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 zhor May 10, 2001 07:49

Exact solutions

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.

 Giuseppe Casillo May 10, 2001 12:08

Re: Exact solutions

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.

 zhor May 11, 2001 06:22

Re: Exact solutions

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?

 Giuseppe Casillo May 13, 2001 02:05

Re: Exact solutions

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.

 TOT KTO 3HAET May 13, 2001 11:26

Re: Exact solutions

Tat iz weri goot anser.

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