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The nascent coffee ring with arbitrary droplet contact set: an asymptotic analysis

Moore, Matthew; Vella, Dominic; Oliver, James

Authors

Dominic Vella

James Oliver



Abstract

We consider the effect of droplet geometry on the early-stages of coffee ring formation during the evaporation of a thin droplet with an arbitrary simple, smooth, pinned contact line. We perform a systematic matched asymptotic analysis of the small-capillary number, large-solutal P├ęclet number limit for two different evaporative models: a kinetic model, in which the evaporative flux is effectively constant across the droplet, and a diffusive model, in which the flux is singular at the contact line. For both evaporative models, solute is transported to the contact line by a capillary flow in the droplet bulk while, local to the contact line, solute diffusion counters advection. The resulting interplay leads to the formation of the nascent coffee-ring profile. By exploiting a coordinate system embedded in the contact line, we solve explicitly the local leading-order problem, deriving a similarity profile (in the form of a gamma distribution) that describes the nascent coffee-ring. Notably, for an arbitrary contact-line geometry, the ring characteristics change due to the concomitant asymmetry in the shape of the droplet free surface, the evaporative flux (for diffusive evaporation) and the mass flux into the contact line. We utilize the asymptotic model to determine the effects of contact-line geometry on the growth of the coffee ring for a droplet with an elliptical contact set. Our results offer mechanistic insight into the effect of contact-line curvature on the development of the coffee-ring from deposition up to jamming of the solute; moreover our model predicts when finite concentration effects become relevant.

Citation

Moore, M., Vella, D., & Oliver, J. (2022). The nascent coffee ring with arbitrary droplet contact set: an asymptotic analysis. Journal of Fluid Mechanics, 940, Article A38. https://doi.org/10.1017/jfm.2022.251

Journal Article Type Article
Acceptance Date Mar 4, 2022
Online Publication Date Apr 12, 2022
Publication Date Jun 10, 2022
Deposit Date Mar 25, 2022
Publicly Available Date Oct 13, 2022
Journal Journal of Fluid Mechanics
Print ISSN 0022-1120
Publisher Cambridge University Press
Peer Reviewed Peer Reviewed
Volume 940
Article Number A38
DOI https://doi.org/10.1017/jfm.2022.251
Public URL https://hull-repository.worktribe.com/output/3954407

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Copyright Statement
This article has been published in a revised form in Journal of Fluid Mechanics https://dx.doi.org/10.1017/jfm.2022.251. This version is free to view and download for private research and study only. Not for re-distribution, re-sale or use in derivative works. © The Author(s), 2022.





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