Edge states control droplet break-up in sub-critical extensional flows

TL;DR

This study investigates how edge states influence droplet break-up in sub-critical extensional flows, revealing that a specific unstable equilibrium controls the break-up process and path, aligning with experimental observations.

Contribution

It introduces a numerical method to identify the edge state in droplet dynamics, elucidating the role of basin boundaries in sub-critical flow-induced break-up.

Findings

Edge state controls droplet break-up.

Basin boundary separates stable and unstable outcomes.

Path to break-up matches end-pinching mechanism.

Abstract

A fluid droplet suspended in an extensional flow of moderate intensity may break into pieces, depending on the amplitude of the initial droplet deformation. In subcritical uniaxial extensional flow the non-breaking base state is linearly stable, implying that only a finite amplitude perturbation can trigger break-up. Consequently, the stable base solution is surrounded by its finite basin of attraction. The basin boundary, which separates initial droplet shapes returning to the non-breaking base state from those becoming unstable and breaking up, is characterized using edge tracking techniques. We numerically construct the edge state, a dynamically unstable equilibrium whose stable manifold forms the basin boundary. The edge state equilibrium controls if the droplet breaks and selects a unique path towards break-up. This path physically corresponds to the well-known end-pinching mechanism. Our results thereby rationalize the dynamics observed experimentally [Stone & Leal, J. Fluid Mech. 206, 223 (1989)]