Basic physics and analytic theory of multi-scale dispersion: A review

This review revisits Taylor dispersion in shear flows. Since the classical theory relies on long-time asymptotic expansions of mean concentration, it inevitably overlooks the multi-scale dynamics inherent in early regimes. To address the entire evolution of shear dispersion, we unveil its physical nature as a set of superposable diffusive processes from a Lagrangian perspective—An approach naturally yields an analytic streamwise dispersion model. By formulating an equivalent diffusivity tensor characterized by spatial anisotropy and temporal nonlinearity, we successfully decouple the dynamics of advection and diffusion in laminar and turbulent flows across timescales. Further, we examine the dispersion of active-matter systems, including suspensions of motile micro-organisms and synthetic particles. Special emphasis is placed on the interplay between rotational dynamics and shear dispersion, particularly regarding the anomalous scaling. Finally, we discuss the fundamental significance of analytic theory for shear dispersion in complex systems, with applications spanning from pollutant transport and the hydrodynamic focusing of micro-algae, to the coordinated control of micro-robots.