Nonlinear dynamics and breakup of free-surface flows

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. [S0034-6861(97)00303-6] CONTENTS

Motility induced changes in viscosity of suspensions of swimming microbes in extensional flows

“…The values of X given by similarity solutions under different conditions have been summarized by McKinley and Tripathy 19 . When inertia is important and Oh ≪ 1, X = 0.5912 is the value most likely to be observed in an experiment 19,25,26 and hence t * 1 2 → 0.7135 as Oh → 0. When viscosity is dominant on the other hand, X = 0.7127 19,27 and t * 1 2 → 7.0522 Oh as Oh → ∞.…”

Nonlinear travelling waves on a spiralling liquid jet

“…Downstream of breakup point, the jet solution no longer has physical meaning, since the jet in that region will have broken up into droplets, which cannot be described by this approach, as is also the case in other works. [6][7][8]11 The solutions obtained can be put on the physical plane x − z in the following way:…”

“…These equations (13) and (14) were also obtained before for straight axisymmetric jets by other authors. 6,7,11 However, the rotation of the container Ω enters the formulation via the initial conditions for the above system of equations, and so this system is fundamentally different to those for straight jets. We shall see in the following sections how the solutions to these equations depend critically upon the rotation of the container, and how therefore the solution to this system of equations is different to the solution of the straight jet equations.…”

Newtonian pizza: spinning a viscous sheet

“…The topic of finite-time rupture has been treated by many authors, such as Erneux & Davis (1993), Ida & Miksis (1996) and Vaynblat, Lister & Witelski (2001a) for viscous sheets and Papageorgiou (1995), Eggers (1997) and Renardy (2001) for viscous threads. Vaynblat, Lister & Witelski (2001b) have shown that lubrication models for van der Waals driven rupture of a sheet and capillary driven rupture of a thread both give rise to families of first-kind similarity solutions.…”