Batchelor, G.K., Note on Class of Solutions of the Navier-Stokes Equations Representing Steady Rotationally — Symmetric Flow, Q. J. Mech. Appl. Math., 1951, vol. 4, pp. 29–41.
Stewartson, K., On the Flow between Two Rotating Coaxial Disks, Proc. Cambridge Philos. Soc., 1953, vol. 5, pp. 333–341.
Holodniok, M., Kubicek, M., and Hlavacek, V., Computation of the Flow between Two Rotating Coaxial Disk: Multiplicity of Steady-State Solutions, J. Fluid Mech., 1981, vol. 108, pp. 227–240.
Brady, J.F. and Durlofsky, L., On Rotating Disk Flow, J. Fluid Mech., 1987, vol. 175, pp. 363–394.
Watson, L.T. and Wang, C.Y., Deceleration of a Rotating Disc in a Viscous Fluid, Phys. Fluids, 1979, vol. 22, pp. 2267–2269.
Lavrent’eva, O.M., Flow of a Viscous Fluid in a Layer on a Rotating Plane, Prikl. Mekh. Tekh. Fiz., 1989, no. 5, pp. 41–48.
Agrawal, H.L., A New Exact Solution of the Equations of Viscous Motion with Axial Symmetry, Q. J. Mech. Appl. Math., 1957, vol. 10, pp. 42–44.
Terrill, R.M. and Cornish, J.P., Radial Flow of a Viscous, Incompressible Fluid between Two Stationary Uniformly Porous Discs, Z. Angew. Math. Phys., 1973, vol. 24, pp. 676–688.
Aristov, S.N., Stationary Cylindrical Vortex in a Viscous Fluid, Dokl. Akad. Nauk, 2001, vol. 377, no. 4, pp. 477–480 [Dokl. Phys. (Engl. Transl.), vol. 46, no. 4, pp. 251–253].
Aristov, S.N. and Gitman, I.M., Viscous Flow between Two Moving Parallel Disks. Exact Solutions and Stability Analysis, J. Fluid Mech., 2002, vol. 464, pp. 209–215.
Polyanin, A.D., Handbook of Linear Partial Differential Equations for Engineers and Scientists, Boca Raton, FL: Chapman & Hall/CRC, 2002.
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Original Russian Text © S.N. Aristov, D.V. Knyazev, A.D. Polyanin, 2009, published in Teoreticheskie Osnovy Khimicheskoi Tekhnologii, 2009, Vol. 43, No. 5, pp. 547–566.
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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. Theor Found Chem Eng 43, 642 (2009). https://doi.org/10.1134/S0040579509050066
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DOI: https://doi.org/10.1134/S0040579509050066