Annular jets are encountered in many industrial processes. Their stability has been studied in the contexts of ink-jet printing (Hertz and Hermanrud, 1983; Sanz and Meseguer, 1985), encapsulation (Lee and Wang, 1989; Kendall, 1986), gas absorption (Baird and Davidson, 1962), and atomization (Crapper, Dombrowski, and Pyott, 1975; Lee and Chen, 1991; Shen and Li, 1996; Villermaux, 1998). Shen and Li analyzed the spatial-temporal instability of an annular liquid jet surrounded by an inviscid gas. Hu and Joseph (1989) investigated the temporal instability of a three-layered liquid core-annular flow. The instability of annular layeres has been used to model the formation of liquid bridges in microairways in lungs (Newhouse and Pozrikidis, 1992). The related problems of liquid bridge instability are reviewed by Alexander (1998). Annular jet instability is also of considerable theoretical interest because it includes many other flow instabilities as special cases (Meyer and Weihs, 1987). Moreover, it serves to establish knowledge of the fluid physics of flows with two distinctive curved fluid-fluid interfaces subjected to different shear forces, capillary forces, and inertial forces under variable gravitational conditions.

An Annular Jet

Consider the flow of a fluid in an annulus enclosing another fluid, which is surrounded by yet another fluid inside a circular pipe of radius Rω as shown in Figure 10.1. The axis of the pipe aligns with the direction of the acceleration due to gravity g. All three fluids are incompressible.

The phenomena of the breakup of liquid sheets and jets are encountered in nature as well as in various industrial applications. A good understanding of these phenomena requires a sound basic scientific knowledge of the dynamics of flows involving interfaces between different fluids. This book is the outcome of the author's inquiry into this fundamental knowledge. My understanding of the subject matter has been consolidated gradually through direct and indirect collaborations with my students and colleagues. The objective and scope of this book in the context of related existing works are explained in Chapter 1. Chapters 2 to 5 are devoted to exposition of the onset of sheet breakup. Chapters 6 to 10 discuss jet breakup. A perspective of the challenging aspects of the subject, including the nonlinear evolution subsequent to the onset of instability and nanojets, is sketched in Chapter 11. Some additional topics related to the breakup of a liquid body into smaller parts are discussed in the epilogue. Readers are expected to have the equivalent of at least an undergraduate background in science or engineering. In the theoretical development I have strived for mathematical rigor, numerical accuracy, and rational approximation. However, mathematics has not been used just for the sake of mathematics. I have depended on comparisons between different theories and experiments to establish physical concepts. Practical applications of the concepts are pointed out in appropriate places. The references relevant to each chapter are listed at the end of the chapter.

Nonuniform liquid sheets are encountered in various industrial applications including radiation cooling in space (Chubb et al., 1994), and paper making (Soderberg and Alfredson, 1998). Many of the works cited below were motivated by applications in surface coating, fuel spray formation, nuclear safety, and other industrial processes. The spatial variation of sheet thicknesses in these applications is necessitated by the conservation of mass flow across the cross section perpendicular to the flow direction. For example, the thickness of a planar sheet of constant width must decrease in the flow direction due to the gravitational acceleration. Consequently the local Weber number, based on the local thickness and velocity, changes spatially. In particular We may be greater than one in part of the sheet and smaller than one in the rest. If one locally applies the concept of absolute and convective instability in a uniform sheet, then part of the sheet may experience convective instability, while the remaining part may experience absolute instability. Depending on the relative location of the regions of We > 1 and We < 1, one would expect different physical consequences to the entire flow. The objective of this chapter is to elucidate the effect of the spatial variation of We on the dynamics of sheet breakup by properly applying the concept developed in the previous chapter for a sheet of uniform thickness.

A liquid jet emanating from a nozzle or orifice exhibits richly varied phenomena that depend on the orifice geometry, the inlet condition before the jet is emanated, and the environmental situation into which the jet is issued. A liquid jet cannot escape the ultimate fate of breakup because of hydro-dynamic instability. The breakup possesses two major regimes: large drop formation and fine spray formation. These two regimes are controlled by distinctively different physical forces, and between them there exist intermediate regimes. All the regimes arise from a subtle dynamic response of the jet to the disturbances.

