IOP PUBLISHING REPORTS ON PROGRESS IN PHYSICS Rep. Prog. Phys. 71 (2008) 036601 (79pp) doi:10.1088/0034-4885/71/3/036601 Physics of liquid jets Jens Eggers1 and Emmanuel Villermaux2,3 1 School of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, UK 2 IRPHE, Universite´ de Provence, Aix-Marseille I, Technopoleˆ de Chateau-Gombert,ˆ 49, rue Fred´ eric´ Joliot-Curie 13384 Marseille Cedex 13, France Received 9 August 2007, in final form 14 December 2007 Published 21 February 2008 Online at stacks.iop.org/RoPP/71/036601 Abstract 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. (Some figures in this article are in colour only in the electronic version) This article was invited by Professor G Gillies. Contents 1. Introduction 2 4. Breakup 37 1.1. Scope and motivation 2 4.1. Overview 37 1.2. History 3 4.2. Asymptotics of viscous breakup 37 2. Describing jets 6 4.3. Other scalings and crossover 39 2.1. Essential parameters 6 4.4. Outer fluid: logarithmic scaling 42 2.2. Hydrodynamic description 8 4.5. Noise 44 2.3. Numerics: Navier–Stokes methods 10 4.6. Continuation through the singularity 47 2.4. Long-wavelength descriptions 11 4.7. Controlling breakup 48 3. Physical mechanisms and small perturbations 15 5. Sprays 51 3.1. Capillarity and the circular geometry: the 5.1. Jets everywhere: ligaments production and Plateau argument 15 dynamics 52 3.2. Capillary instability 15 5.2. Fragmentation scenarios 59 3.3. Weakly non-linear theories 17 5.3. Drop size distributions 61 3.4. A quiescent external medium 17 5.4. Origin of roughness: the case study of merging 3.5. Viscous slowing 18 jets. 65 3.6. Absolute, convective and temporal instability 20 6. Non-Newtonian effects 66 3.7. Longitudinal stretch 22 6.1. Flexible polymers 66 3.8. Shear at the interface 25 6.2. Shear-thinning fluids 70 3.9. Charged jets 28 6.3. Other non-Newtonian behaviour 71 3.10. Ferrofluids in a magnetic field 30 6.4. Surfactants 71 3.11. Other body forces 31 7. Perspectives 72 3.12. Liquid intact length 34 Acknowledgments 73 3.13. Gravitational collapse 35 References 73 3 Also at: Institut Universitaire de France. 0034-4885/08/036601+79$90.00 1 © 2008 IOP Publishing Ltd Printed in the UK Rep. Prog. Phys. 71 (2008) 036601 J Eggers and E Villermaux List of symbols 1. Introduction 1.1. Scope and motivation Q flow rate A jet is a stream of matter having a more or less columnar g acceleration of gravity shape. They are encountered in an extremely large variety of h jet radius situations, spanning a broad range of physical length scales, ρ density hence the wide scope of this review. γ surface tension Jets occur on the scale of the universe as well as on λ wavelength subatomic length scales and have attracted attention both for x dimensionless wave number their potential practical use and for their heuristic interest, We Weber number demonstrating some key phenomena of physics and applied Oh Ohnesorge number mathematics. Our focus will be on the breakup of jets, ν kinematic viscosity most often driven by surface tension. In the first stage, one Bo Bond number investigates the jet’s stability; in the case of instability one is ultimately interested in the resulting fragment sizes. The v velocity motivation is essentially twofold: κ mean curvature On one hand, these studies are motivated by practical σ stress tensor questions and applications. Among them, illustrated p pressure below, one can mention: understanding and explaining the φ velocity potential large-scale structure of the universe and the support of n normal vector galaxy clusters (figure 1), improving and optimizing liquid t tangent vector jet propulsion, diesel engine technology, manufacturing λ viscosity ratio (figure 2), agricultural sewage and irrigation (figure 3), powder η shear viscosity technology, ink-jet printing (figure 4), medical diagnostics or k wave number DNA sampling and nuclear fission. Jets are also present in our m azimuthal wave number everyday environment in kitchens, showers, pharmaceutical ω growth rate sprays and cosmetics, and are also used for our entertainment (figure 3), and for our security, for example to inflate air bags ρ outer density a or to help firemen. pa outer pressure On the other hand, jet dynamics probes a wide range of δ penetration depth physical properties, such