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

Published online by Cambridge University Press:  17 September 2020

Antoine Renaud Open the ORCID record for Antoine Renaud [Opens in a new window]  and
Jacques Vanneste Open the ORCID record for Jacques Vanneste [Opens in a new window]
Antoine Renaud
Affiliation:
School of Mathematics and Maxwell Institute for Mathematical Sciences, University of Edinburgh, King's Buildings, EdinburghEH9 3FD, UK
Jacques Vanneste*
Affiliation:
School of Mathematics and Maxwell Institute for Mathematical Sciences, University of Edinburgh, King's Buildings, EdinburghEH9 3FD, UK
*
†Email address for correspondence: J.Vanneste@ed.ac.uk
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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 $St \ll 1$ and $Pe \gg 1$. Starting from the Maxey–Riley model, we consider the distinguished scaling $St \, Pe = O(1)$ 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 $\bar {D}$ characterising long-time dispersion. This expression quantifies how $\bar {D}$, proportional to $Pe^{-1/2}$ when inertia is neglected, increases for particles heavier than the fluid and decreases for lighter particles. In particular, when $St \gg Pe^{-1}$, we find that $\bar {D}$ is proportional to $St^{1/2}/(\log ( St \, Pe))^{1/2}$ for heavy particles and exponentially small in $St \, Pe$ for light particles. We verify our asymptotic predictions against numerical simulations of the particle dynamics.

JFM classification

Mathematical Foundations: General fluid mechanics

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JFM Papers
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Journal of Fluid Mechanics , Volume 903 , 25 November 2020 , A2
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© The Author(s), 2020. Published by Cambridge University Press

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