Exact solution of Navier-Stokes equations - The theory of functions of a complex variable under Dirac-Pauli representation and its application in fluid dynamis. II
type: Item
instance of: scholarly article
title: Exact solution of Navier-Stokes equations - The theory of functions of a complex variable under Dirac-Pauli representation and its application in fluid dynamis. II
zbMATH Open document ID: 0627.76033
DOI: 10.1007/BF01899554
published in: Applied Mathematics and Mechanics. (English Edition)
publication date: 1986-01-01T00:00:00Z
review text: [For part I see the review above (Zbl 0627.76032).] The author makes an attempt to obtain the exact solution of Navier-Stokes equations which is an unsolved problem. He uses the theory of complex functions under the Dirac-Pauli representation, and the Kaluza `Ghost' coordinate to transform the Navier-Stokes equation into a pair of nonlinear equations for two unknown complex flow functions \(\psi\) and \({\bar \psi}\). With a suitable transformation and \(\phi =\psi +{\bar \psi}\), this pair reduces to the following nonlinear equation with only one real unknown function \[ (\frac{\partial}{\partial t}-4\nu \frac{\partial^ 2}{\partial y\partial \bar y})\frac{\partial^ 2\phi}{\partial y\partial \bar y}=i(\frac{\partial \phi}{\partial \bar y}\quad \frac{\partial}{\partial y}-\frac{\partial \phi}{\partial y}\quad \frac{\partial}{\partial \bar y})\frac{\partial^ 2\phi}{\partial y\partial \bar y}. \] Since it is the sum of two complex conjugate equations, it turns out after suitable algebraic transformation that \(\phi\) satisfies \[ (\frac{\partial}{\partial t}-4\nu \frac{\partial^ 2}{\partial y\partial \bar y})\phi =-2i\phi \frac{\partial \phi}{\partial y}. \] This can be further simplified into the classical Burgers equation \[ \phi_ t+2i\phi \quad \phi_{\xi}-4\nu \phi_{\xi \xi}=0,\quad \xi =y+\bar y. \] With the aid of the Cole-Hopf transformation \(\phi =4i\nu (1/\omega)(\partial \omega /\partial \xi)\), the Burgers equation becomes \(\omega_ t=4\nu\omega_{\xi \xi}\). This Cole-Hopf transformation can be written as the Bäcklund transformation \[ \omega_{\xi}=- \frac{i}{4\nu}\phi \omega,\quad \omega_ t=(-i\phi_{\xi}-\frac{\phi^ 2}{4\nu})\omega. \] Finally, the author claims that the exact solution of the Navier-Stokes equation can be determined by the linear equation \[ 2(\frac{\partial}{\partial t}-4\nu \frac{\partial^ 2}{\partial y\partial \bar y})\frac{\partial^ 2\psi}{\partial y\partial \bar y}=2i({\bar \phi}\frac{\partial}{\partial \phi}-\phi \quad \frac{\partial}…
Mathematics Subject Classification ID: 76D05
Mathematics Subject Classification ID: 35Q30
zbMATH DE Number: 4020675
zbMATH Keywords: Cole-Hopf transformation
zbMATH Keywords: diffusion equation
zbMATH Keywords: Bäcklund transformation
zbMATH Keywords: exact solution
zbMATH Keywords: Dirac-Pauli representation
author: Hui-chuan Shen
reviewed by: Lokenath Debnath
MaRDI profile type: Publication
cites work: The theory of functions of a complex variable under Dirac-Pauli representation and its application in fluid dynamics. I
cites work: Q3247887
cites work: Q3661114
cites work: Q3913734
description: scientific article; zbMATH DE number 4020675