Lagrangian chaos

Other interesting questions concern the modification of the front geometry as a consequence of advection. In particular, one may ask if the presence of Lagrangian chaos has a role in the front dynamics. We shall briefly discuss this problem in the framework of the geometrical optics limit. 

A direct consequence of Lagrangian chaos is the exponential growth of passive scalar gradients and material fines [1, 4] A (passive) material fine of initial length q for large times grows as... 

Let us now switch to the effects of Lagrangian chaos on the asymptotic dynamics of front propagation. An immediate consequence of Eq. (47) is that the asymptotic front length (45) behaves as Lf Vq 1 for values of vo small enough. Indeed,... 

It is worth remarking that even if the scaling [Eq. (48)] holds when chaos is present, in general it is not peculiar of chaotic flows. For instance, for the shear flow (ux = U sin(y), iiy 0) one has Vf = U + vo- On the other hand, since Vf LfVo [8], even if the shear flow is not chaotic, Lf 1 /vq for U/vo 1. From the previous discussion, it seems that the front length dependence on vo is not an unambiguous effect of chaos on the asymptotic dynamics. But, if we look at Fig. 8, the spatial complexity of the front in the presence of Lagrangian chaos is evident. 

Although it is difficult to induce turbulence (so-called Eulerian chaos) in microchannels, the mixing performance obtained in low Reynolds number flow regimes can be enhanced via the chaotic advection mechanism (or so-called Lagrangian chaos). Chaotic advection occurs in regular, smooth (from a Eulerian viewpoint)... 

Results demonstrate that when agitators are switched the slope of the pathline becomes discontinuous. We will see later in this chapter how this mechanism may produce an essentially stochastic response in the Lagrangian sense. Aref termed this chaotic advection, which he suggested to be a new intermediate regime between turbulent and laminar advection. The chaos has a kinematic origin, it is temporal—that is, along trajectories associated with the motion of individual fluid particles. Chaos is used in the sense of sensitivity of the motion to the initial position of the particle, and exponential divergence of adjacent trajectories. 

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