Results from the filament model

When Da 1, all the modes in the sum decay in time towards the C = 0 state, but for Da 1 at least the mode with n = 0 grows, and (7 = 0 becomes unstable to small localized perturbations. To obtain a nonlinear approximation of the steady filament solution close to the bifurcation point Dac = 1, a perturbation approximation can be used by introducing 

Collecting terms of order e gives (the prime denotes derivative with 

Up to this order, the filament width is controlled by the interplay between advection and diffusion wp = Id = 1, or in the units of (7.2), wp = Id = s/D/A, as in the case of a passive scalar. But this is only valid near the bifurcation, i.e. when Da — lei.  

To obtain an asymptotic solution in the Da 1 limit we focus on the region around the stationary front and introduce a new rescaled coordinate with the origin at the center of the front a o(Da) (which in general will not coincide with the coordinate origin, determined by the strain flow fixed point)  

Substituting into (7.3) and writing o(Da) aoDa with (3 and ao to be determined, gives 

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