Geometry of Liquid Jets

Citing the experiment of Bidone, Rayleigh (1945, p. 355) stated, “Thus in the case of an elliptical aperture, with major axis horizontal, the sections of the jet taken at increasing distances gradually lose their ellipticity until at a certain distance the section is circular. Further out the section again assumes ellipticity, but now with major axis vertical.” This statement is illustrated in Figure 6.1, which was taken from Taylor (1960), who also carried out the experiment. The phenomenon was understood as the vibration of a jet enclosed in an envelope of constant tension about its equilibrium configuration with a circular cross section. However, Taylor (1960) demonstrated that the phenomenon can still be predicted without the surface tension in the absence of gravity. With gravity, if the jet is issued vertically downward, it will accelerate.

Overview

When a dense fluid is ejected into a less dense fluid from a narrow slit whose thickness is much smaller than its width, a sheet of fluid can form. When the fluid is ejected not from a slit but from a hole, a jet forms. The linear scale of a sheet or jet can range from light years in astrophysical phenomena (Hughes, 1991) to nanometers in biological applications (Benita, 1996). The fluids involved range from a complex charged plasma under strong electromagnetic and gravitational forces to a small group of simple molecules moving freely with little external force. The fluid sheet and jet are inherently unstable and breakup easily. The dynamics of liquid sheets was first investigated systematically by Savart (1833). Platou (1873) sought the nature of surface tension through his inquiry of jet instability. Rayleigh (1879) illuminated his jet stability analysis results with acoustic excitation of the jet. In some modern applications of the instability of sheets and jets, it is advantageous to hasten the breakup, but in other applications suppression of the breakup is essential. Hence knowledge of the physical mechanism of breakup, aside from its intrinsic scientific value, is very useful when one needs to exploit the phenomenon to the fullest extent. Recent applications include film coating, nuclear safety curtain formation, spray combustion, agricultural sprays, ink jet printing, fiber and sheet drawing, powdered milk processing, powder metallurgy, toxic material removal, and encapsulation of biomedical materials.

In the previous chapters we saw that if We < Q−1, the disturbance of wave numbers smaller than a cut-off wave number is unstable at the onset of instability. The cut-off wave number increases with Q. The capillary force is then shown with linear theories to be responsible for the onset of instability in the presence or absence of fluid viscosities. Subsequent to the onset, the amplitude of disturbances grows rapidly and the neglected nonlinear terms in the linear theory are no longer negligible. Thus the nonlinear evolution of disturbances that lead to the eventual pinching off of drops from a liquid jet can only be described with nonlinear theories. Similarly the pinching off of small droplets from the interface caused by interfacial pressure and shear fluctuations at the onset of instability, when We > Q−1, requires nonlinear theories to describe. Experimental observations of the nonlinear phenomena are presented first.

Experiments

Capillary Pinching

Linear theory predicts that unstable disturbances of different wavelengths grow at different rates and different natural frequencies corresponding to the different wavelengths. Figure 11.1 shows the nonlinear evolution of the disturbances when external sinusoidal forcings are introduced at three different natural frequencies. The forcing frequency for Figure 11.1(a) corresponds to k = 0.683, which is close to the Rayleigh's most amplified disturbance. Figures 11.1(b) and (c) correspond to the cases of lower forcing frequencies corresponding to k = 0.25 and k = 0.075, respectively. The disturbances of wavelengths shorter than that of the fastest growing disturbance appear to grow more slowly as they are convected downstream from the nozzle exit, as predicted by linear theory.

In the previous chapters, we investigated the fairly well studied phenomena of breakup of liquid sheets and liquid jets. The basic flows were assumed to be steady in the continuum theories. Also, they were either of infinite or of semi-infinite extent in the flow direction. Physically such infinite and semi-infinite steady jets or sheets cannot exist, as predicted by stability analysis. The analytical predictions enjoyed fairly good agreement with many known experiments. However, breakup of a liquid body into smaller parts often takes place under an unsteady situation from the beginning. The examples include the formation of satellites and subsatellites from the ligaments after detaching themselves from the main drops, the formation of drops from a dripping faucet, shaped-charge jets, the formation of micro-drops by external forcing, intermittent fuel sprays, and the phenomenon of jet branching induced by external excitation. These are the subjects to be touched upon in this last chapter.

