Dispersion of inertial particles in cellular flows in the small-Stokes, large-Péclet regime

Abstract

We investigate the transport of inertial particles by cellular flows when advection dominates over inertia and diffusion, that is, for Stokes and Péclet numbers satisfying $\mathrm{St}\ll 1$ and $\mathrm{Pe}\gg 1$. Starting from the Maxey--Riley model, we consider the distinguished scaling $\mathrm{St}\mathrm{Pe}=O(1)$ 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 $\overline{D}$ characterising long-time dispersion. This expression quantifies how $\overline{D}$, proportional to ${\mathrm{Pe}}^{-1/2}$ when inertia is neglected, increases for particles heavier than the fluid and decreases for lighter particles. In particular, when $\mathrm{St}\gg {\mathrm{Pe}}^{-1}$, we find that $\overline{D}$ is proportional to ${\mathrm{St}}^{1/2}/(\log (\mathrm{St}\mathrm{Pe}){)}^{1/2}$ for heavy particles and exponentially small in $\mathrm{St}\mathrm{Pe}$ for light particles. We verify our asymptotic predictions against numerical simulations of the particle dynamics.

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