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Updated: Jun 14, 2026

The Diffusion of Passive Tracers in Laminar Shear Flow
Published on: May 1, 2018
Diffusion in a two-dimensional channel with curved midline and varying width: reduction to an effective
1Department of Physics, Colorado State University, Fort Collins, Colorado 80523, USA.
Abstract:
Diffusion in a narrow two-dimensional channel with a midline that need not be straight and a width that may vary is reduced to an effective one-dimensional equation of motion. This equation takes the form of the Fick-Jacobs equation with a spatially varying effective diffusivity. The effective diffusivity includes a contribution that comes from the slope of the midline as well as the usual term stemming from variations in the channel width along the length of the channel. Our derivation of our equation of motion is completely rigorous and is based on an asymptotic expansion in a small dimensionless parameter that characterizes the channel width. For a channel that has a straight midline or wall, our equation of motion reduces to Zwanzig's equation [R. Zwanzig, J. Phys. Chem. 96, 3926 (1992)]. Our derivation therefore provides a rigorous proof of the validity of the latter equation. Finally, the equation of motion is solved analytically for channels with curved midline and constant width.
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