Department of  Mathematics Spring 2006 - Math 231 Topics in Applied Mathematics

An Introduction to the Theory of Free Surfaces:
Soap Bubbles, Soap Films, and Related Surface Tension Driven Flows

"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)

• R. Finn, Equilibrium Capillary Surfaces, Springer,  New York Berlin Heidelberg Tokyo, 1986.
•  Cyril Isenberg, The Science of Soap Films and Soap Bubbles,  Dover,  New York, 1992.
• U. Dierkes, S. Hildebrandt, A. Kuster, O. Wohlrab, Minimal Surfaces I, Springer -Verlag,  Berlin, 1992.
• Stanley Middleman, Modeling Axisymmetric Flows: Dynamics of Films, Jets, and Drops, Academic Press,  San Diego, 1995.
•  Johannes C. C. Nitsche, Lectures on Minimal Surfaces, Springer-Verlag, New York,  1989.
• Michael Struwe, Variational Methods,  Springer-Verlag, Heidelberg, 1996
• J. Eggers, Nonlinear dynamics and breakup of free-surface flows,  Rev. of Modern Physics., 69 no. 3  (1997), 865-929.
• P.  K. Notz and O. A. Basaran, Dynamics of drop formation in an electric field,  J. of Colloid and Interface Science,  213  (1999) 218-237.
• B. Ambravaneswaran, H. J. Subramani, S. D. Phillips, and O. A. Basaran, Dripping-Jetting Transitions in a Dripping Faucet,  Phys. Rev. Letters, 93 No. 3 (2004).