Nonlinear dynamics and breakup of free-surface flows

TLDR

Surface‑tension‑driven flows have long fascinated scientists, and recent experiments and applications have shifted focus to the nonlinear dynamics near the breakup singularity, where one‑dimensional equations reveal universal scaling laws that allow the Navier‑Stokes equations to be continued uniquely through the singularity. The paper aims to review the theoretical development and recent experimental work on free‑surface breakup. The review synthesizes theoretical developments and experimental findings to identify remaining unsolved problems. High‑viscosity flows exhibit noise‑driven instabilities that form a nested superposition of identical singularities, while low‑viscosity flows display rich scaling.

Abstract

Surface-tension-driven flows and, in particular, their tendency to decay spontaneously into drops have long fascinated naturalists, the earliest systematic experiments dating back to the beginning of the 19th century. Linear stability theory governs the onset of breakup and was developed by Rayleigh, Plateau, and Maxwell. However, only recently has attention turned to the nonlinear behavior in the vicinity of the singular point where a drop separates. The increased attention is due to a number of recent and increasingly refined experiments, as well as to a host of technological applications, ranging from printing to mixing and fiber spinning. The description of drop separation becomes possible because jet motion turns out to be effectively governed by one-dimensional equations, which still contain most of the richness of the original dynamics. In addition, an attraction for physicists lies in the fact that the separation singularity is governed by universal scaling laws, which constitute an asymptotic solution of the Navier-Stokes equation before and after breakup. The Navier-Stokes equation is thus continued uniquely through the singularity. At high viscosities, a series of noise-driven instabilities has been observed, which are a nested superposition of singularities of the same universal form. At low viscosities, there is rich scaling behavior in addition to aesthetically pleasing breakup patterns driven by capillary waves. The author reviews the theoretical development of this field alongside recent experimental work, and outlines unsolved problems.