Universal properties of chaotic transport in the presence of diffusion

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Abstract
The combined, finite time effects of molecular diffusion and chaotic advection on a finite distribution of scalar are studied in the context of time periodic, recirculating flows with variable stirring frequency. Comparison of two disparate frequencies with identical advective fluxes indicates that diffusive effects are enhanced for slower oscillations. By examining the geometry of the chaotic advection in both high and low frequency limits, the flux function and the width of the stochastic zone are found to have a universal frequency dependence for a broad class of flows. Furthermore, such systems possess an adiabatic transport mechanism which results in the establishment of a “Lagrangian steady state,” where only the asymptotically invariant core remains after a single advective cycle. At higher frequencies, transport due to chaotic advection is confined to exchange along the perimeter of the recirculating region. The effects of molecular diffusion on the total transport are different in these two cases and it is argued and demonstrated numerically that increasing the diffusion coefficient (in some prescribed range) leads to a dramatic increase in the transport only for low frequency stirring. The frequency dependence of the total, long time transport of a limited amount of scalar is more involved since faster stirring leads to smaller invariant core sizes.