Department of Mathematics, **Machine** VP

School of Mathematical Sciences

**Co-Investigator** B I Henry

Department of Applied Mathematics,

School of Mathematics, University of New South Wales

**Pattern Formation in Laplacian Fields and Zero-Noise Limit of
Three-Dimensional Laplacian Growth Processes**

These projects involve the investigation of basic models for pattern formation at far from equilibrium conditions. Such models have been proposed in the past to describe a wide range of physical phenomena. Of most relevance here is a simple but beautiful series of viscous fingering experiments by Ben-Jacob et al in an etched radial Hele Shaw cell. In these experiments, a less viscous fluid (such as air or water) is pumped into a more viscous fluid (such as glycerol) between two narrowly spaced glass plates. The resulting patterns are nearly two-dimensional in nature and vary in shape according to the applied pressure and spacing of the plates. In this case a triangular grid is etched onto the surface of one of the plates, resulting in roughly six-fold symmetric anisotropic patterns. The basic observed morphologies range between compact faceted shapes at low pressure to elongated dendrites at high pressure. However, in between is a rather puzzling region of patterns in which the dominating growth is at 30 degrees to the lattice axes. Project r50 aims to simulate the basic equations describing the Hele Shaw cell experiments. In this case the governing equations are numerically solved on a planar triangular grid. On the other hand, project r05 involves the simulation of corresponding pattern formation in three dimensions.

**What are the basic questions addressed?**

The governing equations describing the evolution of patterns in the etched Hele Shaw cell experiments, including a microscopic effective surface tension and microscopic surface relaxations, have not been fully simulated previously. Our approach follows our previous work in which we introduced and implemented numerical schemes for simulating the zero-noise deterministic limit of discrete Laplacian growth processes. Beyond the possible simulation of the equations, the basic question to be addressed was if we could recover the full range of experimentally observed morphologies.

**What are the results to date and the future of the work?**

After a systematic series of simulations covering the full range of parameter values, corresponding to plate separation and applied pressure, we were able to reproduce the full range of observed morphologies. This work is currently being extended to predict the patterns to be observed in experiments involving etched square and hexagonal grids.

In three dimensions we have simulated the most basic growth model--diffusion-limited aggregation--in the zero-noise limit with an effective surface tension. We find that the growth evolves from stable needle fingers which split as the surface tension is increased. We are currently developing theoretical methods to predict the observed behaviour.

**What computational techniques are used and why is a supercomputer
required?**

Our numerical simulations involve the solution of Laplace's equation with moving boundary conditions on a discrete lattice. The evolving patterns become large in time and need to be explored throughout a wide range of model parameters.

**Publications**

*Diffusion-limited aggregation with Eden growth surface kinetics*,
M T Batchelor and B I Henry, Physica A, **203**, 566-582 (1994).

*Pattern formation in an etched radial Hele Shaw cell*, M T Batchelor and
B I Henry, Physical Review E, submitted.