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Abstract
The Taylor dispersion coefficient D¯C (above and beyond the purely molecular contribution D) arising during Poiseuille-like flow through noncircular cylindrical ducts of large aspect ratio (but otherwise arbitrary cross section) is theoretically calculated. Our study is motivated by an attempt to rationalize the remarkable fact that D¯C for a rectangular duct does not reduce to its well-known flat-plate value, D¯ ∞ C = 202H2/105D, in the limit where the aspect ratio tends to infinity [V¯ = mean velocity in duct, H = distance between plates]. Slender-body theory is used to express both the fluid velocity and concomitant Taylor dispersion fields as perturbation expansions in the small parameter ε = (aspect ratio)−1 ≪ 1, following which D¯C is computed to the lowest order in e. This dispersivity is found to scale with the longer of the two characteristic duct dimensions for all cross sections save the rectangular, for which D¯C scales with the shorter dimension. This surprising result is explained by noting that, in either case, D¯C scales with that cross-sectional dimension along which the lateral fluid velocity gradients are greatest in extent.
As an easily claimed bonus, we also calculate the Hagen-Poiseuille pressure-drop/flow-rate coefficient for these high aspect ratio ducts.