**Principal Investigator**
Murray T. Batchelor **Project **r05

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

**Zero-noise Limit of Three-dimensional Laplacian
Growth Processes **

This project involves 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. In previous projects we have investigated the numerical solution of a set of equations governing a simple but beautiful series of viscous fingering experiments 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-dimensionsal 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. The basic equations are also of relevance to the formation of snow crystals. This project involves the simulation of corresponding pattern formation in three dimensions, for which experimental results also exist.

**What are the basic questions addressed?**

The governing equations describing the evolution of patterns in these models have not been fully simulated previously. Our approach follows our earlier 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 questions to be addressed involved the nature of the basic growth morphologies. Once revealed, it was intended to form the basis for theoretical analysis.

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

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 found that as for the two-dimensional case, the growth evolves from stable needle fingers which split as the surface tension is increased. In this phase of the work we found the complete morphological description in the absence of surface tension. The simulations revealed a stable cluster consisting of six fourfold symmetric (almost planar) arms with a staircase structure along each arm. Empirical expressions were obtained for; the shape of the arms in the vicinity of the tips, the scaling of the arm lengths and the scaling of the tip growth rates. The empirical expressions were used to derive results for the growth rates along the arms of the cluster and for the step lengths in the needle staircase.

These results were in good agreement with the numerical simulations and provided a lower bound to the experimental values. Future work will involve a similar analysis with surface tension.

**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**

*Growth and form of zero-noise diffusion-limited-aggregation
on the cubic lattice,* M.T. Batchelor and
B.I. Henry, Physica A **, **in press