Compressible Navier Stokes in cylinder coordinates

[schettel]

Hello,

I need the Navier Stokes equations for compressible flow (Newtonian fluid would be ok) in cylinder coordinates. Can anybody help?

Thanks

 

[dextercioby]

Why don't u compute them??

http://scienceworld.wolfram.com/physics/Navier-StokesEquations.html

It's not what u asked for,i'm trying to get you to work...

Daniel.

 

[dextercioby]

To make the work easier,use this

http://mathworld.wolfram.com/CylindricalCoordinates.html

Daniel.

 

[FredGarvin]

Ugh...I get queesy looking at that dex.

How about this:
Radial Direction:
$$\rho (\frac{\partial v_r}{\partial t} + v_r \frac{\partial v_r}{\partial r} + \frac{v_\theta}{r} \frac{\partial v_r}{\partial \theta} - \frac{v^2_\theta}{r} + v_z \frac{\partial v_r}{\partial z}) = -\frac{\partial p}{\partial r} + \rho g_r + \mu [\frac{1}{r} \frac{\partial}{\partial r}(r \frac{\partial v_r}{\partial r}) - \frac{v_r}{r^2} +\frac{1}{r^2} \frac{\partial^2 v_r}{\partial \theta^2} - \frac{2}{r^2} \frac{\partial v_\theta}{\partial \theta} + \frac{\partial^2 v_r}{\partial z^2}]$$

Holly crud that's a lot of typing.

Angular (theta) Direction:
$$\rho (\frac{\partial v_\theta}{\partial t} + v_r \frac{\partial v_\theta}{\partial r} + \frac{v_\theta}{r} \frac{\partial v_\theta}{\partial \theta} + \frac{v_r v_\theta}{r} + v_z \frac{\partial v_\theta}{\partial z}) = -\frac{1}{r} \frac{\partial p}{\partial \theta} + \rho g_\theta + \mu [\frac{1}{r} \frac{\partial}{\partial r}(r \frac{\partial v_\theta}{\partial r}) - \frac{v_\theta}{r^2} +\frac{1}{r^2} \frac{\partial^2 v_\thata}{\partial \theta^2} - \frac{2}{r^2} \frac{\partial v_r}{\partial \theta} + \frac{\partial^2 v_\theta}{\partial z^2}]$$

Z Direction:
$$\rho (\frac{\partial v_z}{\partial t} + v_r \frac{\partial v_z}{\partial r} + \frac{v_\theta}{r} \frac{\partial v_z}{\partial \theta} + v_z \frac{\partial v_z}{\partial z}) = -\frac{\partial p}{\partial z} + \rho g_z + \mu [\frac{1}{r} \frac{\partial}{\partial r}(r \frac{\partial v_z}{\partial r}) + \frac{1}{r^2} \frac{\partial^2 v_z}{\partial \theta^2} + \frac{\partial^2 v_z}{\partial z^2}]$$

 

[dextercioby]

That still doesn't help him too much.U assume the fluid to have an incompressible flow...


Daniel.

 

[schettel]

That's right, unfortunately. Thanks for the typing, anyway. And thanks for the links. I'll take it home on the weekend and try to figure it out myself. I'm bad at maths, though.

 

[dextercioby]

I'm sorry,but you haven't asked for some kindergarten stuff.You need to know what a gradient,curl,divergence,tensor,partial derivative,cylindric coordinate,... are.

I am urging you to read the construction of these equations in the 6-th volume of Landau & Lifschitz theoretical physics course:"Fluid Mechanics",Pergamon Press.Any edition.

Daniel.

 

[FredGarvin]

You know...I didn't realize you had asked for compressible flow. My oops again. I really must learn how to read. Oh well. I had a nice exercise in LATex.