Robin M. Baur

Harvey Mudd College Mathematics 2005

Thesis Proposal: Rupture in Thin Fluid Films
Thesis Advisor: Prof. Andrew J. Bernoff
Second Reader: Prof. Jon Jacobsen
First Presentation: An Overview of Thin Fluid Films
Annotated Bibliography: rmbaur-2005-annbib.pdf
Midyear Report: An Investigation of Rupture in Thin Fluid Films
Completed Thesis (PDF): The Analysis of Rupture in Thin Fluid Films
Completed Thesis (LaTeX source): Complete tar archive of necessary files

Rupture in Thin Fluid Films

The partial differential equation
ht + (hnxxx)x = 0
models the behavior of thin fluid films. This PDE is generally intractable to finding general solutions, but it is possible to characterise its behavior under certain conditions (e.g. certain classes of boundary or initial conditions) using energy methods and numerical simulation. R.S. Laugesen in particular has discovered various energies that are dissipated for the boundary value problem with periodic boundary conditions.

We are concerned with the behavior of a thin fluid meniscus; that is, a thin film in a cell with a set contact angle, so that hx = α and hxxx = 0. Many prior results are particular to the boundary data and will need to be reproduced for our selection of data.

It can be shown for thin films in a cell with periodic boundary conditions that the film cannot rupture (that is, the thickness of the film cannot become 0) if the exponent n is larger than 3.5. Numerical results suggest that in fact the film cannot rupture if n is larger than 2.5. Furthermore, it is hypothesized that the critical value of the exponent above which the film cannot rupture is 2, but this remains only an hypothesis.

Some Preliminary Numerics

Since a fair amount of insight can be gained in this problem from numerical simulations, we present here some images that were output by the numerical code that has been written for this thesis.

The code used here was written in MATLAB using the ode15s routine to solve the system of ODEs that results from discretizing the above PDE with respect to the spatial variable.

The following pictures are for n = 1 after a time t = .0195. Note that on either side of the central droplet, the film has touched down. The first image is of the entire cell in which we observe the film; the second image is zoomed in to better demonstrate the rupture.

Graph showing a film for <em>n</em>=1
after a time <em>t</em>=0.0195 s; view of the entire cell.
Graph showing a film for <em>n</em>=1
after a time <em>t</em>=0.0195 s; zoomed in cell to show rupture.

Now observe the film with the same initial conditions, but with n = 2. Note that, in the first picture, with t = .0195 as above for n = 1, the film is not even close to rupture, and that even if we allow the time to run as far as t = .1, as in the second image below, the film has still not ruptured.

Graph showing a film for <em>n</em>=2
after a time <em>t</em>=0.0195 s.
Graph showing a film for <em>n</em>=2
after a time <em>t</em>=0.1 s.

It is often preferable to use refinement analysis to determine whether a film is approaching a finite or infinite time singularity. The plots below exhibit the minimum thickness of the film on a logarithmic scale against the time step at which these minima occur (on a linear scale). The first plot below is the minimum plot for the film with n = 1, for which we have ending snapshots above. The second plot is for the film above with n = 2.

Minimum plot for a film with <em>n</em>=1.
Minimum plot for a film with <em>n</em>=2.

Note that in the first minimum plot, we can see the minimum thickness of the film quickly approaching 0, demonstrating the finite-time singularity. In the second minimum plot, we can see that the minimum thickness approaches 0 much more slowly, indicating either an infinite-time singularity or no singularity at all.