Yesterday I answered this question, using analysis of forces from this web page.
But today I started having doubts regarding the completeness of this analysis. It occurred to me this approach does not take into account any centripetal forces caused by the changing direction of the fluid flow. Simply put:
$\vec{F}=m\frac{d\vec{v}}{dt}$.
To analyse this, consider the following geometry for a $90^0$ bend in a pipe lying in a horizontal $(x,y)$ plane:
Assumptions:
- turbulent plug flow with constant fluid speed $v$, fluid density $\rho$, cross-section of pipe $A$, bend radius $R$.
- $\frac{R}{A}\gg 1$
Right now I am only concerned with centripetal forces needed to keep the flow on its rotational trajectory.
Take an infinitesimal element at $\theta$:
$dm=\rho A Rd\theta$.
The centripetal forces are:
$dF=-\frac{v^2 dm}{R}=-\rho A v^2 d\theta$
$dF_x=-\rho A v^2\cos\theta d\theta$.
$dF_y=-\rho A v^2\sin\theta d\theta$.
So these would be the forces the pipe needs to exert on the fluid:
$F_x=-\rho A v^2\int_0^{\frac{\pi}{2}}\cos\theta d\theta=-\rho A v^2$.
$F_y=-\rho A v^2\int_0^{\frac{\pi}{2}}\sin\theta d\theta=-\rho A v^2$.
If this is correct then I obviously need to amend my answer. So my question is: is this correct?