The integral invariant of the equations of motion of a viscous gas

Exact solutions to generalized plane Beltrami--Trkal and Ballabh flows

“…Various solutions are obtained, particularly, by the variable separation method. All these solutions of equation 6offer exact solutions to the Navier-Stokes equations by formulas (7) and (10). The obtained solutions will be nonstationary, but with fixed streamlines.…”

“…The essence of this method is the rearrangement of the initial equations to the form convenient for integration. As applied to the Navier-Stokes equations, this line of research is discussed in studies where new forms of writing the equations make it possible to obtain previously unknown invariants and hidden symmetries of the constitutive equations [3][4][5][6][7][8][9][10]. One of the methods for representing the motion equation (the Aristov-Pukhnachev method [6,8,9]) has been introduced to computational fluid dynamics [8,9].…”

“…One of the methods for representing the motion equation (the Aristov-Pukhnachev method [6,8,9]) has been introduced to computational fluid dynamics [8,9]. The numerical solutions of the Navier-Stokes axisymmetric equations were tested by conventional procedures [8,9], which took no account of the invariant Helmholtz theorems and their extensions [10].…”

Towards understanding the algorithms for solving the Navier–Stokes equations

“…,Aristov and Pukhnachev 2004, Pukhnachev 2004, Moshkin and Poochinapan 2010, Moshkin et al 2010, Golubkin et al 2015 in which new forms of presenting the equations allow us to obtain previously unknown invariants and hidden symmetries of the defining relations to which they are devoted by. One of the ways of representing the equations of a rotationally symmetric and plane motion, Aristov-Pukhnachev method(Aristov and Pukhnachev 2004, Pukhnachev 2004, Moshkin and Poochinapan 2010, Moshkin et al 2010 is introduced into the practical computational fluid dynamics (Moshkin and Poochinapan 2010, Moshkin et al 2010), because in article(Aristov and Pukhnachev 2004) the methods for setting boundary conditions for a fairly wide class of areas that a fluid may occupy have been developed.…”