One class of partially invariant solutions of the Navier-Stokes equations

Meleshko, Sergey V.; Пухначев, В. В.

doi:10.1007/bf02468516

Cited by 43 publications

(19 citation statements)

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“…Solution (22) is a generalization of the classical Karman solution of Navier-Stokes equations [41,42]. The theoretical-group nature of this solution was detected in [43]. It turned out that this is a steady solution of a partially invariant submodel of Navier-Stokes equations with respect to a 5-parameter group defined by the operators X 1 , X 2 , Y 1 , Y 2 , X 12 .…”

Section: Motion In a Half-space Induced By Plane Rotationmentioning

confidence: 99%

Analysis of the Models of Motion of Aqueous Solutions of Polymers on the Basis of Their Exact Solutions

Frolovskaya

Пухначев

2018

Polymers

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Abstract:The qualitative properties of solutions of a hereditary model of motion of aqueous solutions of polymers, its modification in the limiting case of short relaxation times, and a similar second grade fluid model are studied. Unsteady shear flows are considered. In the first case, their properties are similar to those of motion of a usual viscous fluid. Other models can include weak discontinuities, which are retained in the course of fluid motion. Exact solutions are found by using the group analysis of the examined systems of equations. These solutions describe the fluid motion in a gap between coaxial rotating cylinders, the stagnation point flow, and the motion in a half-space induced by plane rotation (analog of the Karman vortex). The problem of motion of an aqueous solution of a polymer in a cylindrical tube under the action of a streamwise pressure gradient is considered. In this case, a flow with straight-line trajectories is possible (analog of the Hagen-Poiseuille flow). In contrast to the latter, however, the pressure in the flow considered here depends on all three spatial variables.

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Section: Motion In a Half-space Induced By Plane Rotationmentioning

confidence: 99%

Analysis of the Models of Motion of Aqueous Solutions of Polymers on the Basis of Their Exact Solutions

Frolovskaya

Пухначев

2018

Polymers

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“…Self-similar, invariant, partially invariant, and certain other exact solutions of the Navier-Stokes equations including those with generalized separation o f variables were considered, for example, in [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]. Below, the term "exact solutions" is used according to the definition given in [14, p. 10].…”

Section: Class Of Motions Of the Viscous Incompressible Fluid Under Cmentioning

confidence: 99%

“…For γ = 0, the structure of exact solution (5) and system (6)-(9) was obtained in [12] from other reasons by the investigation of the class of partially invariant solutions (the case of α = β = γ = 0 was considered in [7]). In [12], the group classification of system (6)-(9) was carried out for γ = 0, which resulted in singling out two types of time dependences for the determining functions: (i) α and β are constant, and (ii) α and β are proportional to t −2 (the exact solutions of the Navier-Stokes equations with a reasonably simple structure correspond to these dependences).…”

Section: Class Of Motions Of the Viscous Incompressible Fluid Under Cmentioning

confidence: 99%

New classes of exact solutions and some transformations of the Navier-Stokes equations

Aristov

Polyanin

2010

Russ. J. Math. Phys.

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New classes of exact solutions of the three-dimensional unsteady Navier-Stokes equations containing arbitrary functions and parameters are described. Various periodic and other solutions, which are expressed through elementary functions are obtained. The general physical interpretation and classification of solutions is given.Keywords: Navier-Stokes equations, exact solutions, periodic solutions, three-dimensional equations, unsteady equations. Class of motions of the viscous incompressible fluid under considerationSelf-similar, invariant, partially invariant, and certain other exact solutions of the Navier-Stokes equations including those with generalized separation o f variables were considered, for example, in [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]. Below, the term "exact solutions" is used according to the definition given in [14, p. 10].Three-dimensional unsteady motions of a viscous incompressible fluid are described by the Navier-Stokes and continuity equations:

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“…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 [18][19][20][21][22][23][24][25].…”

Section: Invariant and Partially Invariant Solutionsmentioning

confidence: 99%

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

Khamrod

2013

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The purpose of this paper is to find the admitted Lie group of the reduction of the Navier-Stokes equations U t s y U t s y y sU t s y y s U t s y sUwhere s z y  using the basic Lie symmetry method. This equation is constructed from the Navier-Stokes equations rising to a partially invariant solutions of the Navier-Stokes equations. Two-dimensional optimal system is determined for symmetry algebras obtained through classification of their subalgebras. Some invariant solutions are also found.

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One class of partially invariant solutions of the Navier-Stokes equations

Cited by 43 publications

References 1 publication

Analysis of the Models of Motion of Aqueous Solutions of Polymers on the Basis of Their Exact Solutions

Analysis of the Models of Motion of Aqueous Solutions of Polymers on the Basis of Their Exact Solutions

New classes of exact solutions and some transformations of the Navier-Stokes equations

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

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