Boundary Dynamics of Sweeping Interface

TL;DR

This paper introduces a novel boundary dynamic model for interfaces that sweep space to gather material, linking it to crystal growth phenomena and revealing complex, unstable behaviors through simulations.

Contribution

It formulates a new boundary dynamic based on a diffusion limit of the Mullins-Sekerka problem, providing insights into interface stability and evolution.

Findings

Steady finger solutions exist for certain speeds

Numerical simulations show these solutions are unstable

Interface exhibits complex time-dependent behavior

Abstract

A new type of boundary dynamics is proposed to describe the interface that sweeps space to collect distributed material. Based upon geometrical consideration on a simple physical process representing a certain experiment, the dynamics is formulated as the small diffusion limit of Mullins-Sekerka problem of crystal growth. It is demonstrated that a steadily extending finger solution exists for a finite range of propagation speed, but numerical simulations suggest they are unstable and the interface shows a complex time development.