Abstract
New classes of exact solutions of three-dimensional nonstationary Navier-Stokes equations are described. These solutions contain arbitrary functions. Many periodic solutions (both with respect to the spatial coordinate and with respect to time) and aperiodic solutions are obtained, which can be expressed in terms of elementary functions. A Crocco-type transformation is presented, which reduces the order of the equation for the longitudinal component of the velocity. Problems concerning the nonlinear stability/instability of the solutions thus obtained are investigated. It turns out that a specific feature of many solutions of the Navier-Stokes equations is their instability. It is shown that instability can take place not only for rather large Reynolds numbers but also for arbitrarily small ones (and can be independent of the velocity profile of the fluid). A general physical interpretation and classification of solutions is given.
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References
L. G. Loitsyanskii, Mechanics of Liquids and Gases (Begell House, New York, 1996).
A. D. Polyanin and V. F. Zaitsev, Handbook of Nonlinear Partial Differential Equations (Boca Raton: Chapman & Hall/CRC Press, 2004).
P. G. Drazin and N. Riley, The Navier-Stokes Equations: A Classification of Flows and Exact Solutions (Cambridge Univ. Press, Cambridge, 2006).
C. Y. Wang, “Exact Solutions of the Unsteady Navier-Stokes Equations,” Appl. Mech. Rew. 42(11), 269–282 (1989).
W. I. Fushchich, W. M. Shtelen, and S. L. Slavutsky, “Reduction and Exact Solutions of the Navier-Stokes Equations,” J. of Phys. A 24, 971–984 (1991).
N. H. Ibragimov, CRC Handbook of Lie Group to Differential Equations, Vol. 2 (CRC Press, Boca Raton, 1995).
D. K. Ludlow, P. A. Clarkson, and A. P. Bassom, “Nonclassical Symmetry Reductions of the Three-Dimensional Incompressible Navier-Stokes Equations,” J. of Phys. A 31, 7965–7980 (1998).
D. K. Ludlow, P. A. Clarkson, and A. P. Bassom, “Nonclassical Symmetry Reductions of the Two-Dimensional Incompressible Navier-Stokes Equations,” Stud. in Appl. Math. 103, 183–240 (1999).
S. V. Meleshko and V. V. Pukhnachev, “On a Class of Partially Invariant Solutions of the Navier-Stokes Equations,” Prikl. Mekh. Tekhn. Fiz. 40(2), 24–33 (1999) [J. Appl. Mech. Tech. Phys. 40 (2), 208–216 (1999)].
A. D. Polyanin, “Exact Solutions to the Navier-Stokes Equations with Generalized Separation of Variables,” Dokl. Phys. 46(10), 726–731 (2001).
S. N. Aristov and I. M. Gitman, “Viscous Flow Between Two Moving Parallel Disks: Exact Solutions and Stability Analysis,” J. Fluid Mech. 464, 209–215 (2002).
R. E. Hewitt and P. W. Duck, M. Al-Azhari, “Extensions to Three-Dimensional Flow in a Porous Channel,” Fluid Dynam. Res. 33, 17–39 (2002).
E. C. Dauenhauer and J. Majdalani, “Exact Self-Similarity Solution of the Navier-Stokes Equations for a Porous Channel with Orthogonally Moving Walls,” Phys. Fluids 15, 1485–1495 (2003).
S. V. Meleshko, “A Particular Class of Partially Invariant Solutions of the Navier-Stokes Equations,” Nonlinear Dynam. 36(1), 47–68 (2004).
A. D. Polyanin, V. F. Zaitsev, and A. I. Zhurov, Methods of Solving Nonlinear Equations of Mathematical Physics and Mechanics (Fizmatlit, Moscow, 2005).
V. V. Pukhnachev, “Symmetries in Navier-Stokes Equations,” Uspekhi Mekhaniki 4(1), 6–76 (2006).
C. C. Lin, “Note on a Class of Exact Solutions in Magnetohydrodynamics,” Arch. Ration. Mech. Anal. 1, 391–395 (1958).
L. D. Landau and E. M. Lifshitz, Fluid Mechanics, 3-rd ed. (Nauka, Moscow, 1986) (2nd English Ed., Butterworth-Heinemann, Oxford, 1987).
M. A. Gol’dshtik and V. N. Shtern, Hydrodynamical Stability and Turbulence (Nauka, Novosibirsk, 1977).
Hydrodynamics and Nonlinear Instabilities, C. Godréche and P. Manneville, Eds.(Cambridge Univ. Press, Cambridge, 1998).
A. D. Polyanin, A. M. Kutepov, A. V. Vyazmin, and D. A. Kazenin, Hydrodynamics, Mass and Heat Transfer in Chemical Engineering (London: Taylor & Francis, 2002).
F. Calogero, “A Solvable Nonlinear Wave Equation,” Stud. Appl. Math. 70(3), 189–199 (1984).
M. V. Pavlov, “The Calogero Equation and Liouville-Type Equations,” Theoret. and Math. Phys. 128(1), 927–932 (2001).
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Financially supported by Russian Foundation for Basic Research (under grants nos. 08-01-00553, 08-08-00530, 07-01-96003-r_ural_a, and 09-01-00343).
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Aristov, S.N., Polyanin, A.D. New classes of exact solutions and some transformations of the Navier-Stokes equations. Russ. J. Math. Phys. 17, 1–18 (2010). https://doi.org/10.1134/S1061920810010012
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DOI: https://doi.org/10.1134/S1061920810010012