Hele-shaw fingering

Description:

Hele-Shaw fingering occurs when one introduces a low viscous fluid (water) into a thin plate filled with a high viscous fluid (glycerin) through a small hole, see this video. The outcome of such an experiment exhibits pointy fingers, hence the name, see also the figure (A) below.

This type of phenomenon reveals a morphological unstable interface and a multiplicity of solutions to an a priori symmetrical problem. How is a particular solution obtained? What is the relationship of the macroscopic smooth differential equation with the underlying microscopic phenomena? How do particles moving in a somewhat random manner produce these unique patterns? These interesting questions can be analysed using probablistic methods on self-organising random models, such as,

1. Diffusion Limited Aggregation (DLA),
2. Internal Diffusion Limited Aggregation (IDLA),
3. Sandpiles,
4. Border aggregation models (BA).

For example, it is well known (due to Levine and Peres, 2008) that the scaled limit of an IDLA evolution is a Hele-Shaw flow. While on the other hand, it is conjectured that the BA model on $\mathbb{Z}^2$, see figure (B) above, is an "inversion" of a classical DLA model, (see Levine and Peres, 2007). In this project, you can concentrate on the phase transitions and modelling aspects (with room for concrete simulations) of one or more of the above-mentioned random growth and self-organising processes.

Keywords:

phase transition, random growth models, DLA, IDLA, Sandpiles, BA, random walks.

Prerequisites:

Basic knowledge of stochastic processes and martingales is desirable, which can be picked up according to the requirement of the project.

Resources:

Wikipedia: Hele-Shaw flow.

L. Levine and Y. Peres: Internal erosion and exponent of $\frac{3}{4}$, 2007.

L. Levine and Y. Peres: Spherical Asymptotics for the Rotor-Router Model in $\mathbb{Z}^d$.

A. Jarai : Sandpile models.

D. Thacker and S. Volkov: Border aggregation model.