# Exact solutions

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 May 10, 2001, 07:49 Exact solutions #1 zhor Guest   Posts: n/a It is interesting to know the exact (analytical) solutions of the incompressible Navier-Stokes equations (2-D and 3-D). I am familiar with the solutions which are, e.g., in the articles: Kim&Moin (1985), Ethier et al. (1994), Sheu et al. (1996). May be someone knows and other exact solutions, not considering of course the well-known ones for a pipe (1-D)? What strict theorems are known about the existence and uniqueness for the NS equations (steady state and transient)? Thank you.

 May 10, 2001, 12:08 Re: Exact solutions #2 Giuseppe Casillo Guest   Posts: n/a for 2D and 3D one can use the complex space thery. For 2D one have ixi=-1 in 3D,the same,ixi=-1 and jxj=-1;this products veryfing the equation DDy=0 whit DD the laplace's operator.

 May 11, 2001, 06:22 Re: Exact solutions #3 zhor Guest   Posts: n/a Thank you, but could you write in more details. What means 'ixi'? And except Laplace operator in the Navier-Stokes there are also other terms: D(uv)/Dx, Dp/Dx, Du/Dt. What to do with them?

 May 13, 2001, 02:05 Re: Exact solutions #4 Giuseppe Casillo Guest   Posts: n/a ixi means the product of the versor i for his same. The Navier-Stoke's equation in the incompressible flow reduce to Laplace's equation whit the ipotesys of the viscous terms are trascurable.This thinks divide our fisic space into two region:1 near our body where we use the N.S.'s equation, 2 far a body where we use the laplace's equation. Into the space complex we have for a point 2D P=x+i*y 3D P=x+i*y+j*z for a function 2D f=f1+i*f2 3D f=f1+i*f2+j*f3 The trigonometric rappresentation (you must use this because you have pair terms)give us 2D P=r*(cos t +i*sin t) 3d P=r*(cos t + i*sin t )* (cos f +j * sin f) I think the product i*j is a new axis. If you calculate the derivative (you can to do egual the derivative along the axis increment) you have flxl=fkxk and flxk=-fkxl, whit fl a l component of f,flxl a partial derivative of fl rispect to xl axys (remember if i*i=-1 you have i=-1/i).When you have obtined this you can veryfi DDf=0 for all f into the space complex.

 May 13, 2001, 11:26 Re: Exact solutions #5 TOT KTO 3HAET Guest   Posts: n/a Tat iz weri goot anser.

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