Stability of similarity solutions of viscous thread pinch-off

TL;DR

This paper analyzes the linear stability of similarity solutions in viscous thread pinch-off, finding the primary solution stable and explaining satellite droplet formation, with implications for understanding breakup dynamics.

Contribution

It provides the first linear stability analysis of similarity solutions in viscous thread breakup, identifying stability properties relevant to experimental observations.

Findings

The main similarity solution is linearly stable with a complex eigenvalue.

The second similarity solution is linearly unstable.

Numerical simulations confirm the stability results and behavior near unstable solutions.

Abstract

In this paper we compute the linear stability of similarity solutions of the breakup of viscous liquid threads, in which the viscosity and inertia of the liquid are in balance with the surface tension. The stability of the similarity solution is determined using numerical continuation to find the dominant eigenvalues. Stability of the first two solutions (those with largest minimum radius) is considered. We find that the first similarity solution, which is the one seen in experiments and simulations, is linearly stable with a complex nontrivial eigenvalue, which could explain the phenomenon of break-up producing sequences of small satellite droplets of decreasing radius near a main pinch-off point. The second solution is seen to be linearly unstable. These linear stability results compare favorably to numerical simulations for the stable similarity solution, while a profile starting near the unstable similarity solution is shown to very rapidly leave the linear regime.