Effective diffusion in laminar convective flows

Abstract

The effective diffusion coefficient D* of a passive component, such as test particles, dye, temperature, magnetic flux, etc., is derived for motion in periodic two-dimensional incompressible convective flow with characteristic velocity v and size d in the presence of an intrinsic local diffusivity D. Asymptotic solutions for effective diffusivity D*(P) in the large P limit, with P∼ vd/D, is shown to be of the form D*=cDP1/2 with c being a coefficient that is determined analytically. The constant c depends on the geometry of the convective cell and on an average of the flow speed along the separatrix. The asymptotic method of evaluation applies to both free boundary and rough boundary flow patterns and it is shown that the method can be extended to more complicated patterns such as the flows generated by rotating cylinders, as in the problem considered by Nadim, Cox, and Brenner [J. Fluid Mech. 164, 185 (1986)]. The diffusivity D* is readily calculated for small P, but the evaluation for arbitrary P requires numerical methods. Monte Carlo particle simulation codes are used to evaluate D* at arbitrary P, and thereby describe the transition for D* between the large and small P limits.

Suppression of chaotic dynamics and localization of two-dimensional electrons by a weak magnetic field

“…Our approach to calculating D is close in spirit to the ones used for calculation of the diffusion coefficient in planar periodic vortical flows, e.g., Rayleigh-Bénard cells. 32 …”

Front speed enhancement in cellular flows

“…(1), which reduces to the equation for a passive scalar in a cellular flow. This is a well studied problem, the solution of which is [23][24][25]:…”

“…(10) [10,14]. The renormalized diffusivity D eff accounts for the process of diffusion from cell to cell as a result of the nontrivial interaction of advection and molecular diffusion [23][24][25]. The renormalized reaction time τ eff amounts to the time that it takes for a single cell to be filled by inert material, and depends on the interaction of advection and production.…”

Front propagation in laminar flows

“…which, in spite of their apparent simplicity, show intriguing behaviour [12][13][14][15]. For a given structure of the flow field, we aim to explore the dependence of V f on relevant parameters, such as the typical flow velocity U and the production time-scale τ r .…”