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Physical meaning of $\frac{\pi }{8}$ in Poiseuille's equation
Recently I read about the Poiseuille's equation which relates the flow rate of a viscous fluid to coefficient of viscosity ($\nu$), pressure per unit length($\frac{P}{l}$) and radius of the tube ($r$) in which the fluid is flowing. The equation is
$\frac{V}{t}=\frac{\pi P{r}^{4}}{8\nu l},$
where $V$ denotes volume of the fluid.
One thing which I don't understand in this equation is that why do we have the constant term as $\frac{\pi }{8}$ . Is there some physical significance of this specific number here or is it just a mathematical convention?
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pomastitxz27r
Poiseuille's law is the result of doing a a force balance on the fluid, applying Newton's law of viscosity between the fluid velocity gradient radially and the shear stress, solving for the fluid axial velocity distribution, and integrating the velocity distribution to get the volumetric flow rate. The $\pi /8$ comes in from integrating the velocity over the circular area.
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Azzalictpdv
The appearance of the irrational $\pi$ comes from the assumption, that the pipe is perfectly circular.
This in itself is of course a nonphysical assumption, so it is not surprising to get a irrational result.