A Fick-Jacobs equation for channels over 3D curves

TL;DR

This paper derives new formulas for the effective diffusion coefficient in narrow 3D channels by projecting the diffusion equation along a curve, incorporating geometric properties of the channel's cross sections and curvature.

Contribution

It introduces a generalized Fick-Jacobs equation for 3D channels over curves, linking effective diffusion to geometric moments and curvature, including effects of rotating cross sections.

Findings

Derived formulas express effective diffusion in terms of geometric moments and curvature.

Rotating cross sections with offsets significantly influence the effective diffusion coefficient.

The approach extends classical Fick-Jacobs models to more complex 3D channel geometries.

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.