  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.
