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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: [email protected]
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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
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 903 , 25 November 2020 , A2
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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References

REFERENCES

Auton, T. R., Hunt, J. C. R. & Prud'Homme, M. 1988 The force exerted on a body in inviscid unsteady non-uniform rotational flow. J. Fluid Mech. 197, 241–257.CrossRefGoogle Scholar
Childress, S. 1979 Alpha-effect in flux ropes and sheets. Phys. Earth Planet. Inter. 20 (2), 172–180.CrossRefGoogle Scholar
Childress, S. & Soward, A. M. 1989 Scalar transport and alpha-effect for a family of cat's-eye flows. J. Fluid Mech. 205, 99–133.CrossRefGoogle Scholar
Daitche, A. & Tél, T. 2011 Memory effects are relevant for chaotic advection of inertial particles. Phys. Rev. Lett. 107, 244501.CrossRefGoogle ScholarPubMed
DLMF 2019 NIST Digital Library of Mathematical Functions. http://dlmf.nist.gov/, Release 1.0.24 of 2019-09-15, f. (ed. Olver, W. J., Olde Daalhuis, A. B., Lozier, D. W., Schneider, B. I., Boisvert, R. F., Clark, C. W., Miller, B. R., Saunders, B. V., Cohl, H. S. & McClain, M. A.).Google Scholar
Evans, L. C. 2013 An Introduction to Stochastic Differential Equations. American Mathematical Society.CrossRefGoogle Scholar
Ferry, J. & Balachandar, S. 2001 A fast Eulerian method for disperse two-phase flow. Intl J. Multiphase Flow 27 (7), 1199–1226.CrossRefGoogle Scholar
Freidlin, M. & Wentzell, A. 2012 Random Perturbations of Dynamical Systems, 3rd edn.Springer.CrossRefGoogle Scholar
Haller, G. & Sapsis, T. 2008 Where do inertial particles go in fluid flows? Physica D 237 (5), 573–583.CrossRefGoogle Scholar
Haynes, P. H. & Vanneste, J. 2014 Dispersion in the large-deviation regime. Part 2: cellular flow at large Péclet number. J. Fluid Mech. 745, 351–377.CrossRefGoogle Scholar
Heinze, S. 2003 Diffusion-advection in cellular flows with large Péclet numbers. Arch. Rat. Mech. Anal. 168, 329–342.CrossRefGoogle Scholar
Langlois, G. P., Farazmand, M. & Haller, G. 2015 Asymptotic dynamics of inertial particles with memory. J. Nonlinear Sci. 25, 1225–1255.CrossRefGoogle Scholar
Majda, A. J. & Kramer, P. R. 1999 Simplified models for turbulent diffusion: theory, numerical modelling, and physical phenomena. Phys. Rep. 314 (4), 237–574.CrossRefGoogle Scholar
Manton, M. J. 1974 On the motion of a small particle in the atmosphere. Boundary-Layer Meteorol. 6, 487–504.CrossRefGoogle Scholar
Martins Afonso, M., Mazzino, A. & Muratore-Ginanneschi, P. 2012 Eddy diffusivities of inertial particles under gravity. J. Fluid Mech. 694, 426–463.CrossRefGoogle Scholar
Maxey, M. R. & Riley, J. J. 1983 Equation of motion for a small rigid sphere in a nonuniform flow. Phys. Fluids 26 (4), 883–889.CrossRefGoogle Scholar
Novikov, A., Papanicolaou, G. & Ryzhik, L. 2005 Boundary layers for cellular flows at high Péclet numbers. Commun. Pure Appl. Maths 867–922, 563–580.Google Scholar
Pavliotis, G. A. & Stuart, A. M. 2005 Periodic homogenization for inertial particles. Physica D 204 (3), 161–187.CrossRefGoogle Scholar
Pavliotis, G. A., Stuart, A. M. & Band, L. 2006 Monte Carlo studies of effective diffusivities for inertial particles. In Monte Carlo and Quasi-Monte Carlo Methods 2004 (ed. Niederreiter, H. & Talay, D.), pp. 431–441. Springer.CrossRefGoogle Scholar
Pavliotis, G. A., Stuart, A. M. & Zygalakis, K. C. 2009 Calculating effective diffusivities in the limit of vanishing molecular diffusion. J. Comput. Phys. 228 (4), 1030–1055.CrossRefGoogle Scholar
Rosenbluth, M. N., Berk, H. L., Doxas, I. & Horton, W. 1987 Effective diffusion in laminar convective flows. Phys. Fluids 30 (9), 2636–2647.CrossRefGoogle Scholar
Rubin, J., Jones, C. K. R. T. & Maxey, M. 1995 Settling and asymptotic motion of aerosol particles in a cellular flow field. J. Nonlinear Sci. 5 (4), 337–358.CrossRefGoogle Scholar
Sapsis, T. & Haller, G. 2010 Clustering criterion for inertial particles in two-dimensional time-periodic and three-dimensional steady flows. Chaos 20 (1), 017515.CrossRefGoogle ScholarPubMed
Shraiman, B. I. 1987 Diffusive transport in a Rayleigh–Bénard convection cell. Phys. Rev. A 36, 261–267.CrossRefGoogle Scholar
Soward, A. M. 1987 Fast dynamo action in a steady flow. J. Fluid Mech. 180, 267–295.CrossRefGoogle Scholar
Vergassola, M. & Avellaneda, M. 1997 Scalar transport in compressible flow. Physica D 106 (1), 148–166.CrossRefGoogle Scholar