Author:
Malone George H.,Hutchinson Thomas E.,Prager Stephen
Abstract
A variational method involving minimization of the energy dissipation rate which was previously developed for transport in polymer systems is applied here to flow of a continuum solvent through a thin membrane. The membrane is represented by an array of spherical particles undergoing Brownian motion, subject to various interactions with one another and with the motion of the solvent. General upper bounds on the solvent permeability of the membrane are obtained in terms of equilibrium distribution functions, and applications of the method are illustrated for the case where membrane elements are confined to a plane. Calculations which treat all beads equivalently give permeability estimates whose dependence on the number n of beads per unit area of membrane has the form, at low n,
\[
\kappa = (6\pi\eta n)^{-1}(1-\alpha n+\ldots),
\]
where η is the solvent viscosity and α is a constant. More-elaborate trials which allow the drag on a bead to be influenced by the distribution of other beads in the vicinity give the stronger bounding estimate
\[
\kappa = (6\pi\eta n)^{-1}(1+\alpha^{\prime}n\ln n+\ldots).
\]
Comparison with a self-consistent field approach suggests that this logarithmic behaviour is the true first-order correction.
Publisher
Cambridge University Press (CUP)
Subject
Mechanical Engineering,Mechanics of Materials,Condensed Matter Physics
Cited by
10 articles.
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