Helmensdorfer, Sebastian (2012) A model for the behavior of fluid droplets based on mean curvature flow. SIAM Journal on Mathematical Analysis, Vol.44 (No.3). pp. 1359-1371. doi:10.1137/110824905 ISSN 0036-1410.

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Abstract

The authors of [W. D. Ristenpart et al., Nature, 461 (2009), pp. 377–380] have observed the following remarkable phenomenon during their experiments. If two oppositely charged droplets of fluid are close enough, at first they attract each other and eventually touch. Surprisingly after that the droplets are repelled from each other, if the initial strength of the charges is high enough. Otherwise they coalesce and form a big drop, as one might expect. We present a theoretical model for these observations using mean curvature flow. The local asymptotic shape of the touching fluid droplets is that of a double cone, where the angle corresponds to the strength of the initial charges. Our model yields a critical angle for the behavior of the touching droplets, and numerical estimates of this angle agree with the experiments. This shows, contrary to general belief (see [W. D. Ristenpart et al., Nature, 461 (2009), pp. 377–380] and [W. D. Ristenpart et al., Phys. Rev. Lett., 103 (2009), 164502]), that decreasing surface energy can explain the phenomenon. To determine the critical angle within our model, we construct appropriate barriers for the mean curvature flow. In [Comm. Partial Differential Equations, 20 (1995), pp. 1937–1958] Angenent, Chopp, and Ilmanen manage to show the existence of one-sheeted and two-sheeted self-expanding solutions with a sufficiently steep double cone as an initial condition. Furthermore they provide arguments for nonuniqueness even among the one-sheeted solutions. We present a proof for this, yielding a slightly stronger result. Using the one-sheeted self-expanders as barriers, we can determine the critical angle for our model.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science, Engineering and Medicine > Science > Mathematics
Library of Congress Subject Headings (LCSH):	Drops -- Mathematical models
Journal or Publication Title:	SIAM Journal on Mathematical Analysis
Publisher:	Society for Industrial and Applied Mathematics
ISSN:	0036-1410
Official Date:	2012
Dates:	Date Event 2012 Published
Volume:	Vol.44
Number:	No.3
Page Range:	pp. 1359-1371
DOI:	10.1137/110824905
Status:	Peer Reviewed
Publication Status:	Published
Access rights to Published version:	Restricted or Subscription Access
Date of first compliant deposit:	22 December 2015
Date of first compliant Open Access:	22 December 2015
Funder:	Leverhulme Trust (LT)

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