Saffman-Taylor Instability

## Capstone (Math 451H) Spring 2002: Instabilities of two-fluid flow in Hele-Shaw geometry

When a low viscosity fluid (for example, water) is injected into a more viscous one, such as glycerin, an instability occurs. The figure below shows an example of this instability (often called Saffman-Taylor instability) in the Hele-Shaw geometry, where blue water is injected into a colorless glycerin. One important reason for studying this problem is that it is closely related to many technologically relevant ones, such as flow in porous media.

Main purpose of this project was to understand the characteristics of the emerging patterns. A variety of techniques were used to achieve this goal:

• a number of experiments were performed, where fluids, injection rates, and separation between the plates were varied. As an example, the movie below shows the influence of the injection rate on the pattern formation
• students derived a governing equation (Darcy's law) that models the dynamics in Hele-Shaw flow. Then, they performed linear stability analysis, and compared their results to the experimental ones;
• numerical simulations of the governing equation were performed. The results of this simulations were then compared to both experiments, and the linear stability analysis;
• in a different, but related direction, discrete simulations based on random walkers (diffusion limited aggregation) were formulated. These simulations produced elaborate patterns, shown in the figures below. The main difference between the different results is the `sticking' rules determining how random walkers stick to the interface. In this manner, the resulting patterns model the experimental ones resulting in the experiments where surface tension between the fluids is varied. These simulations were presented by their author, Tsezar Seman, at the 13th Annual Saint Joseph's University Sigma Xi Student Research Symposium in Philadelphia, April 2002.