Liquid sheet instability

Abstract This chapter relates to the liquid sheets and their instabihty. Liquid sheet instability is due to the interaction between the hquid and its surrounding fluid. When the amplitude of a perturbation grows and reaches a critical value, sheet is disintegrated forming liquid ligaments. Here, the linear and nonlinear instability of an inviscid and viscous liquid sheet is discussed, showing the effect of the aerodynamic forces on the growth rate of the initially small perturbations. Other effects, such as the effect of initial velocity profile on the instability are also discussed. 

Keywords Liquid sheet instability Nonlinear sheet instability Sinuous and dilational disturbances of a liquid sheet Thinning liquid sheet Three dimensional instability of liquid sheets Viscous sheets... 

Taylod205 also conducted mathematical analysis of the generation of ripples by wind blowing over a viscous fluid. Using a relationship between the growth of the amplitude of disturbance waves and the surface stress, Taylor derived a criterion for the instability of waves. In Taylor s instability theory, the disintegration of a liquid sheet/film is visualized as a process in which droplets are detached from the liquid surface with a wave of optimum amplitude. The diameter of the most frequent droplets is then formulated as a function of air velocity over the liquid surface, liquid density, surface tension and viscosity, as well as air density. 

Although a liquid sheet may leave the nozzle with some perturbations, the principal cause of the instabilities is the interaction of the sheet with the high-velocity air streams whereby rapidly growing waves are imposed on the sheet. Disintegration may occur when the amplitude of these waves reaches a critical value. Each full sinusoidal wave is initially distorted to yield two half-waves of very similar forms. The constant stretching of the half-waves increases... 

Flat Sheets. Generally, the interface between a liquid sheet and air can be perturbed by aerodynamic, turbulent, inertial, surface tension, viscous, acoustic, or electrical forces. The stability of the sheet and the growth rate of unstable disturbances are determined by the relative magnitude of these forces. Theoretical and experimental studies 255112561 on disintegration mechanisms of flat sheets showed that the instability and wave formation at the interface between the continuous and discontinuous phases are the maj or factors leading to... 

Linear stability theories have also been applied to analyses of liquid sheet breakup processes. The capillary instability of thin liquid sheets was first studied by Squire[258] who showed that instability and breakup of a liquid sheet are caused by the growth of sinuous waves, i.e., sideways deflections of the sheet centerline. For a low viscosity liquid sheet, Fraser et al.[73] derived an expression for the wavelength of the dominant unstable wave. A similar formulation was derived by Li[539] who considered both sinuous and varicose instabilities. Clark and DombrowskF540 and Reitz and Diwakar13161 formulated equations for liquid sheet breakup length. 

York, Stubbs, and Tek (27C) have developed a theory for the breakup of thin liquid sheets as produced by tangential nozzles. A force balance between the inter-facial tension and aerodynamic forces is the basic concept, and the criteria for stability and instability are determined from it. Wave formation at the interface is of major importance in the subsequent breakup. 

Fan Sprays, It was demonstrated around the 1950s that instability theory can be used to analyze the wave growth on a thin liquid sheet (18). This analysis predicted the existence of an optimum wavelength at which a wave would grow rapidly. This optimum wavelength, X, corresponds to a condition that leads to liquid sheet disintegration. It can be expressed as in equation 2 ... 

Hollow-Cone Sprays. In swid atomizers, the liquid emerges from the exit orifice in the form of a conical sheet. As the liquid sheet spreads radially outward, aerodynamic instability immediately takes place and leads to the formation of waves which subsequently disintegrate into ligaments and droplets. Figure 3 illustrates the breakup process in an annular liquid sheet. 

There have been numerous studies on the temporal and spatial instability of liquid sheet [1-40]. This chapter is mainly on the temporal instability. Among these, Dombrowski and his coworkers [8-16] conducted extensive studies on the factors influencing the breakup of sheets and obtained information on the wave motions of high velocity sheets. More recent analyses are provided by Senecal et al. [20], and Rangel and Sirignano [21], This chapter provides only... 

Dombrowski and John [12] combined a linear model for temporal instability and a sheet breakup model for an inviscid liquid sheet in a quiescent inviscid gas, to predict the ligament and droplet sizes after breakup. The schematic of their wavy sheet is reproduced in Fig. 3.7. The equation of motion of the neutral axis mid-way... 

Fig. 3.7 Wave instability of a thinning liquid sheet [12, Fig. 4] (Courtesy of Elsevier)...

Ibrahim [19] developed a power series mathematical solutiOTi for the problem of instability of an inviscid liquid sheet of parabolic velocity profile emanated from a nozzle into an inviscid gas. The results show that for both antisymmetrical and symmetrical disturbances departure from uniformity of the velocity profile causes the instability to be reduced. It has been suggested that jet instability may be affected by the relaxation of the velocity profile that takes place once the liquid exits the nozzle and is no longer constrained by its wall. The variation of the growth rate with wave number at We = 10,000, p = 0.01, = 0,0.1,0.3,0.4, and = 96 are shown in Fig. 3.8 for antisymmetrical disturbances. The results of Fig. 3.8 indicate... 

The linear theory does not provide a means for the liquid sheet to breakup, because during the sinuous mode of instability, the distance between the two sides of the sheet remains a constant value. Therefore, it is not possible to predict the breakup length of a sheet... 

Dombrowski N, Johns WR, The aerodynamic instability and disintegration of viscous liquid sheets, Chem, Eng. Sci. 18, 203-214, 1963. 

Ibrahim EA, Akpan ET, Three-dimensional instability of viscous liquid sheets. Atom. Sprays 6, 649-665, 1996. 

Altimira M, Riras A, Ramos JC, Anton R, Linear spatial instability of viscous flow of a liquid sheet through gas. Physics of Fluids 22, 1-11, 2010. 

Jazayeri S, Li X, Nonlinear instability of plane liquid sheets, J. Fluid Mech. 406, 281-308,... 

Lozano A, GarcTa-Olivares A, Dopazo C, The instability growth leading to a liquid sheet breakup, Phys. Fluids 10(9), 2188-2197, 1998. 

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