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 $\mathrm{ St} \ll 1$ 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.
Categories: physics.flu-dyn