A Fick-Jacobs equation for channels over 3D curves
Abstract
The purpose of this paper is to provide new formulas for the effective diffusion coefficient of a generalized Fick-Jacobs equation for narrow 3-dimensional channels. The generalized Fick-Jacobs equation is obtained by projecting the 3-dimensional diffusion equation along the normal directions of a curve in three dimensional space that roughly resembles the narrow channel. The projection (or dimensional reduction) is achieved by integrating the diffusion equation along the cross sections of the channel contained in the planes orthogonal to the curve. We show that the resulting formula for the associated effective diffusion coefficient can be expressed in terms of the geometric moments of the channel's cross sections and the curve's curvature. We show the effect that a rotating cross section with offset has on the effective diffusion coefficient.
Keywords
Cite
@article{arxiv.1406.3524,
title = {A Fick-Jacobs equation for channels over 3D curves},
author = {Carlos Valero Valdes and Rafael Herrera Guzman},
journal= {arXiv preprint arXiv:1406.3524},
year = {2014}
}
Comments
Effective diffusion formula for the twisted ellipse was corrected in this version. The cardioid case formula removed and numerical integration used instead. Jacob's changed to Jacobs