Chaotic advection and diffusion

Although advection plays a key role in bringing together fluid regions of different properties it cannot produce a spatially uniform concentration field without the help of small scale diffusive transport. In this section we discuss the combined effects of chaotic advection and diffusion. Consider an initial-value problem starting with a spatially non-uniform concentration field, e.g. a small localized patch superimposed on an otherwise uniform background. In a closed flow on a 

Snapshots of a concentration field obtained numerically by solving the advection-diffusion equation with the velocity field (2.66) are shown in Fig. 2.16. 

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In the presence of advection and diffusion in a bounded domain the concentration field becomes more and more uniform in space. The distribution can be characterized by its variance, that in the case of chaotic advection decays exponentially in time. 

Taylor s dispersion is one of the most well-known examples of the role of transport in dispersing a flow carrying a dissolved solute. The simplest setting for observing it is the injection of a solute into a slit channel. The solute is transported by Poiseuille s flow. In fact this problem could be studied in three distinct regimes (a) diffusion-dominated mixing, (b) Taylor dispersion-mediated mixing and (c) chaotic advection. 

Other micromixers based on various principles have also been constructed. These principles include vortex [492], eddy diffusion [493-501,654,955], rotary stirring [502], turbulence [495,503], EK instability [504—506], chaotic advection [248,507-513], magnetic stirring [514], bubble-induced acoustic mixing [515], and piezoelectric actuation [516,517]. 

Jones, S.W. Young, W.R. Shear dispersion and anomalous diffusion by chaotic advection. J. Fluid Mech. 1994, 280, 149-172. 

For fully developed three-dimensional turbulence one can identify different regimes. Below the Kolmogorov scale the velocity field is smooth and chaotic advection dominates, therefore the FSLE is the same as the standard Lyapunov exponent and is independent of 5. When 5 is within the inertial range, we have A(5) 5 2/3. If the size of the domain is much larger than the integral scale this is followed by a diffusive regime where A(5) 5 2 for 6 > L. Thus the FSLE... 

P.E. Arratia and J.P. Gollub. Predicting the progress of diffusively limited chemical reactions in the presence of chaotic advection. Phys. Rev. Lett., 96(2) 024501, 2006. 

M. Chertkov and V. Lebedev. Boundary effects on chaotic advection-diffusion chemical reactions. Phys. Rev. Lett., 90 134501, 2003. 

C. Lopez and E. Hernandez-Garcfa. The role of diffusion in the chaotic advection of a passive scalar with finite lifetime. Eur. Phys. J. B, 28 353, 2002. 

T.H. Solomon, E.R. Weeks, and H.L. Swinney. Chaotic advection in a two-dimensional flow Levy flights and anomalous diffusion. Physica D, 76 70-84, 1994. 

Y.-K. Tsang, T.M. Antonsen, and E. Ott. Exponential decay of chaotically advected passive scalars in the zero diffusivity limit. Phys. Rev. E, 71 066301, 2005. 

Passive micromixers rely on the mass transport phenomena provided by molecular diffusion and chaotic advection. These devices are designed with a channel geometry that increases the surface area between the different fluids and decreases the diffusion path. By contrast, the enhancement of chaotic advection can be realized by modifying the design to allow the manipulation of the laminar flow inside the channels. The modified flow pattern is characterized by a shorter diffusion path that improves the mixing velocity. In this section, an overview of the different types of passive micromixers is provided. Mixed phase passive micromixers can be categorized as ... 

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