Eigenvalue–eigenfunction analysis of infinitely fast reactions and micromixing regimes in regular and chaotic bounded flows

Abstract

We analyze the dynamics of a single irreversible reaction A+ B→ Products, occurring in a bounded incompressible flow. Within the limits of infinitely fast kinetics, the system is reduced to an advection–diffusion equation for the scalar φ, representing the difference between the reactant concentrations. By the linearity of the governing PDE, the system evolution is determined by the properties of the eigenvalue–eigenfunction spectrum associated with the advection–diffusion operator. In particular, the dependence of the dominant eigenvalue Λ—yielding the time-scale controlling the asymptotic reactant decay—as a function of the molecular diffusivity, D , for different stirring protocols is analyzed. We find Λ∼ D α , where the exponent α∈[0,1] depends upon the kinematic features of the stirring flow. When the kinematics is regular within most of the flow domain (e.g. two-dimensional autonomous flows or time-periodic protocols possessing large quasiperiodic islands) a purely diffusive scaling, α=1 settles as D→0 . The singular scaling α=0 is found in the case of globally chaotic kinematics, whereas mixed regimes, 0< α<1, occur in flows that are characterized by the coexistence of quasiperiodic and chaotic behavior. The analysis of spectral properties of the advection–diffusion operator provides a new classification of micromixing regimes, and new mixing indices for quantifying homogenization performances in the presence of diffusion.