as liquid surface tension, viscosity ηa outer viscosity or non-Newtonian rheology and density contrast with its e film thickness environment. Jets are also sensitive, on very small scales σ elongation rate (typically nanometres) to thermal fluctuations (see figure 5). ρ1 liquid density On very large scales, on the other hand, gravitational ρ2 gas density interactions are important. The basic flow state can be both v1 liquid velocity laminar and turbulent. The carrying fluid can be electrically charged, or magnetic. Nearly all classical physics comes into v2 gas velocity E electric field play in jet dynamics, and articulating the different effects in a sound picture remains in several cases a challenging exercise. + permittivity of free space 0 For all these applications or academic situations, the σ surface charge 0 recurrent questions are: will the jet break, and if so, how V electric potential long will it take? How sensitive is the jet to, e.g. background N magnetic Bond number turbulence, or the presence of a dense, viscous outer medium in S Swirl number relative motion? How does viscosity affect the ultimate stages CD drag coefficient of the separation between two droplets? After breakup, how constant of gravity disperse in size will the fragments be? If they are, how can the G ξ ligament diameter size distribution be made narrower, or broader? L ligament length Experimentally, various laboratory techniques, and notably high-speed digital cinematography in recent years, νs kinematic viscosity of solvent have revived the subject. Minute details of the breakup process νp kinematic viscosity of polymer can now be documented in real time, as well as the structure b extensibility parameter of polymer of the resulting spray, thus providing us with a rich source of λ time scale of polymer p information for comparison with theory. νe extensional viscosity On the analytical side, the most basic tool is linear stability - surfactant concentration analysis around the cylindrical base state. However, there are Pe Peclet number many important features of the break-up process for which β surface activity number non-linear effects are dominant. Numerically, this remains 2 Rep. Prog. Phys. 71 (2008) 036601 J Eggers and E Villermaux Figure 1. Studies of the large-scale structure of the cosmos indicate that the universe consists mostly of voids (90%), with filaments and sheets of galaxies comprising the rest. In the image on the left (Smithsonian Astrophysical Observatory, 1993), each of the 11 000 dots are individual galaxies. Our own Milky Way galaxy is at the centre. The outer radius is 450 million light years away. Obstruction by the plane of the Milky Way caused the missing pie-shaped sectors. The image above is less than 5% of the distance to the edge of the observable universe. (Right) Numerical simulation of the ‘filamentarization’ of the mass support in the universe, interacting by gravitation, and the subsequent breakup of filaments. Figure 2. (Left) A jet of tap water falling into a sink. The jet is too thick and its falling time too short for breakup to occur, yet it has become rough. The continuous jet hits the sink floor, where it expands radially in the form of a thin sheet boarded by a hydraulic jump. (Right) Higher speed water jets are also used to cut tissues, meat, and even metal plates. an extremely hard problem: one either has to focus on a 1.2. History detailed description of individual breakup events, using very high resolution. On the other hand, the complex geometry of The earliest study of the behaviour of jets and of breakup was a spray is hardly captured by existing codes due to the many by Leonardo da Vinci in the Codex Leicester (cf figure 6). The degrees of freedom. The final, highly non-linear stages of same work also contains thoughts on the cohesion of fluids and breakup can be understood making use of scale invariance; its role for the formation of drops [2]: an obvious example is visible from the conical shape of a ‘How water has tenacity in itself and cohesion French baguette (figure 4). The interpretation of the resulting between its particles. This is seen in the process drop size distribution, inherently large to jet breakup, requires of a drop becoming detached from the remainder, statistical tools.