Dispersion of inertial particles in cellular flows in the small-Stokes, large-P\'eclet regime

We investigate the transport of inertial particles by cellular flows when advection dominates over inertia and diffusion, that is, for Stokes and P'eclet numbers satisfying $\mathrm{S}\mathrm{t}\ll 1$ and $\mathrm{P}\mathrm{e}\gg 1$. Starting from the Maxey–Riley model, we consider the distinguished scaling $\mathrm{S}\mathrm{t}\mathrm{P}\mathrm{e}=O(1)$ and derive an effective Brownian dynamics approximating the full Langevin dynamics. We then apply homogenisation and matched-asymptotics techniques to obtain an explicit expression for the effective diffusivity $\bar{D}$ characterising long-time dispersion. This expression quantifies how $\bar{D}$, proportional to ${\mathrm{P}\mathrm{e}}^{-1\mathrm{/}2}$ when inertia is neglected, increases for particles heavier than the fluid and decreases for lighter particles. In particular, when $$\mathrm{S}\mathrm{t}$ \gg $\mathrm{P}\mathrm{e}$^{-1}$,wefindthat$\overline{D}$isproportionalto$$\mathrm{S}\mathrm{t}$^{1/2}/(\log ( $\mathrm{S}\mathrm{t}$ , $\mathrm{P}\mathrm{e}$))^{1/2}$ for heavy particles and exponentially small in $\mathrm{S}\mathrm{t}\mathrm{P}\mathrm{e}$ for light particles. We verify our asymptotic predictions against numerical simulations of the particle dynamics.