Viscous laminar fluid flow in a pipe is described by the well known Hagen-Poiseuille relation. According to this relation, the velocity of the fluid is directly proportional to the pressure difference between the ends of the pipe and the square of the radius of the pipe and inversely proportional to its length. But as this length becomes vanishingly small, this relation predicts that the velocity of the flow approaches infinity, which obviously is unphysical. The origin of this paradox is discussed, and an extension of the Hagen-Poiseuille relation is presented that is valid for all values of the length of the pipe.

## REFERENCES

1.

The usual name for this relation is the Poiseuille law or the Hagen-Poiseuille law, but the term “law” is a misnomer because this relation is not a law of physics. For a brief history of this relation, see

O.

Darrigol

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, Oxford

, 2005

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–144

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). In this paper, the authors derive the H-P relation as valid for arbitrary values of the pipe lengthland radius a, by making the usual incorrect assumption that the velocity of the fluid has only a longitudinal component near the end where the fluid enters the pipe.4.

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. Tipler mentions that the H-P relation ceases to be valid at sufficiently large velocities because the flow becomes turbulent. But as we have shown here, the breakdown occurs already in the laminar regime.12.

For a viscous fluid, the pressure at the hole inside the vessel is somewhat less than the static pressure, a fact well known to hydraulic engineers who introduce a “fudge” factor ξ and write $v=\xi 2gh$. This factor, however, is ignored in physics textbooks, although already 300 years ago Newton had observed that the horizontal velocity of a water jet from a hole on the side of tank is less than the value predicted by Torricelli's relation. In the second edition (1713) of the

*Principia*, Book 2, Prop. 36, case 3, Newton wrote: “I found that when the height of the standing water above the hole was 20 in. and the height of the hole above a plane parallel to the horizon was also 20 in., the stream of the water gushing forth would fall upon the plane at a distance of about 37 in., taken from a perpendicular that was dropped from the plane to the hole. For in the absence of resistance the stream would have had to fall upon the plane at a distance of 40 in., the latus rectum of the parabolic stream being 80 in.” Hence, Newton had found that ξ = 37/40, which I have verified to be correct for a hole of radius*a*≈ 0.26 cm.13.

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For example, see

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. Lautrup also presents an approximation to the entry length l_{e}based on the diffusion of momentum due to viscosity, and obtains $le/a=(1/8)\u2009Re$.15.

A similar relation was introduced by Gotthilf Hagen to analyse his experiment of viscous fluid flow, in

G.

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In Jean Poiseuille's analysis of his experiments, a contribution to the pressure proportional to

*v*^{2}was not included. SeeJ.

Poiseullie

, “Recherches expérimentales sur le mouvement des liquides dans les tubes de très petit diamètre

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9

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(1844

).© 2014 American Association of Physics Teachers.

2014

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