Group properties and new solutions of Navier-Stokes equations

Abstract

Using the machinery of Lie theory (groups and algebras) applied to the Navier-Stokes equations a number of exact solutions for the steady state are derived in (two) three dimensions. It is then shown how each of these generates an infinite number of time-dependent solutions via (three) four arbitrary functions of time. This algebraic structure also provides the mechanism to search for other solutions since its character is inferred from the basic equations.

Analysis of fluid equations by group methods

“…Thus any solution of the steady equations generates an infinity of time-dependent solutions. This idea was exploited in Boisvert et al [5] and Nucci [6].…”

“…The determination of the full group requires extremely lengthy calculations. Detailed calculations can be found in Ames [1], Ovsiannikov [3], and for the Navier-Stokes equations in Boisvert [4] (see also Boisvert et al [5]). Algebraic programming packages for determining these groups have been developed by Schwarz using REDUCE [9], by Roseneau and Schwarzmeier using MACSYMA [10] and CINO in Russia (see Ovsiannikov [3], p. 57).…”

Group method analysis of unsteady free-convective laminar boundary-layer flow on a nonisothermal vertical flat plate

“…Similarity analysis has been applied intensively by Gabbert [12]. For additional discussions on group transformations, one consults Ames [1][2][3], Eisenhart, Bluman and Cole [6], Boisvert et al [7], Moran and Gaggioli [24,25] and [27]..…”

Optimal System and Invariant Solutions on ((U<sub>yy</sub>(t,s,y)－U<sub>t</sub>(t,s,y))y－2sU<sub>sy</sub>(t,s,y))y＋（s<sup>2</sup>＋1)U<sub>ss</sub>(t,s,y)＋2sU<sub>s</sub>=0

“…Several articles [5][6][7][8][9][10][11] are devoted to invariant solutions of the Navier-Stokes equations. While partially invariant solutions of the Navier-Stokes equations have been less studied, there has been substantial progress in studying such classes of solutions of inviscid gas dynamics equations [12][13][14][15][16][17][18][19].…”