scholarly journals On exact solutions of quasi-hydrodynamic system that don't satisfy the Navier-Stokes and Euler systems

2021 ◽  
pp. 5-15
Author(s):  
Вера Владимировна Григорьева ◽  
Юрий Владимирович Шеретов
Keyword(s):  
Exact Solutions ◽  
Euler Equations ◽  
Stokes System ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Sonic Velocity ◽  
Navier Stokes Equations ◽  
Euler Systems ◽  
New Class ◽  
Hydrodynamic System

Квазигидродинамическая система была предложена Шеретовым Ю.В. в 1993 году. Известные точные решения этой системы в подавляющем большинстве случаев удовлетворяют либо уравнениям Навье-Стокса, либо уравнениям Эйлера. В настоящей работе описан новый класс точных решений квазигидродинамической системы, которые не удовлетворяют ни уравнениям Навье-Стокса, ни уравнениям Эйлера. Соответствующие точные решения системы Навье-Стокса получаются из построенных решений предельным переходом при $c_s\to +\infty$, где $c_s$ - скорость звука в жидкости. The quasi-hydrodynamic system was proposed by Sheretov Yu.V. in 1993. The known exact solutions of this system in the overwhelming majority of cases satisfy either the Navier-Stokes equations or the Euler equations. This paper describes a new class of exact solutions of quasi-hydrodynamic system that satisfy neither the Navier-Stokes equations, nor the Euler equations. The corresponding exact solutions of the Navier-Stokes system are obtained from the constructed solutions by passing to the limit at $c_s\to +\infty$, where $c_s$ is the sonic velocity in the fluid.

Uniqueness of the exact solutions of the Navier—Stokes equations having null nonlinearity

2006 ◽  
Vol 136 (6) ◽  
pp. 1303-1315 ◽  
Author(s):  
Sun-Chul Kim ◽  
Hisashi Okamoto
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Exact Solutions ◽  
Euler Equations ◽  
Stokes Equations ◽  
Elliptic Partial Differential Equations ◽  
Navier Stokes ◽  
Classical Solutions ◽  
Overdetermined System ◽  
Navier Stokes Equations

We consider an overdetermined system of elliptic partial differential equations arising in the Navier–Stokes equations. This analysis enables us to prove that the well-known classical solutions such as Couette flows and others are the only solutions that satisfy both the stationary Navier–Stokes and Euler equations.

scholarly journals A new class of non-helical exact solutions of the Navier-Stokes equations

2020 ◽  
Vol 24 (4) ◽  
pp. 762-768
Author(s):  
Vitalii Petrovich Kovalev ◽  
Eugenii Yurevich Prosviryakov
Keyword(s):  
Exact Solutions ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
New Class

Приведен новый класс точных решений уравнений Навье-Стокса. Эти решения описывают нестационарные трехмерные по скоростям и двумерные по координатам течения вязкой несжимаемой жидкости. Процедура построения точного решения обобщает метод Тркала, предложенный для изучения винтовых течений. Новый класс точных решений позволяет описывать невинтовые течения (вектор скорости образует ненулевой угол с вектором завихренности) и течения жидкости, существующие конечное время.

Development of mixing and isotropy in inviscid homogeneous turbulence

1982 ◽  
Vol 120 ◽  
pp. 155-183 ◽  
Author(s):  
Jon Lee
Keyword(s):  
Dynamical Systems ◽  
Lower Order ◽  
Stokes System ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Navier Stokes ◽  
Fourier Space ◽  
Kolmogorov Entropy ◽  
Navier Stokes Equations ◽  
Constant Energy

We have investigated a sequence of dynamical systems corresponding to spherical truncations of the incompressible three-dimensional Navier-Stokes equations in Fourier space. For lower-order truncated systems up to the spherical truncation of wavenumber radius 4, it is concluded that the inviscid Navier-Stokes system will develop mixing (and a fortiori ergodicity) on the constant energy-helicity surface, and also isotropy of the covariance spectral tensor. This conclusion is, however, drawn not directly from the mixing definition but from the observation that one cannot evolve the trajectory numerically much beyond several characteristic corre- lation times of the smallest eddy owing to the accumulation of round-off errors. The limited evolution time is a manifestation of trajectory instability (exponential orbit separation) which underlies not only mixing, but also the stronger dynamical charac- terization of positive Kolmogorov entropy (K-system).

On the Generalized Navier-Stokes Equations

2003 ◽  
Author(s):  
Moustafa El-Shahed ◽  
Ahmed Salem
Keyword(s):  
Exact Solutions ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Hankel Transforms ◽  
Fourier Sine ◽  
Special Cases

In this paper, we present a general Inodel of the classical Navier-Stokes equations. With the help of Laplace, Fourier Sine transforms, finite Fourier Sine transforms, and finite Hankel transforms, an exact solutions for three different special cases have been obtained.

On the energy dissipative spatial discretization of the barotropic quasi-gasdynamic and compressible Navier–Stokes equations in polar coordinates

2018 ◽  
Vol 33 (3) ◽  
pp. 199-210 ◽  
Author(s):  
Alexander Zlotnik
Keyword(s):  
Stokes System ◽  
Stokes Equations ◽  
Physical Property ◽  
Polar Coordinates ◽  
Navier Stokes ◽  
Uniform Mesh ◽  
Viscous Stress ◽  
Spatial Discretization ◽  
Navier Stokes Equations ◽  
Viscous Stress Tensor

Abstract The barotropic quasi-gasdynamic system of equations in polar coordinates is treated. It can be considered a kinetically motivated parabolic regularization of the compressible Navier–Stokes system involving additional 2nd order terms with a regularizing parameter τ > 0. A potential body force is taken into account. The energy equality is proved ensuring that the total energy is non-increasing in time. This is the crucial physical property. The main result is the construction of symmetric spatial discretization on a non-uniform mesh in a ring such that the property is preserved. The unknown density and velocity are defined on the same mesh whereas the mass flux and the viscous stress tensor are defined on the staggered meshes. Additional difficulties in comparison with the Cartesian coordinates are overcome, and a number of novel elements are implemented to this end, in particular, a self-adjoint and positive definite discretization for the Navier–Stokes viscous stress, special discretizations of the pressure gradient and regularizing terms using enthalpy, non-standard mesh averages for various products of functions, etc. The discretization is also well-balanced. The main results are valid for τ = 0 as well, i.e., for the barotropic compressible Navier–Stokes system.

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