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

We study an instantaneous bimolecular chemical reaction in a two-dimensional chaotic, incompressible and closed Navier-Stokes flow. Areas of well mixed reactants are initially separated by infinite gradients. We focus on the initial regime, characterized by a well-defined one-dimensional contact line between the reactants. The amount of reactant consumed is given by the diffusive flux along this line, and hence relates directly to its length and to the gradients along it. We show both theoretically and numerically that the probability distribution of the modulus of the gradient of the reactants along this contact line multiplied by {\kappa} does not depend on the diffusion {\kappa} and can be inferred, after a few turnover times, from the joint distribution of the finite time Lyapunov exponent {\lambda} and the frequency 1/{\tau} . The equivalent time {\tau} measures the stretching time scale of a Lagrangian parcel in the recent past, while {\lambda} measures it on the whole chaotic orbit. At smaller times, we predict the shape of this gradient distribution taking into account the initial random orientation between the contact line and the stretching direction. We also show that the probability distribution of the reactants is proportional to {\kappa} and to the product of the ensemble mean contact line length with the ensemble mean of the inverse of the gradient along it. Besides contributing to the understanding of fast chemistry in chaotic flows, the present study based on a Lagrangian stretching theory approach provides results that pave the way to the development of accurate sub- grid parametrizations in models with insufficient resolution for capturing the length scales relevant to chemical processes, for example in Climate-Chemsitry Models.