Dimensional reduction of a general advection–diffusion equation in 2D channels

Abstract

Diffusion of point-like particles in a two-dimensional channel of varying width is studied. The particles are driven by an arbitrary space dependent force. We construct a general recurrence procedure mapping the corresponding two-dimensional advection-diffusion equation onto the longitudinal coordinate x. Unlike the previous specific cases, the presented procedure enables us to find the one-dimensional description of the confined diffusion even for non-conservative (vortex) forces, e.g. caused by flowing solvent dragging the particles. We show that the result is again the generalized Fick-Jacobs equation. Despite of non existing scalar potential in the case of vortex forces, the effective one-dimensional scalar potential, as well as the corresponding quasi-equilibrium and the effective diffusion coefficient [Formula: see text] can be always found.

Active microrheology in corrugated channels: Comparison of thermal and colloidal baths

“…In this work, we present simulations and a theoretical model of active microrheology in corrugated channels, studying the effect of the corrugation amplitude: a tracer particle is pulled with a constant force through a colloidal bath confined in the channel. The model is based on the Fick-Jacobs approximation [25][26][27][28]31] for an isolated tracer in a corrugated channel, and it is validated by simulations without the colloidal bath (only the solvent, as a thermal bath, is considered). The results of the simulations with the colloidal bath follow the same trends of the isolated tracer, although quantitative differences are noticed.…”

“…This results again in a one-dimensional problem, where the external potentials (or confining walls) can be introduced. This description was improved by Zwanzig [26], who introduced a space-dependent diffusion coefficient, and further developed in different works [27][28][29][30][31].…”

Special issue on transport in narrow channels

“…This approach is used in two of the present contributions: Chávez et al [2] develop a theory for the position-dependent effective diffusivity for two-dimensional asymmetric varying-width channels, and for three-dimensional curved midline tubes, formed by straight walls. Kalinay and Slanina [8] present a formalism reducing the general 2D advection-diffusion equation onto the longitudinal coordinate, also valid for nonconservative forces. The result is a generalized Fick-Jacobs equation.…”