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A model for the behavior of fluid droplets based on mean curvature flow
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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. 13591371. ISSN 00361410

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Official URL: http://dx.doi.org/10.1137/110824905
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 onesheeted and twosheeted selfexpanding solutions with a sufficiently steep double cone as an initial condition. Furthermore they provide arguments for nonuniqueness even among the onesheeted solutions. We present a proof for this, yielding a slightly stronger result. Using the onesheeted selfexpanders as barriers, we can determine the critical angle for our model.
Item Type:  Journal Article 

Subjects:  Q Science > QA Mathematics 
Divisions:  Faculty of 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:  00361410 
Official Date:  2012 
Volume:  Vol.44 
Number:  No.3 
Page Range:  pp. 13591371 
Identification Number:  10.1137/110824905 
Status:  Peer Reviewed 
Publication Status:  Published 
Access rights to Published version:  Restricted or Subscription Access 
Funder:  Leverhulme Trust (LT) 
References:  [AI] S. Angenent, D. Chopp, and T. Ilmanen, A computed example of nonuniqueness of mean 
URI:  http://wrap.warwick.ac.uk/id/eprint/49572 
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