Jens Eggers

Physics of liquid jets

Jets, i.e. collimated streams of matter, occur from the microscale up to the large-scale structure of the universe. Our focus will be mostly on surface tension effects, which result from the cohesive properties of liquids. Paradoxically, cohesive forces promote the breakup of jets, widely encountered in nature, technology and basic science, for example in nuclear fission, DNA sampling, medical diagnostics, sprays, agricultural irrigation and jet engine technology. Liquid jets thus serve as a paradigm for free-surface motion, hydrodynamic instability and singularity formation leading to drop breakup. In addition to their practical usefulness, jets are an ideal probe for liquid properties, such as surface tension, viscosity or non-Newtonian rheology. They also arise from the last but one topology change of liquid masses bursting into sprays. Jet dynamics are sensitive to the turbulent or thermal excitation of the fluid, as well as to the surrounding gas or fluid medium. The aim of this review is to provide a unified description of the fundamental and the technological aspects of these subjects.

Drop Formation in a One-Dimensional Approximation of the Navier-Stokes Equation

We consider the viscous motion of a thin axisymmetric column of fluid with a free surface. A one-dimensional equation of motion for the velocity and the radius is derived from the Navier-Stokes equation. We compare our results with recent experiments on the breakup of a liquid jet and on the bifurcation of a drop suspended from an orifice. The equations form singularities as the fluid neck is pinching off. The nature of the singularities is investigated in detail.

Coalescence of liquid drops

When two drops of radius R touch, surface tension drives an initially singular motion which joins them into a bigger drop with smaller surface area. This motion is always viscously dominated at early times. We focus on the early-time behaviour of the radius rm of the small bridge between the two drops. The flow is driven by a highly curved meniscus of length 2rm and width rm around the bridge, from which we conclude that the leading-order problem is asymptotically equivalent to its two-dimensional counterpart. For the case of inviscid surroundings, an exact two-dimensional solution (Hopper 1990) shows that / r 3 m and rm (t= )l n [t=(R)]; and thus the same is true in three dimensions. We also study the case of coalescence with an external viscous fluid analytically and, for the case of equal viscosities, in detail numerically. A signicantly dierent structure is found in which the outer-fluid forms a toroidal bubble of radius / r 3=2 m at the meniscus and rm (t=4 )l n [t=(R)]. This basic dierence is due to the presence of the outer-fluid viscosity, however small. With lengths scaled by R a full description of the asymptotic flow for rm(t) 1 involves matching of lengthscales of order r 2 m , r 3=2 m , rm, 1 and probably r 7=4 m .

The Beads-on-String Structure of Viscoelastic Threads

By adding minute concentrations of a high-molecular-weight polymer, liquid jets or bridges collapsing under the action of surface tension develop a characteristic shape of uniform threads connecting spherical fluid drops. In this paper, high-precision measurements of this beads-on-string structure are combined with a theoretical analysis of the limiting case of large polymer relaxation times and high polymer extensibilities, for which the evolution can be divided into two distinct regimes. For times smaller than the polymer relaxation time over which the beads-on-string structure develops, we give a simplified local description, which still retains the essential physics of the problem. At times much larger than the relaxation time, we show that the solution consists of exponentially thinning threads connecting almost spherical drops. Both experiment and theoretical analysis of a one-dimensional model equation reveal a self-similar structure of the corner where a thread is attached to the neighbouring drops.

Theory of drop formation

The motion of an axisymmetric column of Navier–Stokes fluid with a free surface is considered. Due to surface tension, the thickness of the fluid neck goes to zero in finite time. After the singularity, the fluid consists of two halves, which constitute a unique continuation of the Navier–Stokes equation through the singular point. The asymptotic solutions of the Navier–Stokes equation are calculated, both before and after the singularity. The solutions have scaling form, characterized by universal exponents as well as universal scaling functions, which are computed without adjustable parameters.

Drop dynamics after impact on a solid wall: Theory and simulations

We study the impact of a fluid drop onto a planar solid surface at high speed so that at impact, kinetic energy dominates over surface energy and inertia dominates over viscous effects. As the drop spreads, it deforms into a thin film, whose thickness is limited by the growth of a viscous boundary layer near the solid wall. Owing to surface tension, the edge of the film retracts relative to the flow in the film and fluid collects into a toroidal rim bounding the film. Using mass and momentum conservation, we construct a model for the radius of the deposit as a function of time. At each stage, we perform detailed comparisons between theory and numerical simulations of the Navier–Stokes equation.

Breakdown of scaling in droplet fission at high Reynolds number

In this paper we address the shape of a low-viscosity fluid interface near the breaking point. Experiments show that the shape varies dramatically as a function of fluid viscosity. At low viscosities, the interface develops a region with an extremely sharp slope, with the steepness of the slope diverging with vanishing viscosity. Numerical simulations demonstrate that this tip forms as a result of a convective instability in the fluid; in the absence of viscosity this instability results in a finite time singularity of the interface far before rupture (in which the interfacial curvature diverges). The dynamics before the instability roughly follow the scaling laws consistent with predictions based on dimensional analysis, though these scaling laws are violated at the instability. Since the dynamics after rupture is completely determined by the shape at the breaking point, the time dependences of recoiling do not follow a simple scaling law. In the process of demonstrating these results, we present detailed comparisons between numerical simulations and experimental drop shapes with excellent agreement.

Inviscid coalescence of drops

We study the coalescence of two drops of an ideal fluid driven by surface tension. The velocity of approach is taken to be zero and the dynamical effect of the outer fluid (usually air) is neglected. Our approximation is expected to be valid on scales larger than

Sand as Maxwell's Demon

\ell_{\nu} = \rho\nu^2/\sigma

TWO FLUID DROP SNAP-OFF PROBLEM : EXPERIMENTS AND THEORY

, which is 10 nm for water. Using a high-precision boundary integral method, we show that the walls of the thin retracting sheet of air between the drops reconnect in finite time to form a toroidal enclosure. After the initial reconnection, retraction starts again, leading to a rapid sequence of enclosures. Averaging over the discrete events, we find the minimum radius of the liquid bridge connecting the two drops to scale like