Graduation Year

2005

Document Type

Open Access Senior Thesis

Degree Name

Bachelor of Science

Department

Mathematics

Reader 1

Andrew J. Bernoff

Reader 2

Jon T. Jacobsen

Abstract

The behavior of a fluid with a thin capillary meniscus can be modelled on a one-dimensional domain Ω = [−L, L] by the thin film equation ht = −(hnhxxx)x with boundary conditions hx(±L) = ±α (giving a fixed contact angle) and hxxx(±L) = 0 (prohibiting mass flux). It is desirable to know whether or not such a film experiences rupture; that is, whether there exists some x0, t0 (with t0 possibly ∞) such that h(x0, t0) = 0, corresponding to the appearance of a dry spot. We approach this problem using energy methods, which use the conservation or dissipation of quantities such as mass, surface area, coating energy, and other more abstract quantities to describe the behavior of the fluid. We present a brief analysis of the behavior of some of these energies, as well as a proof that, given certain assumptions, rupture cannot occur in a thin capillary meniscus for n > 4 and, in more restricted cases, for n > 7/2. We also show that rupture must occur for 0 < n < 1/2. We describe the asymptotic behavior of the regions in which rupture occurs. We also describe the numerical implementation of this problem and the advantages and drawbacks of using certain prewritten solvers in MATLAB and new implementations of θ-weighted schemes and the Newton-Raphson method. We propose uses of these numerical methods to make further progress on the problem.

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