Green's theorem or divergence theorem?

[hermano]

Hi,

I want to calculate the total flux but I'm not sure if I have to use Green's theorem (2D) or the divergence theorem (3D). The equation below is a modified Reynolds equation describing the air flow in the clearance of porous air bearing.

$\frac{\partial}{\partial\theta}(PH^3 \frac{\partial P}{\partial\theta}) +(\frac{R}{L})^2 \frac{\partial}{\partial Y}(PH^3 \frac{\partial P}{\partial Y}) - \Lambda \frac{\partial}{\partial\theta}PH + \Psi P (\frac{\partial P'}{\partial Z})_{Z=1} = 0$

P is the pressure in the clearance of the air bearing, P' is the pressure in the porous media. However, at Z=1 P' must be equal to P for continuity.

This equation can be rewritten using the divergence vector operation:

$\nabla\bullet\left[PH^3 \frac{\partial P}{\partial\theta} + (\frac{R}{L})^2 PH^3 \frac{\partial P}{\partial Y} - \Lambda PH + \Psi P P'\right] = 0$

Solving this equation with a numerical method (i.e. finite difference) can be done by first simplifying the equation with Green's theorem for flux or the divergence theorem. Because I'm interested in the flow in the \(\displaystyle \theta\) - Y direction I want to solve the equation applying Green's theorem (2D). However, in the first equation there is also a gradient in the Z-direction namely $\Psi P (\frac{\partial P'}{\partial Z})_{Z=1}$. So my question is: Can I calculate the flux with Green's theorem or do I have to use the divergence theorem because of the pressure gradient in the Z-direction?

 

[Valerie Park]

Thanks in advance!Since the equation involves derivatives in 3 dimensions, you should use the divergence theorem. The divergence theorem allows you to calculate the flux of a vector field over a closed surface by calculating the divergence of the vector field within the volume enclosed by the surface. Green's theorem is only applicable for 2 dimensional systems.