Observed in many physical systems, coarsening is an orderly decrease in the
number of localized structures, such as particles, drops, shear bands, solitons,
or point defects. Coarsening is a type of pattern
formation in which the characteristic length scale between features grows
while the total number of features decreases. These phenomena have
been studied in many problems and several mathematical techniques for
modeling these phenomena have been developed. This dissertation examines the
aggregation of drops in the thin film equation, where drops may coarsen
through two general mechanisms: collision and collapse. A series of
simplifications to model this process is developed. Slender-body
asymptotics is applied to the
Navier-Stokes equations for fluid motion in order to derive the Reynolds
lubrication equation. The lubrication equation is in turn simplified to
a coarsening dynamical system (CDS) model for interacting
drops through solvability conditions for a perturbation about a drop-type steady
state. Lastly, the dynamical system is averaged into an ensemble model to
describe the dynamics of the distribution of drop sizes.
The ensemble model takes the form of an integro-differential equation for
the distribution function, much like the model of Ostwald ripening proposed
by Lifshitz and Slyozov. A convenient choice of scaling yields an
intermediate asymptotic self-similar solution. This solution is compared
to numerical simulations of the ensemble model and histograms of drop
masses from the CDS model. The early-time dynamics before
similarity are explored by varying the initial distribution of drop
sizes. Interesting far-from-similarity ``stairstep'' behavior is observed
in the coarsening rate when the initial distribution has a very small
variance. A well-chosen initial condition with a fractal-like structure is
shown to replicate the stairstep behavior.
At very long times, the mean drop size grows large, requiring the inclusion
of gravity in the model. The CDS model parameters are modified as a result of
the dependence of drop shapes on both size and gravity. The new dynamical
system predicts the coarsening rate slowing from a power law to an inverse
logarithmic rate. The energy liberated by each coarsening event is shown
to approach a gravity-dependent constant as the mean drop mass increases.
This suggests a reason for the coarsening slow-down. / Dissertation
| Identifer | oai:union.ndltd.org:DUKE/oai:dukespace.lib.duke.edu:10161/595 |
| Date | 15 April 2008 |
| Creators | Gratton, Michael B. |
| Contributors | Witelski, Thomas P. |
| Source Sets | Duke University |
| Language | en_US |
| Detected Language | English |
| Type | Dissertation |
| Format | 2279914 bytes, application/pdf |
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