scholarly journals Computer-assisted proof for the stationary solution existence of the Navier–Stokes equation over 3D domains

2022 ◽  
pp. 106223
Author(s):  
Xuefeng Liu ◽  
Mitsuhiro T. Nakao ◽  
Shin’ichi Oishi
Keyword(s):  
Stationary Solution ◽  
Stokes Equation ◽  
Navier Stokes ◽  
Computer Assisted ◽  
Solution Existence ◽  
Navier Stokes Equation ◽  
Computer Assisted Proof

scholarly journals Correct Expression of the Material Derivative in Continuum Physics

2020 ◽  
Author(s):  
Bohua Sun
Keyword(s):  
Stokes Equation ◽  
Velocity Gradient ◽  
Existence Condition ◽  
Form Solution ◽  
Navier Stokes ◽  
Solution Existence ◽  
Navier Stokes Equation ◽  
Material Derivative ◽  
The Solution Existence ◽  
Correct Expression

The material derivative is important in continuum physics. This Letter shows that the expression $\frac{d }{dt}=\frac{\partial }{\partial t}+(\bm v\cdot \bm \nabla)$, used in most literature and textbooks, is incorrect. The correct expression $ \frac{d (:)}{dt}=\frac{\partial }{\partial t}(:)+\bm v\cdot [\bm \nabla (:)]$ is formulated. The solution existence condition of Navier-Stokes equation has been proposed from its form-solution, the conclusion is that "\emph{The Navier-Stokes equation has a solution if and only if the determinant of flow velocity gradient is not zero, namely $\det (\bm \nabla \bm v)\neq 0$.}"

scholarly journals A Conjecture on the Solution Existence of the Navier-Stokes Equation

2020 ◽  
Author(s):  
Bohua Sun
Keyword(s):  
Flow Velocity ◽  
Stokes Equation ◽  
Velocity Gradient ◽  
Existence Condition ◽  
Navier Stokes ◽  
Solution Existence ◽  
Navier Stokes Equation ◽  
The Solution Existence

For the solution existence condition of the Navier-Stokes equation, we propose a conjecture as follows: "\emph{The Navier-Stokes equation has a solution if and only if the determinant of flow velocity gradient is not zero, namely $\det (\bm \nabla \bm v)\neq 0$.}"

scholarly journals Correct Expression of the Material Derivative in Continuum Physics

2020 ◽  
Author(s):  
Bohua Sun
Keyword(s):  
Stokes Equation ◽  
Velocity Gradient ◽  
Existence Condition ◽  
Form Solution ◽  
Navier Stokes ◽  
Solution Existence ◽  
Navier Stokes Equation ◽  
Material Derivative ◽  
The Solution Existence ◽  
Correct Expression

The material derivative is important in continuum physics. This Letter shows that the expression $\frac{d }{dt}=\frac{\partial }{\partial t}+(\bm v\cdot \bm \nabla)$, used in most literature and textbooks, is incorrect. The correct expression $ \frac{d (:)}{dt}=\frac{\partial }{\partial t}(:)+\bm v\cdot [\bm \nabla (:)]$ is formulated. The solution existence condition of Navier-Stokes equation has been proposed from its form-solution, the conclusion is that "\emph{The Navier-Stokes equation has solution if and only if the determinant of flow velocity gradient is not zero, namely $\det (\bm \nabla \bm v)\neq 0$.}"

Wavelet-distributed approximating functional method for solving the Navier-Stokes equation

1998 ◽  
Vol 115 (1) ◽  
pp. 18-24 ◽  
Author(s):  
G.W. Wei ◽  
D.S. Zhang ◽  
S.C. Althorpe ◽  
D.J. Kouri ◽  
D.K. Hoffman
Keyword(s):  
Stokes Equation ◽  
Functional Method ◽  
Navier Stokes ◽  
Navier Stokes Equation

scholarly journals Generalized Navier–Stokes Equations and Dynamics of Plane Molecular Media

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 288
Author(s):  
Alexei Kushner ◽  
Valentin Lychagin
Keyword(s):  
Carbon Dioxide ◽  
Internal Structure ◽  
Stokes Equation ◽  
Hydrogen Cyanide ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Navier Stokes Equations

The first analysis of media with internal structure were done by the Cosserat brothers. Birkhoff noted that the classical Navier–Stokes equation does not fully describe the motion of water. In this article, we propose an approach to the dynamics of media formed by chiral, planar and rigid molecules and propose some kind of Navier–Stokes equations for their description. Examples of such media are water, ozone, carbon dioxide and hydrogen cyanide.

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