Satellite Formation

When a stretched liquid ligament is relaxed, the capillary force associated with the large surface curvature at both ends of the ligament tends to compress and fragment the ligament into small drops. We saw the formation of the ligament during the last stage of nonlinear evolution of instability. The stretching of a liquid ligament submerged in another fluid can be achieved by pure straining or shearing or a combination of both. Figure 12.1 (Stone et al., 1986) shows how a spherical drop is stretched in two purely straining external flows with two different viscosities.

This chapter elucidates the role of interfacial shear on the onset of instability of a cylindrical viscous liquid jet in a viscous gas surrounded by a coaxial circular pipe by using an energy budget associated with the disturbance. It is shown that the shear force at the liquid-gas interface retards the Rayleigh mode instability, which leads to the breakup of the liquid jet into drops of diameter comparable to the jet diameter because of capillary force. On the other hand the interfacial shear and pressure work in concert to cause the Taylor mode instability, which leads the jet to breakup into droplets of diameter much smaller than the jet diameter. While the interfacial pressure plays a slightly more important role than the interfacial shear in amplifying the longer wave spectrum in the Taylor mode, shear stress plays the main role of generating shorter wavelength disturbances.

Basic Flow

Consider the instability of an incompressible Newtonian liquid jet of radius R1. The jet is surrounded by a viscous gas enclosed in a vertical pipe of radius R2, which is concentric with the jet. For the jet to maintain a constant radius, the dynamic pressure gradients in the steady liquid and gas flows must maintain the same constant. This will allow the pressure force difference across the liquid-gas interface to be exactly balanced by the surface tension force as required. Such coaxial flows, which satisfy exactly the Navier–Stokes equations, are given by (Lin and Ibrahim, 1990).

In the previous chapter we mentioned that fluid viscosity might alter the critical Weber number that divides the parameter space into regimes of absolute and convective instability. The effects of gas and liquid viscosities are investigated separately in this chapter, not just to understand each individual effect but also to demonstrate the coupled effect, which is unexpected. In Chapter 3, stability analysis for an inviscid liquid sheet of uniform thickness was applied locally to investigate the stability of gradually thinning liquid sheets. The thinning was either due to axial expansion or gravitational acceleration. The local application of the inviscid theory for a uniform sheet to the two different cases of nonuniform sheets was made judiciously. Likewise the viscous theories given in this chapter can be applied judiciously to a gradually thinning viscous sheet whatever the cause of the thinning. The thinning may be caused by kinematic requirements, gravitational acceleration, or viscous extrusion. The breakup of a viscous liquid sheet in an inviscid gas is expounded in Section 4.1. The effect of gas viscosity is elucidated in Section 4.2. The effects of liquid and gas viscosities on the onset of sheet breakup are summarized in Section 4.3.

A Viscous Sheet in an Inviscid Gas

The basic flow attributed to G. I. Taylor is given in Section 4.1a, and its stability is analyzed in Section 4.1b. The physical mechanism of the sheet breakup is discussed in Section 4.1c, based on energy considerations.

The study of the theory of elastic wave propagation and generation can be a daunting task because of its inherent mathematical complexity. The books on the subject currently available are either advanced or introductory. The advanced ones require a mathematical background and/or maturity generally beyond that of the average seismology student. The introductory ones, on the other hand, address advanced subjects but usually skip the more difficult mathematical derivations, with frequent references to the advanced books. What is needed is a text that goes through the complete derivations, so that readers have the opportunity to acquire the tools and training that will allow them to pose and solve problems at an intermediate level of difficulty and to approach the more advanced problems discussed in the literature. Of course, there is nothing new in this idea; there are hundreds of physics, mathematics, and engineering books that do just that, but unfortunately this does not apply to seismology. Consequently, the student in a seismology program without a strong quantitative or theoretical component, or the observational seismologist interested in a clear understanding of the analysis or processing techniques used, do not have an accessible treatment of the theory. A result of this situation is an ever widening gap between those who understand seismological theory and those who do not. At a time when more and more analysis and processing computer packages are available, it is important that their users have the knowledge required to use those packages as something more than black boxes.