Harmonic moment dynamics in Laplacian growth

TL;DR

This paper investigates the dynamics of harmonic moments in Laplacian growth, demonstrating their linearization of zero surface tension problems, their decay with surface tension, and their practical measurement of surface tension in a laboratory setting.

Contribution

It extends the theory of harmonic moments to include surface tension effects and validates the approach through laboratory experiments, providing a new method to measure surface tension.

Findings

Harmonic moments are conserved in zero surface tension growth.

With surface tension, harmonic moments decay over time.

Laboratory measurements of harmonic moments yield surface tension within 20% of accepted values.

Abstract

Harmonic moments are integrals of integer powers of z = x+iy over a domain. Here the domain is an exterior of a bubble of air growing in an oil layer between two horizontal closely spaced plates. Harmonic moments are a natural basis for such Laplacian growth phenomena because, unlike other representations, these moments linearize the zero surface tension problem (Richardson, 1972), so that all moments except the lowest one are conserved in time. For non-zero surface tension, we show that the the harmonic moments decay in time rather than exhibiting the divergences of other representations. Our laboratory observations confirm the theoretical predictions and demonstrate that an interface dynamics description in terms of harmonic moments is physically realizable and robust. In addition, by extending the theory to include surface tension, we obtain from measurements of the time evolution of the harmonic moments a value for the surface tension that is within 20% of the accepted value.