Taylor-Aris dispersion in high aspect ratio columns of nearly rectangular cross section

We consider Taylor-Aris dispersion in columns of nearly rectangular cross section of large aspect ratio. We generalize the results of [P.C. Chatwin, P.J. Sullivan, The effect of aspect ratio on longitudinal diffusivity in rectangular channels, J. Fluid Mech. 120 (1982) 347-358] and [M.R. Doshi, P.M. Daiyai, W.N. Gill, Three dimensional laminar dispersion in open and closed rectangular conduits, Chem. Eng. Sci. 33 (1978) 795-804] who showed that the effective diffusion rate for perfectly rectangular cross sections is remarkably different than the diffusion rate between two parallel plates - as the aspect ratio goes to infinity, the effective diffusion rate does not approach the effective diffusion rate for two parallel plates. In particular, we examine columns of nearly rectangular cross section having both non-parallel walls and asymmetric ends of arbitrary shape. In particular, this includes geometries common to microfabricated gas chromatography columns. We develop an expression for the effective diffusivity showing the contributions from the walls and the ends, and the relative importance of each. We also discuss the large effect that a small nonuniformity in the middle of the cross section can have on the effective diffusion rate, and how the ends of the cross section can be modified to control the effective diffusion rate.