Lamellar reacting models

The idea of focussing on the transverse profile of filaments (see Sect. 2.7.1) has a long history, specially in the chemical engineering and combustion contexts (Carrier et al., 1975 Ranz, 1979 Ottino, 1994 Bish and Dahm, 1995), where it appears under the names of lamellar models or stretch models. More recently it has been explicitly used by Neufeld (2001), Neufeld et al. (2002b), McLeod et al. (2002), Neufeld et al. (2002c), Szalai et al. (2003), Hernandez-Garcfa et al. (2003), or Cox (2004) and is one of the ideas behind the approaches used in papers such as Jimenez and Martel (1991), Wonhas and Vassilicos (2002), or the ones reviewed in Tel et al. (2005). 

In a reaction-diffusion-advection formulation, the model considers a one-dimensional slice transverse to concentration filaments, and models its evolution by an equation of the type 

Equation (4.13) is a particular case. Ci is a set of concentrations interacting through the reaction terms Ri and vx = (t)x is the, possibly time-dependent, transverse velocity field pointing towards the center of the filament located at x = 0. As in Sect. 2.7.1 this is to be understood as a local Lagrangian description. 

The change of variables (2.92) makes here also the advection term in (5.1) to disappear (we stress that equality of the diffusion coefficients of all species is needed see Kiss et al. (2003b) for an example in which this condition is not fulfilled). In the new coordinates the equations for the concentrations defined by 

The advection terms have disappeared, at the expense of introducing time-dependent reaction rates Ri = s t)2Ri/D. Thus, usually this approach can be used to obtain analytic solutions only when the reaction term can be neglected or eliminated by additional changes of variables. This is the case for example with Eq. (4.13), since the reaction term p t)P can be eliminated by considering the equation for C(x,t) defined by P(x,t) = C(a ,f)exp p(u)duj. All the solutions discussed in Sect. 2.7.1 can be applied to that case. In the following we describe another situation in which one can deal with the reaction term. 

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