End Correction for Slow Viscous Flow through Long Tubes

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.

Optimizing water permeability through the hourglass shape of aquaporins

“…The scaling results accordingly from Stokes' equation: ηΔv = ∇p ⇒ ηv=a 2 ∼ Δp=a, with v ∼ Q=a 2 the typical fluid velocity. Extending this estimate to the converging flow into a cylindrical pore, the previous Sampson's formula provides a very good estimate of the access pressure drop (17), as highlighted by an exact calculation (18). Now, if one considers water transport through a pore, the dissipation occurring in the bulk access regions yields an upper bound to the hydrodynamic permeability.…”

The microscopic analysis of high forchheimer number flow in porous media

“…The pressule drops were measured for a I'ange of Reynolds numbers (10 < Re¿ < Weissberg (1962) and experimentally verified by La Nieve and Bogue (1968) for the "cleeping" flow limit, Astarita and Greco (1968) and Sylvester and Rosen (1970) Astarita and Greco (1968) and Sylvester and Rosen (1970) is the temperatule change in the flow Ioop which affects considelably the acculate determination of these corrections. Dullien and Azz¿m (1973) and Azzam and Dullien (1977).…”

Perméabilité optimale des aquaporines

“…En effet la convergence du liquide à chaque entrée/sortie du canal génère des gradients de vitesse, donc une dissipation visqueuse. Un nouveau terme de résistance, appelé résistance d'entrée R ent , vient donc s'ajouter à la résistance interne du canal [14][15][16]. Cette résistance est proportionnelle à la viscosité du fluide , et inversement proportionnelle au rayon du canal a au cube : R ent = C/a 3 (équation 2) avec C, un pré-facteur qui dépend de la géométrie du canal.…”