Discrete self-similarity in the formation of satellites for viscous cavity break-up

TL;DR

This paper investigates how the breakup pattern of a viscous fluid jet transitions from continuous to discrete self-similarity as the viscosity ratio approaches zero, leading to a cascade of filament and bubble formation.

Contribution

It demonstrates the transition from continuous to discrete self-similarity in viscous jet breakup as viscosity ratio decreases, linking it to a Hopf bifurcation.

Findings

Transition from continuous to discrete self-similarity as viscosity ratio decreases.

Formation of an infinite sequence of filaments and bubbles during breakup.

The transition is explained as a Hopf bifurcation in the governing equations.

Abstract

The breakup of a jet of a viscous fluid with viscosity $\mu _{1}$ immersed into another viscous fluid with viscosity $\mu _{2}$ is considered in the limit when the viscosity ratio $\lambda =\mu _{1}/\mu _{2}$ is close to zero. We show that, in this limit, a transition from ordinary continuous selfsimilarity to discrete selfsimilarity takes place as $\lambda $decreases. The result being that instead of a single point breakup, the rupture of the inner jet occurs through the appearance of an infinite sequence of filaments of decreasing size that will eventually produce infinite sequences of bubbles of the inner fluid inside the outer fluid. The transition can be understood as the result of a Hopf bifurcation in the system of equations modelling the physical problem.