Shear dispersion and anomalous diffusion by chaotic advection

The dispersion of passive scalars by the steady viscous flow through a twisted pipe is both a simple example of chaotic advection and an elaboration of Taylor's classic shear dispersion problem. In this article we study the statistics of the axial dispersion of both diffusive and perfect (non-diffusive) tracer in this system. For diffusive tracer chaotic advection assists molecular diffusion in transverse mixing and so diminishes the axial dispersion below that of integrable advection. As in many other studies of shear dispersion the axial distribution ultimately becomes Gaussian as t → ∞. Thus there is a diffusive regime, but with an effective diffusivity that is enhanced above molecular values. In contrast to the classic case, the effective diffusivity is not necessarily inversely proportional to the molecular diffusivity. For instance, in the irregular regime the effective diffusivity is proportional to the logarithm of the molecular diffusivity. For perfect tracer chaotic advection does not result in a diffusive process, even in the irregular regime in which streamlines wander throughout the cross-section of the pipe. Instead the variance of the axial position is proportional to t in t so that the measured diffusion coefficent diverges like In t. This faster than linear growth of variance is attributed to the trapping of tracer for long times near the solid boundary, where the no-slip condition ensures that the fluid moves slowly. Analogous logarithmic effects associated with the no-slip condition are well known in the context of porous media. A simple argument, based on Lagrangian statistics and a local analysis of the trajectories near the pipe wall, is used to calculate the constants of proportionality before the logarithmic terms in both the large- and infinite-Péclet-number limits.