End Correction for Slow Viscous Flow through Long Tubes

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Abstract
A variational method valid for the creeping-flow regime is used to calculate an upper bound for the pressure drop associated with viscous dissipation near the ends of a long round tube. The result states that this pressure drop, which is an additive correction to that given by the Poiseuille formula, is not greater than 1.154 times the pressure drop through a thin orifice. The orifice pressure drop, in terms of the fluid viscosity μ, the volumetric flow rate Q, and the radius a, is found to be 3μQ/a3, in confirmation of a previous calculation of Roscoe by a different method.

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