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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

$$\frac{V}{t}=\frac{πPr^4}{8\nu l},$$ 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 $\frac{π}{8}$ . Is there some physical significance of this specific number here or is it just a mathematical convention?

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    $\begingroup$ Do you have a reason to think there is a special explanation for this constant, as opposed to just being some dimensionless constant that can't be fixed by dimensional analysis? $\endgroup$ – Andrew Nov 22 '20 at 12:01
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    $\begingroup$ @Andrew I just want to know if there is some . $\endgroup$ – A student Nov 22 '20 at 12:19
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    $\begingroup$ That's fair enough, and I don't know there isn't, but my prior belief (unless there is a specific reason to think otherwise) would be that dimensionless constants in equations like this are essentially random. $\endgroup$ – Andrew Nov 22 '20 at 12:29
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    $\begingroup$ Have you actually looked at a derivation of this law? If you did, then I am pretty sure you cannot miss how the numerical factor comes into the final result. $\endgroup$ – DanielC Nov 22 '20 at 18:13
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    $\begingroup$ Hint: area of tube is $\pi r^2$ and that is where the $\pi$ comes from. $\endgroup$ – John Alexiou Nov 22 '20 at 19:59
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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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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.

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