Compressible Navier Stokes in cylinder coordinates

1. Apr 19, 2005

schettel

Hello,

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

Thanks

2. Apr 19, 2005

dextercioby

3. Apr 19, 2005

dextercioby

4. Apr 19, 2005

FredGarvin

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

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

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

5. Apr 19, 2005

dextercioby

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

Daniel.

6. Apr 19, 2005

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.

7. Apr 19, 2005

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.

8. Apr 19, 2005

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.