Contact:
- Office: Tech M464
- Phone: (00.1) 847.467.3451
- Email: m-gratton@northwestern.edu
- RTG Webpage
Research Interests
Perturbation Methods, Mathematical Modeling and Mathematics in Industry, Numerical Analysis, Fluid Dynamics.
Education
- Ph.D. in Mathematics, Duke University 2008
- M.A. in Mathematics, Duke University, 2004
- B.S. in Mathematics, Harvey Mudd College 2002
See my full CV.
Research
Coarsening of thin liquid films
A one-dimensional thin liquid film dewets. The droplets then communicate through the adsorbed ultra-thin film, causing them to coarsen. Early on, the most unstable coarsening mode is visible, with every other drop vanishing. Later, the dynamics are more interesting, with dilation and translational modes visible for each drop.
M.B. Gratton and T.P. Witelski, Coarsening of unstable thin films subject to gravity, Phys. Rev. E, 77:016301, 2008. [pdf]
M.B. Gratton and T.P. Witelski, Transient and self-similar dynamics in thin film coarsening, Physica D, 238:2380-2394, 2009. [pdf]
Suppressing film rupture
A thin liquid film, ordinarily subject to a dewetting instability, is sheared by a wind stress at the surface. The surface shear has constant magnitude, but rotating direction. For the right rotation rate and shear strength, the film does not dewet, but instead settles into a time-periodic steady-state.
- M.J. Davis, M.B. Gratton, and S.H. Davis, Suppressing van der Waals driven rupture through shear, J. Fluid Mech., 661:522-539, 2010. [pdf]
Current Projects
- Foams: I'm interested in the dynamics of surfactant-free foams. Such foams last for only seconds in the lab. Freezing these foams is of great interest in material science. The fundamental modeling challenge is to discover the separation in timescale between the processes of foam evolution.
- Boundary integral foams: Boundary integral methods are a natural choice for simulating bubbles in Stokes flow, but closely-packed bubbles lead to near singularities in the integrals. I'm looking at methods of retaining accuracy in the face of these near-singularities.
Here is a doubly-periodic foam evolving. The liquid is in the smaller interstitial regions. The bubbles are inviscid and incompressible. T1 topological transitions lower the energy by having four-junctions become three-junctions. Several are visible at different times in this simulation.
Teaching
- Gen Eng 206-4 (EA4): Ordinary differential equations, Spring 2011. Webpage on Blackboard. Office hours: Monday 2-4pm, Thursday 4-5pm
- ESAM 311-2: Partial differential equations, Winter 2011
Math in Industry
I'm interested in industrial problem study groups.
Upcoming:
Math Problems in Industry (MPI) workshop at NJIT, Jun. 2011
Past:
- AIM Workshop on Sustainability Problems in Palo Alto, Jan. 2011
- Claremont Colleges Math-in-Industry Workshop at Harvey Mudd College 2009
- 7th Mathematics in Medicine Study Group at the University of Southampton 2007
- Math Problems in Industry (MPI) workshop, 20th -- 23rd meetings:
- University of Delaware, Department of Mathematical Sciences 2007
- Franklin W. Olin College of Engineering 2006
- Worster Polytechnic Institute, Department of Mathematical Sciences 2005
- University of Delaware, Department of Mathematical Sciences 2004
Tools
Software I've written that might be of general interest:
- Input parameter is a small C++ class I wrote for parsing program options from a file or the command-line. It is similar to Boost's Program Options, but easier to use and less safe :)
- Parameter Study is a python script that generates runs of a program, stepping through a 1D or 2D parameter space. It uses Input Parameter to communicate the new parameter value to the program. It can also generate SGE job files so that the various runs are distributed to a cluster.
- Gnuplot Latex is a Bash script that takes the eps and tex output from the Gnuplot epslatex terminal and combines them into a single PDF.
Fun Stuff
- PyxPlot, by Dominic Ford and others, is a gnuplot-like plotting tool for Linux, but has much better LaTeX support, and an improved function interpreter.
- "Don't worry about people stealing your ideas. If your ideas are any good, you'll have to ram them down people's throats." (as quoted in A Computer Science Reader : Selections from Abacus (1988) by Eric A. Weiss, p. 404)
- About a thousand years ago, instructional methods at the University of Oxford apparently involved beating pupils over the heads with books. Here is documentary evidence:
After all that time, so little has changed...