**II. The Saffman-Taylor Instability.**

This simulation shows the result of the Saffman-Taylor instability on the expansion of a gas bubble into a liquid in a Hele-Shaw cell. The simulation is based upon a boundary integral formulation of the equations of motion and employs the Small-Scale Decomposition to remove high-order time-stepping constraints, the Fast Multipole Method for rapid velocity evaluation, and the GMRES iterative technique for the efficient solution of integral equations.

This animation was produced by Jacek Ossowski (Courant) using OpenGL-based software on a Silicon Graphics Onyx RE2/R8000.

Click on the image for the animation.

These simulations and techniques, applied to this and other problems, are discussed in

*Removing the Stiffness from Interfacial Flows with Surface
Tension*

**Journal of Computational Physics **, Vol. 114, p. 312,
1994.

T.Y. Hou (Caltech), J.S. Lowengrub (Minnesota), and M.J. Shelley
(Courant)

download gzipped postscript
or pdf

III. Bubble Instability in a Time-Dependent Gap. Krzysztof Wlodarski and Michael Shelley

This simulation shows the Saffman-Taylor instability of a drop of fluid, resulting from the lifting of the plate in a Hele-Shaw cell. Again, this simulation is based upon a boundary integral formulation of the equations of motion and employs the Small-Scale Decomposition to remove high-order time-stepping constraints, and the GMRES iterative technique for the efficient solution of integral equations.

Click on the image for the animation.

These and other simulations, as well as a mathematical analysis of the properties of this system, can be found in

*Hele-Shaw Flow and Pattern Formation in a Time-Dependent
Gap*

by M.J. Shelley, F.-R. Tian (Ohio State), and K. Wlodarski,

in** Nonlinearity,** Vol. 10, p. 1471, 1997.

Download gzipped postscript
or PDF

Page setup by Jacek Ossowski (Courant).