Rigorous derivation of a hyperbolic model for Taylor dispersion

In this paper, we upscale the classical convection-diffusion equation in a narrow slit. We suppose that the transport parameters are such that we are in Taylor's regime, i.e. we deal with dominant Péclet numbers. In contrast to the classical work of Taylor, we undertake a rigorous derivation of the upscaled hyperbolic dispersion equation. Hyperbolic effective models were proposed by several authors and our goal is to confirm rigorously the effective equations derived by Balakotaiah et al. in recent years using a formal Lyapounov–Schmidt reduction. Our analysis uses the Laplace transform in time and an anisotropic singular perturbation technique, the small characteristic parameter e being the ratio between the thickness and the longitudinal observation length. The Péclet number is written as $C \epsilon^{-\alpha}$, with \alpha <2. Hyperbolic effective model corresponds to a high Péclet number close to the threshold value when Taylor's regime turns to turbulent mixing and we characterize it by assuming 4/3 <\alpha <2. We prove that the difference between the dimensionless physical concentration and the effective concentration, calculated using the hyperbolic upscaled model, divided by $\epsilon^{2-\alpha}$ (the local Péclet number) converges strongly to zero in L^2-norm. For Péclet numbers considered in this paper, the hyperbolic dispersion equation turns out to give a better approximation than the classical parabolic Taylor model.