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No Existence and Smoothness of Solution of the Navier-Stokes Equation
The Navier-Stokes equation can be written in a form of Poisson equation. For laminar flow in a channel (plane Poiseuille flow), the Navier-Stokes equation has a non-zero source term (∇(2)u(x, y, z) = F(x) (x, y, z, t) and a non-zero solution within the domain. For transitional flow, the velocity pro...
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Formato: | Online Artículo Texto |
Lenguaje: | English |
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MDPI
2022
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Acceso en línea: | https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8947576/ https://www.ncbi.nlm.nih.gov/pubmed/35327850 http://dx.doi.org/10.3390/e24030339 |
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author | Dou, Hua-Shu |
author_facet | Dou, Hua-Shu |
author_sort | Dou, Hua-Shu |
collection | PubMed |
description | The Navier-Stokes equation can be written in a form of Poisson equation. For laminar flow in a channel (plane Poiseuille flow), the Navier-Stokes equation has a non-zero source term (∇(2)u(x, y, z) = F(x) (x, y, z, t) and a non-zero solution within the domain. For transitional flow, the velocity profile is distorted, and an inflection point or kink appears on the velocity profile, at a sufficiently high Reynolds number and large disturbance. In the vicinity of the inflection point or kink on the distorted velocity profile, we can always find a point where ∇(2)u(x, y, z) = 0. At this point, the Poisson equation is singular, due to the zero source term, and has no solution at this point due to singularity. It is concluded that there exists no smooth orphysically reasonable solutions of the Navier-Stokes equation for transitional flow and turbulence in the global domain due to singularity. |
format | Online Article Text |
id | pubmed-8947576 |
institution | National Center for Biotechnology Information |
language | English |
publishDate | 2022 |
publisher | MDPI |
record_format | MEDLINE/PubMed |
spelling | pubmed-89475762022-03-25 No Existence and Smoothness of Solution of the Navier-Stokes Equation Dou, Hua-Shu Entropy (Basel) Article The Navier-Stokes equation can be written in a form of Poisson equation. For laminar flow in a channel (plane Poiseuille flow), the Navier-Stokes equation has a non-zero source term (∇(2)u(x, y, z) = F(x) (x, y, z, t) and a non-zero solution within the domain. For transitional flow, the velocity profile is distorted, and an inflection point or kink appears on the velocity profile, at a sufficiently high Reynolds number and large disturbance. In the vicinity of the inflection point or kink on the distorted velocity profile, we can always find a point where ∇(2)u(x, y, z) = 0. At this point, the Poisson equation is singular, due to the zero source term, and has no solution at this point due to singularity. It is concluded that there exists no smooth orphysically reasonable solutions of the Navier-Stokes equation for transitional flow and turbulence in the global domain due to singularity. MDPI 2022-02-26 /pmc/articles/PMC8947576/ /pubmed/35327850 http://dx.doi.org/10.3390/e24030339 Text en © 2022 by the author. https://creativecommons.org/licenses/by/4.0/Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/). |
spellingShingle | Article Dou, Hua-Shu No Existence and Smoothness of Solution of the Navier-Stokes Equation |
title | No Existence and Smoothness of Solution of the Navier-Stokes Equation |
title_full | No Existence and Smoothness of Solution of the Navier-Stokes Equation |
title_fullStr | No Existence and Smoothness of Solution of the Navier-Stokes Equation |
title_full_unstemmed | No Existence and Smoothness of Solution of the Navier-Stokes Equation |
title_short | No Existence and Smoothness of Solution of the Navier-Stokes Equation |
title_sort | no existence and smoothness of solution of the navier-stokes equation |
topic | Article |
url | https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8947576/ https://www.ncbi.nlm.nih.gov/pubmed/35327850 http://dx.doi.org/10.3390/e24030339 |
work_keys_str_mv | AT douhuashu noexistenceandsmoothnessofsolutionofthenavierstokesequation |