Front propagation in cellular flows

This problem was studied by Abel et al. (2001, 2002) in the case of simple two-dimensional cellular flows defined by a streamfunction of the form 

In the fast reaction case, the time needed for the reaction to be completed within a single cell is limited by advection, therefore the invasion of a cell can be described by an effective reaction rate pe// = U/L = n/Da, and the front speed is 

in this case the front speed enhancement due to the flow is less efficient since the advection is unable to supply fresh reactants to the front at the maximum rate to keep up with the fast reaction. In terms of the flow velocity, in these two cases the front velocity scales as vf C71/4 for slow reactions and vf t/3//4 when the reaction is fast. These theoretical predictions have been confirmed by numerical simulations (Abel et al., 2001, 2002). 

In the so called geometric optics limit when the reaction front is very thin and can be approximated by a sharp discontinuity, the front speed was found to be vf U with a logarithmic correction (Cencini et ah, 2003b). In this limit the case of time-periodic flow was also studied. As a result of the interplay between the spatial periodicity 

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The material is organized as follows. In Section II we discuss particles motion in (both regular and chaotic) laminar flows. In Section III after a brief discussion on some general results that do not depend on the specific properties of the velocity field, we shall analyze front propagation in cellular flows. 

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