# 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 |

E-Mail: | rmbaur@math.hmc.edu |

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

h_{t} + (h^{n}_{xxx})_{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 h_{x} = α and h_{xxx} = 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.

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.

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.

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.