Mixing and Bimolecular Reaction Kinetics in a Plane Poisseulle Flow

Mixing and a nonlinear bimolecular chemical reaction (reactant A + reactant B → product; reaction rate r?=?κc ₁ c ₂) in laminar shear flow are investigated. It is found that asymptotically the dominant balance between the rates of production and dissipation of the mean-squared concentration fluctuations ((sigma_{c_1 }^2 ,sigma_{c_2 }^2)) and cross-covariance of concentration fluctuations ((overline {c_1 c_2 })) occurs under nonreactive and reactive conditions. Longitudinal dispersion of the cross-sectional averages (C ₁, C ₂), and variances and the cross-covariance of reactant concentrations can be asymptotically quantified by the classic Taylor dispersion coefficient (D) even under reactive conditions. The characteristic time-scale (τ) over which molecular diffusion dissipates concentration variance and the cross-covariance of reactant concentrations is also shown to be the same under nonreactive and reactive conditions. A variational estimate of τ is shown to be close to the values inferred from detailed numerical simulation. The production-dissipation balance implies that the cross-sectional averaged reaction rate follows (overline r =kappa_{eff} C_1 C_2 ) and (kappa _{eff} approx kappa left[ {1+2Dtau left( {{partial ln C_1 } mathord{left/ {vphantom {{partial ln C_1 } {partial x}}} right. kern-nulldelimiterspace} {partial x}} right)left( {{partial ln C_2 } mathord{left/ {vphantom {{partial ln C_2 } {partial x}}} right. kern-nulldelimiterspace} {partial x}} right)} right]). The effective reaction rate parameter (κ _eff ) is higher than that of well-mixed batch test reaction rate constant (κ) for initially overlapping species and κ _eff is smaller than κ for initially non-overlapping species.