Nonclassical symmetry reductions of the three-dimensional incompressible Navier-Stokes equations

The nonclassical reduction method as pioneered by Bluman and Cole ( J. Meth. Mech. 18 1025-42) is used to examine symmetries of the full three-dimensional, unsteady, incompressible Navier-Stokes equations of fluid mechanics. The procedure, when applied to a system of partial differential equations, yields reduced sets of equations with one fewer independent variables. We find eight possibilities for reducing the Navier-Stokes equations in the three spatial and one temporal dimensions to sets of partial differential equations in three independent variables. Some of these reductions are derivable using the Lie-group method of classical symmetries but the remainder are genuinely nonclassical. Further investigations of one of our eight forms shows how it is possible to derive novel exact solutions of the Navier-Stokes equations by the nonclassical method.

[1]  M. Kruskal,et al.  New similarity reductions of the Boussinesq equation , 1989 .

[2]  Willy Hereman,et al.  The computer calculation of Lie point symmetries of large systems of differential equations , 1991 .

[3]  Elizabeth L. Mansfield,et al.  Algorithms for the Nonclassical Method of Symmetry Reductions , 1994, SIAM J. Appl. Math..

[4]  W. Miller,et al.  Group analysis of differential equations , 1982 .

[5]  P. Clarkson,et al.  Symmetry reductions and exact solutions of shallow water wave equations , 1994, solv-int/9409003.

[6]  P. Clarkson,et al.  Symmetries and exact solutions of a (2+1)-dimensional sine-Gordon system , 1994, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[7]  B. Cantwell.  Similarity transformations for the two-dimensional, unsteady, stream-function equation , 1978, Journal of Fluid Mechanics.

[8]  Elizabeth L. Mansfield,et al.  Applications of the Differential Algebra Package diffgrob2 to Classical Symmetries of Differential Equations , 1997, J. Symb. Comput..

[9]  P. Clarkson,et al.  Symmetries and exact solutions for a 2 + 1-dimensional shallow water wave equation , 1997 .

[10]  P. Clarkson,et al.  Nonclassical symmetry reductions and exact solutions of the Zabolotskaya–Khokhlov equation , 1992, European Journal of Applied Mathematics.

[11]  Chao-Yang Wang,et al.  Exact Solutions of the Unsteady Navier-Stokes Equations , 1989 .

[12]  W. F. Ames,et al.  Group properties and new solutions of Navier-Stokes equations , 1983 .

[13]  G. I. Barenblatt.  Scaling: Self-similarity and intermediate asymptotics , 1996 .

[14]  Peter A. Clarkson,et al.  Nonclassical symmetry reductions of the Boussinesq equation , 1995 .

[15]  P. Olver.  Applications of lie groups to differential equations , 1986 .

[16]  A. Grauel,et al.  Similarity solutions of the Euler equation and the Navier-Stokes equation in two space dimensions , 1985 .

[17]  R. Popovych,et al.  Symmetry reduction and exact solutions of the Navier-Stokes equations. I , 1994, math-ph/0207016.

[18]  G. Bluman,et al.  Symmetries and differential equations , 1989 .

[19]  Symmetry reduction and exact solutions of the Navier-Stokes equations. II , 1994 .

[20]  W. I. Fushchich,et al.  Reduction and exact solutions of the Navier-Stokes equations , 1991 .

[21]  Peter A. Clarkson,et al.  Nonclassical symmetry reductions for the Kadomtsev-Petviashvili equation , 1991 .