Taylor Dispersion in Curved Channels

The Taylor dispersion problem is to determine the long-time behavior of the concentration density of a passive scalar (contaminant) diffusing in an incompressible channel flow, given an initial injection of contaminant which is slowly varying---order $\epsilon$---in the longitudinal direction. Taylor [Proc. Roy. Soc. London Sect. A, 219 (1953), pp. 186--203] showed that the cross-channel mean of the concentration experiences an enhanced diffusion relative to the mean flow. This is due to mechanical rather than molecular events. Mercer and Roberts [SIAM J. Appl. Math., 50 (1990), pp. 1547--1565] have treated this problem by a formal application of infinite-dimensional center manifold theory and extended the analysis to the case of a curved channel which is slowly varying in the longitudinal direction. It is shown that after the introduction of appropriate multiple scales, their center manifold technique can be used to carry out the Taylor dispersion calculations for a channel whose cross-section has large---order 1---variation in the longitudinal directions. Also, more general underlying flows can be allowed than was heretofore possible. A heuristic argument is given to show the existence of some channel shape and flow field for which effective diffusion is reduced rather than enhanced. A similar advection-diffusion problem but with no channel boundary has been rigorously analyzed by homogenization theory and discussed in a recent paper of Majda and McLaughlin [Stud. Appl. Math., 89 (1993), pp. 245--279]. The two problems are briefly compared.