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The present issue of the Pacific Journal consists of invited
research articles on mathematical problems of capillarity.
A capillary surface is the interface separating two
fluids that lie adjacent to each other and do not mix. In
conjunction with boundary conditions imposed by rigid
“supporting walls”, such interfaces can exhibit
remarkable geometric properties and seemingly strange behavior,
occasionally confounding intuition. The earliest known writing on
the topic, due to Aristoteles, comains basic misconceptions that
apparently went unchallenged for almost 2000 years, when Galileo
addressed them in his Discorsi. Quantitative progress had
to await the later discovery of the Calculus. The
characterization of rise height in a circular cylindrical glass
“capillary tube” dipped into a reservoir of liquid
became a major scientific challenge of the eighteenth century,
and was not achieved during that period. Initial breakthroughs
came in 1805 and 1806 with insights of Thomas Young and
Pierre-Simon de Laplace. Young professed to scorn the
mathematical method but nevertheless introduced the mathematical
concept of mean curvature that now underlies the entire theory.
The framework for the theory achieved a clear definitive form
with the 1830 paper of Gauss, who gained conceptual advantage by
basing his study on an energy principle, in preference to the
force balance conceived by his predecessors. Even so, the Gauss
framework still leaves room for more inclusive discussion, as is
pointed out in the initial article of the present volume.
During almost a century and a half following the Gauss paper
interest for the topic declined, although the physical
foundations continued to be studied on a molecular level by van
der Waals and by his successors. With regard to global
macroscopic problems, those studies led to no changes in the
equations or boundary conditions, which present nonlinearities
that initially defied analysis. Achievements during that time
period were limited to some isolated striking insights due to
Kelvin, Rayleigh and a few others, and some of the explicit
unsolved problems of the time served as an impetus toward
development of modern numerical methods. For the equations that
apply in a gravity field, only a single nontrivial closed form
solution has as yet been discovered, and classical linearizing
procedures have provided little substantive information.
Inspired perhaps by the needs of space technology and of
medicine, and utilizing new insights appearing in geometric
measure theory, an explosion of activity has occurred during the
past thirty-five years, in many directions. New problems have
been attacked, new methods introduced, and discoveries of
basically new nature have appeared. Already during the initial
ten years of that explosion, enough substantive new material had
appeared to justify an entire issue (88:2) of this journal
devoted exclusively to capillarity theory and related problems.
The present issue, about a quarter century later and somewhat
more restrictive as to topics addressed, is intended as a sequel
to that initial one. It will be apparent to those familiar with
the earlier collection that some perspectives and also some
participants have changed, but that the level of activity and the
interest in the problems and in the methods have not lessened.
Nor have the individual results and the new insights become less
striking. That point is of course best made by the papers
themselves, which present their own messages. Unfortunately space
and time limitations have forced us to restrict the number of
papers included here; the present collection should be regarded
as an effort to make accessible in a single location a
representative section of the (considerable) current activity, in
the context of its varying methods and perspectives.
Much of the impetus for this volume developed at the First
International Summer School on Capillarity, held at the
Max-Planck-Institut für Mathematik in den
Naturwissenschaften, in Leipzig, Germany, 2003. A number of the
papers that follow had their origins in intense discussions held
during that gathering, as did early scientific training for
several students who have since continued to successful graduate
degrees.
The reader will perceive that the fortress guarding the inner
mysteries of capillarity is under heavy siege but has not yet
succumbed. We trust that the materials joined together here will
serve as a stimulus leading ultimately to completion of the
conquest.
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