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We study an instantaneous bimolecular chemical reaction in a two-dimensionalchaotic, incompressible and closed Navier-Stokes flow. Areas of well mixedreactants are initially separated by infinite gradients. We focus on theinitial regime, characterized by a well-defined one-dimensional contact linebetween the reactants. The amount of reactant consumed is given by thediffusive flux along this line, and hence relates directly to its length and tothe gradients along it. We show both theoretically and numerically that theprobability distribution of the modulus of the gradient of the reactants alongthis contact line multiplied by {\kappa} does not depend on the diffusion{\kappa} and can be inferred, after a few turnover times, from the jointdistribution of the finite time Lyapunov exponent {\lambda} and the frequency1/{\tau} . The equivalent time {\tau} measures the stretching time scale of aLagrangian parcel in the recent past, while {\lambda} measures it on the wholechaotic orbit. At smaller times, we predict the shape of this gradientdistribution taking into account the initial random orientation between thecontact line and the stretching direction. We also show that the probabilitydistribution of the reactants is proportional to {\kappa} and to the product ofthe ensemble mean contact line length with the ensemble mean of the inverse ofthe gradient along it. Besides contributing to the understanding of fastchemistry in chaotic flows, the present study based on a Lagrangian stretchingtheory approach provides results that pave the way to the development ofaccurate sub- grid parametrizations in models with insufficient resolution forcapturing the length scales relevant to chemical processes, for example inClimate-Chemsitry Models.

Fast chemical reaction in a two-dimensional Navier-Stokes flow: Probability distribution in the initial regime