On shapes and forms: Population balance dynamics of corrugated stirred fronts

Emmanuel Villermaux

doi:10.1016/j.crhy.2018.10.009

Prix Edmond-Brun 2017 de l'Académie des sciences

On shapes and forms: Population balance dynamics of corrugated stirred fronts

Emmanuel Villermaux ^1,^2,³

¹ Aix Marseille Université, CNRS, Centrale Marseille, IRPHE UMR 7342, Marseille, France
² Institut universitaire de France, Paris, France
³ CNRS/MIT/AMU Joint Laboratory “MultiScale Materials Science for Energy and Environment”, MIT Energy Initiative, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

Comptes Rendus. Physique, Prizes of the French Academy of Sciences 2017 / Prix 2017 de l'Académie des Sciences, Volume 19 (2018) no. 5, pp. 306-315.

Abstract
Résumé

We introduce a unified framework to discuss the emergence of corrugations on material interfaces transported by random media. Relating the shape of these interfaces to the stirring field giving birth to it, we formalize a population balance dynamics for the r-elements (segments of length r) needed to cover the interface contour in the course of its deformation. As long as corrugations grow kinematically, shapes change continuously, their fractal dimension $d_{f} (r, t)$ is a non-monotonous function of the scale r, and increases in time t with no bounds. Interface creation and destruction balance, however, in self-propagating fronts like flames, and in fronts smearing by molecular diffusion, through a mixing induced overlap mechanism, leading to a stationary shape. These findings, which help reexamining old observations in a new perspective, also reconcile kinetics with geometry.

Nous introduisons un cadre unifié pour discuter l'émergence de corrugations sur les interfaces matérielles transportées dans des milieux agités aléatoirement. En reliant la forme de ces interfaces au champ d'agitation qui la crée, nous formalisons une dynamique de bilan de population pour les r-éléments (segments de longueur r) nécessaires pour couvrir le contour de l'interface au cours de sa déformation. Tant que les corrugations croissent cinématiquement, les formes changent continuellement, leur dimension fractale $d_{f} (r, t)$ est une fonction non monotone de l'échelle r et augmente dans le temps t sans limite. La création et la destruction d'interfaces s'équilibrent toutefois pour les fronts auto-propagés tels que les flammes, et pour les fronts qui s'étalent par diffusion moléculaire, par un mécanisme de chevauchement induit par le mélange, conduisant à une forme stationnaire. Ces résultats, qui permettent de réexaminer d'anciennes observations dans une nouvelle perspective, réconcilient également la cinétique avec la géométrie.

Published online: 2018-07-01

DOI: 10.1016/j.crhy.2018.10.009

Keywords: Interfaces, Turbulence, Mixing, Growth, Fractals
Mots-clés : Interfaces, Turbulence, Mélange, Croissance, Fractales

Author's affiliations:

Emmanuel Villermaux ^{1, 2, 3}

¹ Aix Marseille Université, CNRS, Centrale Marseille, IRPHE UMR 7342, Marseille, France
² Institut universitaire de France, Paris, France
³ CNRS/MIT/AMU Joint Laboratory “MultiScale Materials Science for Energy and Environment”, MIT Energy Initiative, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

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     title = {On shapes and forms: {Population} balance dynamics of corrugated stirred fronts},
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Emmanuel Villermaux. On shapes and forms: Population balance dynamics of corrugated stirred fronts. Comptes Rendus. Physique, Prizes of the French Academy of Sciences 2017 / Prix 2017 de l'Académie des Sciences, Volume 19 (2018) no. 5, pp. 306-315. doi : 10.1016/j.crhy.2018.10.009. https://comptes-rendus.academie-sciences.fr/physique/articles/10.1016/j.crhy.2018.10.009/

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