Discrete self-similarity in interfacial hydrodynamics and the formation of iterated structures

TL;DR

This paper investigates how discrete self-similarity leads to the formation of iterated structures like drops and filaments in thin viscous films, combining computational and theoretical approaches to reveal underlying patterns and bifurcations.

Contribution

It introduces the concept of discrete self-similarity as a mechanism for pattern formation in interfacial hydrodynamics, linking it to bifurcation theory and providing a new understanding of structure development.

Findings

Iterated structures arise from discrete self-similarity in thin film flows.

Discretely self-similar solutions result from a Hopf bifurcation.

Patterns include infinite sequences of ridges and filaments.

Abstract

The formation of iterated structures, such as satellite and sub-satellite drops, filaments and bubbles, is a common feature in interfacial hydrodynamics. Here we undertake a computational and theoretical study of their origin in the case of thin films of viscous fluids that are destabilized by long-range molecular or other forces. We demonstrate that iterated structures appear as a consequence of discrete self-similarity, where certain patterns repeat themselves, subject to rescaling, periodically in a logarithmic time scale. The result is an infinite sequence of ridges and filaments with similarity properties. The character of these discretely self-similar solutions as the result of a Hopf bifurcation from ordinarily self-similar solutions is also described.