## End Correction for Slow Viscous Flow through Long Tubes

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

Harold L. Weissberg

2004

#### Scholarcy highlights

• 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 quantity 12QΔP represents the work done on the fluid by surface forces at the boundary of the region
• The first equality in Eq follows from the fact that inertial effects are being neglected, i.e., the rate of increase of kinetic energy is negligible in comparison with the dissipation rate
• Note that the oblate spheroidal coordinates discussed in this reference correspond in our notation to t, x, and cos φ, which, in that order, constitute a right‐handed system
• It may be helpful to point out that the coordinates t and x go over to the spherical polar coordinates R and cos θ as t/a becomes large

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