Bi-Laplacian Growth Patterns in Disordered Media

TL;DR

This paper develops a new theoretical framework for bi-Laplacian growth patterns observed in experiments with disordered media, using iterated conformal maps to address the complex fracture phenomena in elastic materials.

Contribution

It introduces an analytic theory for bi-Laplacian growth patterns, filling a gap in understanding fracture in elastic media through conformal mapping techniques.

Findings

Provides a mathematical model for fracture patterns in elastic media

Connects experimental observations with conformal map theory

Advances understanding of bi-Laplacian growth processes

Abstract

Experiments in quasi 2-dimensional geometry (Hele Shaw cells) in which a fluid is injected into a visco-elastic medium (foam, clay or associating-polymers) show patterns akin to fracture in brittle materials, very different from standard Laplacian growth patterns of viscous fingering. An analytic theory is lacking since a pre-requisite to describing the fracture of elastic material is the solution of the bi-Laplace rather than the Laplace equation. In this Letter we close this gap, offering a theory of bi-Laplacian growth patterns based on the method of iterated conformal maps.