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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. 1359-1371. ISSN 0036-1410

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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 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 > 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

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
curvature flow in R3, Comm. Partial Differential Equations, 20 (1995), pp. 1937–1958.
[BB] G. Barles, S. Biton, M. Bourgoing, and O. Ley, Uniqueness results for quasilinear
parabolic equations through viscosity solutions’ methods, Calc. Var., 18 (2003), pp. 159–
179.
[BO] J. Bode, Mean Curvature Flow of Cylindrical Graphs, Ph.D. thesis, Freie Universit¨at
Berlin, 2007; available online from http://www.diss.fu-berlin.de/diss/receive/FUDISS
thesis 000000003363?lang=en.
[BR] K. A. Brakke, The Motion of a Surface by Its Mean Curvature, Math. Notes Princeton,
Princeton University Press, Princeton, NJ, 1978.
[EC] K. Ecker, Regularity Theory for Mean Curvature Flow, Birkh¨auser, Boston, MA, 2004.
[EH] K. Ecker and G. Huisken, Mean curvature evolution of entire graphs, Ann. Mat., 130 (1989),
pp. 453–471.
[LS] O. A. Lady˘zenskaja, V. A. Solonnikov, and N. N. Uralceva, Linear and Quasi-Linear
Equations of Parabolic Type, AMS, Providence, RI, 1968.
[RB] W. D. Ristenpart, J. C. Bird, A. Belmonte, F. Dollar, and H. A. Stone, Noncoalescence
of oppositely charged drops, Nature, 461 (2009), pp. 377–380.
[RB1] W. D. Ristenpart, J. C. Bird, A. Belmonte, and H. A. Stone, Critical angle for electrically
driven coalescence of two conical droplets, Phys. Rev. Lett., 103 (2009), 164502.
[SI] M. Simon, Mean Curvature Flow of Rotationally Symmetric Hypersurfaces, Honours thesis,
Australian National University, 1990.