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Using all cylindrical coordinates, pipe with z-axis vertically upward and radius of $r = G(z)$, flow is incompressible, inviscid and steady, Using appropriate boundary conditions I want to find the wall pressure.Flow is axis symmetric and irrotational, and of the form $(a(r,z), 0, c(r,z))$ in cylindrical coordinates.

The boundary conditions I have are the the fluid is stationary at the wall.

$a(G(z), z) = c(G(z), z) = 0, $

Using Bernoilli's equation for steady flow

$\frac{p}{\rho} + \frac{v^2}{2} + \Omega = G$

$\frac{p}{\rho} + \frac{c^2}{2} = G - \Omega$

$p_w = (G - \Omega - \frac{c^2}{2})\rho$

But I don't think this is correct for some reason.

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If the flow is inviscid you can only satisfy no-penetration at the wall, $a(G(z),z) = 0$. To satisfy the no-slip condition, $c(G(z),z) = 0$, you must have viscosity. – OSE Mar 18 '13 at 22:22
My previous comment only applies if the pipe wall is parallel to the z-axis. In the general case, to satisfy no-penetration, $a(G(z),z) \neq 0$ and $c(G(z),z) \neq 0$. – OSE Mar 19 '13 at 19:33

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