Dispersion by chaotic advection

The relevance of chaotic motion of fluid elements for the mixing of fluids was first recognized by Aref (1984). The key property of the chaotic orbits is their sensitivity to small changes in the initial conditions. Since small perturbations grow fast in time fluid elements initially close to each other end up later in very different regions. This also holds for the time reversed chaotic advection dynamics, so 

A simple numerical example of chaotic advection in a piecewise steady sinusoidal shear flow periodically alternating along the x and y direction is shown in Fig. 2.9, where the velocity field is defined as 

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

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. 

A more exact quantitative characterization of the chaotic advection can be given by considering the relative dispersion of fluid particles. Let us consider two fluid elements moving on trajectories r(t) and r (t). When the distance between them is small compared to the characteristic lengthscale of the velocity field (L) the velocity difference can be approximated by Taylor expansion and the separation S(t) = r (t) — r(t) satisfies the equation... 

To connect the two markedly different scenarios observed in the static and the well-mixed environments, it is natural to analyze the role of increasing mobility (Reichenbach et al., 2007). Karolyi et al. (2005) studied the above competition model combined with dispersion by a chaotic map that represents advection of fluid elements in the alternating sine-flow. By continuously changing the frequency of the chaotic dispersion as a control parameter, it is possible to follow the transitions between the two limiting situations. When the chaotic mixing is much faster than the local population dynamics, the killer and resistant cells gradually disappear from the population and only the sensitive cells survive. This is because the killer cells... 

Other channel geometries to induce chaotic advection in microdroplets have also been developed. Liau et al. [154] proposed the introduction of bumps on one side of the channel wall to promote droplet deformation. The authors proposed that the enhancement of mixing could be addressed to the thinning of the lubricant layer of dispersant fluid and by the interfacial stress induced by the bumps. A similar approach was presented by Liau et al. [154]. However, in this case the bumps were introduced in both the lateral channels walls. Similarly, Tung et al. [155] proposed the introduction of a nonuniform cross-section of the wall to deform the microdroplets and enhance the mixing. 

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