"Enlightenment, however,
must come from an understanding of motives; live mathematical development
springs from
specific natural problems
which can easily be understood, but whose solutions are difficult and demand
new methods of
more general significance."
[R. Courant]
From an early age,
everyone has been fascinated by soap bubbles and soap films. Biologists,
chemists, mathematicians, and
physicists have also been
fascinated with the properties of soap bubbles and soap films. Plateau's
problem, named in honor
of the Belgian physicist
J. A. F. Plateau, is the question of finding the surface of least area
spanned by a given Jordan curve.
Plateau carried out a number
of experiments connected with the phenomenon of capillarity (e.g., every
closed wire bounds
at least one soap film).
The theory of capillarity attaches a potential energy to such a surface
that is proportional to surface
area. By J. Bernoulli's
principle of virtual work, soap films in stable equilibrium correspond
to surfaces of minimal area.
Plateau's problem was a
great challenge to mathematicians. It was solved for numerous special contours,
but a general
solution was found by Douglas
and simultaneously by Rado in the 1930s. Plateau's problem and related
questions provided
the impetus for the development
of new solution techniques in the study of partial differential equations.
In this course, we will begin
with a study of the shapes of liquid drops, bubbles, and soap films using
the Laplace-Young
equation and the equations
of Poisson and Laplace for related problems. We will carry
out a rigorous treatment of
Plateau's Problem and the
partial differential equations for surfaces of constant mean curvature.
Finally, we will consider
the problem of a liquid
jet. A jet of fluid issuing from a circular orifice does not retain its
cylindrical form, breaking up into
droplets instead. We will
consider a variation of this phenomenon when an electric field is
applied, leading us to the study
of the Taylor cone-jet problem.
The Taylor cone-jet problem leads to a number of challenging mathematical
problems in
modeling, analysis,
dynamics, and numerical simulation.
Prerequisites:
A previous course in partial differential equations (such as Math 219)
is useful, but not essential.
Contact the instructor
if you are interested in this course or have questions about your background.
Course: Math 231 Topics
in Applied Mathematics CRN: 56207
Mon: 2:20 - 3:35 PM (Duques 360) & Wed: 3:55 - 5:10 (Gelman
502)
Instructor:
Frank Baginski Office: Old Main 105-C 1922
F Street NW
Phone: (202) 994-6269 Email: baginski@gwu.edu
References (no required textbook)