Discrete Self-Similarity in Interfacial Hydrodynamics and the Formation of Iterated Structures

Area of Science:

• Interfacial hydrodynamics
• Fluid dynamics
• Nonlinear dynamics

Background:

• Iterated structures like drops, filaments, and bubbles are common in interfacial hydrodynamics.
• The origin of these structures in thin viscous films destabilized by long-range forces requires further investigation.

Purpose of the Study:

• To computationally and theoretically study the origin of iterated structures in thin viscous fluid films.
• To demonstrate the link between discrete self-similarity and the formation of these structures.

Main Methods:

• Computational fluid dynamics simulations.
• Theoretical analysis of fluid instabilities.
• Investigation of Hopf bifurcations.

Main Results:

• Iterated structures arise from discrete self-similarity, where patterns repeat under rescaling in a logarithmic time scale.
• An infinite sequence of ridges and filaments with self-similar properties is generated.
• The study describes these solutions as emerging from a Hopf bifurcation.

Conclusions:

• Discrete self-similarity is the key mechanism behind iterated structure formation in destabilized thin viscous films.
• The findings provide a theoretical framework for understanding complex pattern formation in fluid interfaces.
• The research connects nonlinear dynamics concepts like Hopf bifurcations to physical phenomena in fluid mechanics.