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A singular minimization problem for droplet profiles

Published online by Cambridge University Press:  26 September 2008

Amy Novick-Cohen
Amy Novick-Cohen
Affiliation:
Department of Mathematics, Technion-Israel Institute of Technology, Haifa 32000, Israel
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Abstract

A minimization problem for partially wetting droplet profiles is considered, in which Van der Waal's forces have been taken into account via a singular ‘disjoining pressure’. When the singular disjoining pressure is neglected, energy minimization leads to Laplace's equation and Young's equation; once the singular disjoining pressure is included, this is no longer the case. Indeed, the free energy is no longer bounded from below. Introducing the notion of overtaking to compare the energies of configurations whose energies are arbitrarily large and negative, we demonstrate that if a configuration is not convex then it cannot be an absolute minimizer. If profiles are allowed to ‘double-over’ then there does not exist an absolute minimizer. Within the class of profiles which do not double-over, absolute minimizers are shown to exist; these minimizing profiles are not single-valued. The singular minimization problem is shown to be discontinuously dependent on the definition of the wetting profile in the neighbourhood of the contact points; the implications of this discontinuity are discussed.

Type
Research Article
Information
European Journal of Applied Mathematics , Volume 4 , Issue 4 , December 1993 , pp. 399 - 418
Copyright
Copyright © Cambridge University Press 1993

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