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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.
[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 Universita¨t 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, Birkha¨user, Boston, MA, 2004.
[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.