Mixing and bistable dynamics

The behavior of a bistable chemical reaction system under mixing in many respects is quite similar to the autocatalytic case. A standard example is the model studied by Neufeld et al. (2002b), Menon and Gottwald (2005), or Cox and Gottwald (2006), given by 

In the closed flow case we obtain again two different behaviors above and below a critical value Dac. When Da Dac and the localized perturbation added to the (7 = 0 state is large enough the phenomenology is similar to the autocat alytic case and configurations similar to those in Fig. 7.2 are obtained the reaction product 

In summary, the bistable system shows a scenario quite similar to the autocatalytic case. The main difference is the discontinuous character of the transition between the two regimes, associated with the discontinuous behavior of the observed filament at Dac, and the existence of a threshold perturbation needed to initiate it. All these features can also be recovered from the filament model (Eqs. (7.2)- 

Based on similar arguments as for the autocatalytic case the width of a filament of the C = 1 state flanked by sharp fronts can be obtained from the balance between the strain A and the propagation speed of the bistable reaction-diffusion front wp ss v/X. By using the expression (4.27), v = (1 — 2a) kD/2, we find for the stable filament solution 

The parameter dependence is the same as in the autocatalytic case, apart from the difference in the numerical factors which depend on the stability properties of the two stable states. Note that the above solution only exists for a 1/2. 

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