Symmetries and Conservation Laws of Navier—Stokes Equations

1989 ◽  
pp. 65-81
Author(s):  
V. N. Gusyatnikova ◽  
V. A. Yumaguzhin
Keyword(s):  
Conservation Laws ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Navier Stokes Equations

Euler and Navier–Stokes equations in a new time-dependent helically symmetric system: derivation of the fundamental system and new conservation laws

2017 ◽  
Vol 818 ◽  
pp. 344-365 ◽  
Author(s):  
Dominik Dierkes ◽  
Martin Oberlack
Keyword(s):  
Conservation Laws ◽  
Coordinate System ◽  
Stokes Equations ◽  
Time Dependent ◽  
Navier Stokes ◽  
Symmetric System ◽  
Present Contribution ◽  
Navier Stokes Equations ◽  
Constant Pitch ◽  
New Time

The present contribution is a significant extension of the work by Kelbin et al. (J. Fluid Mech., vol. 721, 2013, pp. 340–366) as a new time-dependent helical coordinate system has been introduced. For this, Lie symmetry methods have been employed such that the spatial dependence of the originally three independent variables is reduced by one and the remaining variables are: the cylindrical radius $r$ and the time-dependent helical variable $\unicode[STIX]{x1D709}=(z/\unicode[STIX]{x1D6FC}(t))+b\unicode[STIX]{x1D711}$, $b=\text{const.}$ and time $t$. The variables $z$ and $\unicode[STIX]{x1D711}$ are the usual cylindrical coordinates and $\unicode[STIX]{x1D6FC}(t)$ is an arbitrary function of time $t$. Assuming $\unicode[STIX]{x1D6FC}=\text{const.}$, we retain the classical helically symmetric case. Using this, and imposing helical invariance onto the equation of motion, leads to a helically symmetric system of Euler and Navier–Stokes equations with a time-dependent pitch $\unicode[STIX]{x1D6FC}(t)$, which may be varied arbitrarily and which is explicitly contained in all of the latter equations. This has been conducted both for primitive variables as well as for the vorticity formulation. Hence a significantly extended set of helically invariant flows may be considered, which may be altered by an external time-dependent strain along the axis of the helix. Finally, we sought new conservation laws which can be found from the helically invariant Euler and Navier–Stokes equations derived herein. Most of these new conservation laws are considerable extensions of existing conservation laws for helical flows at a constant pitch. Interestingly enough, certain classical conservation laws do not admit extensions in the new time-dependent coordinate system.

scholarly journals Conservation Laws for Some Systems of Nonlinear Partial Differential Equations via Multiplier Approach

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Rehana Naz
Keyword(s):  
Conservation Laws ◽  
Stokes Equations ◽  
Nonlinear Partial Differential Equations ◽  
Type System ◽  
Coupled System ◽  
Magnetized Plasma ◽  
Navier Stokes ◽  
Local Conservation ◽  
Navier Stokes Equations ◽  
Dependent Variables

The conservation laws for the integrable coupled KDV type system, complexly coupled kdv system, coupled system arising from complex-valued KDV in magnetized plasma, Ito integrable system, and Navier stokes equations of gas dynamics are computed by multipliers approach. First of all, we calculate the multipliers depending on dependent variables, independent variables, and derivatives of dependent variables up to some fixed order. The conservation laws fluxes are computed corresponding to each conserved vector. For all understudying systems, the local conservation laws are established by utilizing the multiplier approach.

Finite surface discretization for incompressible Navier-Stokes equations and coupled conservation laws

2020 ◽  
Vol 423 ◽  
pp. 109790
Author(s):  
Arpiruk Hokpunna ◽  
Takashi Misaka ◽  
Shigeru Obayashi ◽  
Somchai Wongwises ◽  
Michael Manhart
Keyword(s):  
Conservation Laws ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Navier Stokes Equations

Symmetry-invariant conservation laws of partial differential equations

2017 ◽  
Vol 29 (1) ◽  
pp. 78-117 ◽  
Author(s):  
STEPHEN C. ANCO ◽  
ABDUL H. KARA
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Conservation Laws ◽  
Stokes Equations ◽  
Two Dimensions ◽  
Navier Stokes ◽  
Partial Differential ◽  
Navier Stokes Equations ◽  
General Method

A simple characterization of the action of symmetries on conservation laws of partial differential equations is studied by using the general method of conservation law multipliers. This action is used to define symmetry-invariant and symmetry-homogeneous conservation laws. The main results are applied to several examples of physically interest, including the generalized Korteveg-de Vries equation, a non-Newtonian generalization of Burger's equation, theb-family of peakon equations, and the Navier–Stokes equations for compressible, viscous fluids in two dimensions.

