Symmetry reduction and exact solutions of the Navier-Stokes equations. I

Abstract

Ansatzes for the Navier-Stokes field are described. These ansatzes reduce the Navier-Stokes equations to system of differential equations in three, two, and one independent variables. The large sets of exact solutions of the Navier-Stokes equations are constructed.

One Class of Regular Partially Invariant Solutions of the Navier–Stokes Equations

“…The case H = 0 corresponds to spherically symmetric flows. After substituting the representation (19) into the Navier-Stokes equations one has…”

“…Several articles [11][12][13][14][15][16][17][18][19][20][21] are devoted to invariant solutions of the Navier-Stokes equations. 3 While partially invariant solutions of the Navier-Stokes equations have been less studied, 4 there has been substantial progress in studying such classes of solutions of inviscid gas dynamics equations [1,3,6,[22][23][24][25][26][27].…”

“…Short reviews devoted to invariant solutions of the Navier-Stokes equations can be found in[11,13,[19][20][21] 4. Firstly, the approach of partially invariant solutions to the Navier-Stokes equations was applied in[11].…”

Optimal System of Subalgebras for the Reduction of the Navier-Stokes Equations

“…Classification of infinite-dimensional subalgebras of this algebra was studied in [6]. 3 Short reviews devoted to invariant solutions of the Navier-Stokes equations can be found in [14][15][16]. 4 Firstly the approach of partially invariant solutions to the NavierStokes equations was applied in [17].…”

Symmetry analysis for Whitham-Broer-Kaup equations

“…Given any subgroup of the symmetry group, one can write down the equations for the similarity solution with respect to this subgroup. This reduced system is of fewer variables and easier to solve generally [1,2,3,4,5]. But a Lie group (or Lie algebra) usually contains infinitely many subgroups (or subalgebras) of the same dimension, it is not usually feasible to list all possible similarity solutions.…”