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Green's theorem or divergence theorem?

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November 30, 2011, 16:11	Green's theorem or divergence theorem?	#1
Hermano New Member Join Date: Jul 2010 Posts: 10 Rep Power: 15	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 $\theta$ - 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?

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