scholarly journals Influence of the Angular Momentum in Problems Continuum Mechanics

2021 ◽  
Vol 16 ◽  
pp. 1-8
Author(s):  
Evelina Prozorova
Keyword(s):  
Conservation Laws ◽  
Continuum Mechanics ◽  
Stokes Equations ◽  
Theory Of Elasticity ◽  
Navier Stokes ◽  
Physical Parameters ◽  
Navier Stokes Equations ◽  
Potential Flows ◽  
Balance Of Forces ◽  
And Mathematics

- For continuum mechanics a model is proposed, that is built with consideration outside the integral term when deriving conservation laws using the Ostrogradsky-Gauss theorem. Performed analysis shows discrepancy between accepted classical conservation laws and classical theoretical mechanics and mathematics. As a result, the theory developed for potential flows was extended to flows with significant gradients of physical parameters. We have proposed a model that takes into account the joint implementation of the laws for balance of forces and angular momentums. It does not follow from the Boltzmann equation that the pressure in the Euler and Navier-Stokes equations is equal to one third of the sum the pressures on the corresponding coordinate axes. The vector definition of pressure is substantiated. It is shown that the symmetry condition for the stress tensor is one of the possible conditions for closing the problem. An example of solving the problem of the theory of elasticity is given

Symmetries and conservation laws of Navier-Stokes equations

1989 ◽  
Vol 15 (1-2) ◽  
pp. 65-81 ◽  
Author(s):  
V. N. Gusyatnikova ◽  
V. A. Yumaguzhin
Keyword(s):  
Conservation Laws ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Navier Stokes Equations

scholarly journals Role of Conservation Laws in the Development of Nonequilibrium and Emergence of Turbulence

2018 ◽  
Vol 14 (2) ◽  
pp. 7682-7690
Author(s):  
Ludmila Ivanovna Petrova
Keyword(s):  
Conservation Laws ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Coordinate Systems ◽  
Navier Stokes Equations ◽  
Equations Of Mathematical Physics ◽  
Physical Quantities ◽  
Numerical And Analytical Methods ◽  
Turbulent Pulsations

It turns out that the equations of mathematical physics, which consist equations of the conservation laws for energy, linear momentum, angular momentum, and mass, possess additional, hidden, properties that enables one to describe not only a variation of physical quantities (such as energy, pressure, density) but also processes such as origination of waves, vortices, turbulent pulsations and other ones. It is caused by the conservation laws properties. In present paper the development of nonequilibrium in gasdynamic systems, which are described by the Euler and Navier-Stokes equations, will be investigated.  Under studying the consistence of conservation laws equations, from the Euler and Navier-Stokes equations it can be obtained the evolutionary relation for entropy (as a state functional).  The evolutionary relation possesses a certain peculiarity, namely, it turns out to be nonidentical. This fact points out to inconsistence of the conservation law equations and noncommutativity of conservation laws. Such a nonidentical relation discloses peculiarities of the solutions to the Navier-Stokes equations due to which the Euler and Navier-Stokes equations can describe the processes the development of nonequilibrium and emergence of vortices and turbulence. It has been shown that such processes can be described only with the help of two nonequivalent coordinate systems or by simultaneous using numerical and analytical methods.

scholarly journals Invariant analysis, exact solutions and conservation laws of (2+1)-dimensional time fractional Navier–Stokes equations

2021 ◽  
Vol 477 (2250) ◽  
pp. 20210220
Author(s):  
Xiaoyu Cheng ◽  
Lizhen Wang
Keyword(s):  
Conservation Laws ◽  
Exact Solutions ◽  
Stokes Equations ◽  
Power Series Expansion ◽  
Expansion Method ◽  
Optimal System ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
One Dimensional ◽  
Fractional Differential Operator

In this paper, we investigate the exact solutions and conservation laws of (2 + 1)-dimensional time fractional Navier–Stokes equations (TFNSE). Specifically, Lie symmetries and corresponding one-dimensional optimal system for TFNSE in Riemann–Liouville sense are obtained. Then, based on the admitted symmetries and optimal system, we reduce these equations to one-dimensional equations or (1 + 1)-dimensional fractional partial differential equations (PDEs) with the help of Erdélyi–Kober fractional differential operator and compound variable transformation. In addition, we solve the reduced PDEs applying power series expansion method and invariant subspace method, respectively. Furthermore, the conservation laws of TFNSE are derived using new Noether theorem.

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