Abstract
In this chapter, we introduce a method of imposing asymmetric conditions on the velocity vector with respect to independent spatial variables and a moving-frame method to solve the three-dimensional Navier–Stokes equations. Seven families of unsteady rotating asymmetric solutions with various parameters are obtained. In particular, one family of solutions blows up on a moving plane, which may be used to study abrupt high-speed rotating flows. Using Fourier expansion and two families of our solutions, one can obtain discontinuous solutions that may be useful in the study of shock waves. Another family of solutions is partially cylindrical invariant, containing two parameter functions of t, which may be used to describe an incompressible fluid in a nozzle. Most of our solutions are globally analytic with respect to spatial variables.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Brutyan, M.A., Krapivsky, P.L.: Exact solutions of Navier–Stokes equations describing the evolution of a vortex structure in generalized shear flow. Comput. Math. Math. Phys. 32, 270–272 (1992)
Bytev, V.O.: Invariant solutions of the Navier–Stokes equations. Zh. Prikl. Mekh. Tekh. Fiz. 6, 56 (1972)
Cao, B.: Asymmetric and moving-frame approaches to MHD equations. Acta Math. Sin. 28(1), 1–36 (2012)
Gryn, V.I.: Exact solutions of Navier–Stokes equations. J. Appl. Math. Mech. 55, 301–309 (1991)
Kapitanskii, L.B.: Group analysis of Navier–Stokes equations and Euler equations with rotational symmetry and new exact solutions of these equations. Dokl. Akad. Nauk SSSR 243, 901 (1978)
Landau, L.: A new exact solutions of Navier–Stokes equations. Dokl. Akad. Nauk SSSR 43, 286–288 (1944)
Leipnik, R.B.: Exact solutions of Navier–Stokes equations by recursive series of diffusive equations. C. R. Math. Acad. Sci. 18, 211–216 (1996)
Polyanin, A.D.: Exact solutions of the Navier–Stokes equations with generalized separation of variables. Dokl. Phys. 46, 726–731 (2001)
Pukhnachev, V.V.: Invariant solutions of Navier–Stokes equations describing motions with free boundary. Dokl. Akad. Nauk SSSR 202, 302 (1972)
Shen, H.-C.: The theory of functions of a complex variable under Dirac-Pauli representation and its application in fluid dynamics I. Appl. Math. Mech. 7, 391–411 (1986a)
Shen, H.-C.: Exact solutions of Navier–Stokes equations—the theory of functions of a complex variable under Dirac–Pauli representation and its application in fluid dynamics II. Appl. Math. Mech. 7, 557–562 (1986b)
Vyskrebtsov, V.G.: New exact solutions of Navier–Stokes equations for axisymmetric self-similar fluid flows. J. Math. Sci. (N.Y.) 104, 1456–1463 (2001)
Xu, X.: Asymmetric and moving-frame approaches Navier–Stokes equations. Q. Appl. Math. 67, 163–193 (2009a)
Yakimov, A.Yu.: Exact solutions of Navier–Stokes equations in the presence of a vortex singularity on a ray. Dokl. Akad. Nauk SSSR 276, 79–82 (1984)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Xu, X. (2013). Navier–Stokes Equations. In: Algebraic Approaches to Partial Differential Equations. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-36874-5_9
Download citation
DOI: https://doi.org/10.1007/978-3-642-36874-5_9
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-36873-8
Online ISBN: 978-3-642-36874-5
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)