Paper ID ICLASS SURFACE WAVES ON LIQUID SHEETS EMERGING FROM AIR-ASSIST ATOMIZERS

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1 ICLASS-006 Aug.7-Sept.1, 006, Kyoto, Japan Paper ID ICLASS06-69 SURFACE WAVES ON LIQUID SHEETS EMERGING FROM AIR-ASSIST ATOMIZERS V. Sivadas 1 and A. L. N. Moreira 1 Singapore Stanford Partnership, School of Civil & Env. Eng, Nanyang Techn Univ, Republic of Singapore, vsdas@ntu.edu.sg Instituto Superior Tecnico, Department Mechanical Engineering, Lisbon, Portugal, moreira@dem.ist.utl.pt ABSTRACT This paper addresses the physical mechanisms of instability of -D water-air interfaces at room temperature and further development to the critical state at which break up occurs. High-speed imaging techniques reveal that the flow structure moves from high amplitude sinusoidal waved to an entwined pattern, as the velocity of the surrounding air increases. A Weber number classification based on liquid inertia and aerodynamic shear allows elucidate the forces controlling the flow-field and shows that, with increasing air velocity, shear forces at the liquid-gas interface becomes significant compared to the liquid inertia force, and has pronounced effects on the wave characteristics. Gravity forces, associated with the boundary conditions, increase the amplitude of the wave and its non-linear mode of break-up. Except near the rim, remaining liquid surface is ruptured owing to the normal component of air momentum at the interface. The rim region, which is under the predominant control of surface tension forces, will breed capillary waves, and its interference can give rise to the interlaced wave pattern. The results also show that as a consequence of enhanced aerodynamic shear, the subsequent break-up process of entwined waves tends to become quasi-linear. Moreover, for the reported flow conditions, convective instability is the source of primary break-up. Keywords: liquid atomization, air-assist atomizer, liquid sheet 1. INTRODUCTION The investigation on liquid surface instability dates back to Kelvin [1], and according to the Kelvin Helmholtz theory, surface waves in water, under the influence of surface tension and gravity, are generated due to interfacial shear and static pressure fluctuation. Subsequent studies of Taylor [] and Rayleigh [3] gave emphasis on the perturbation growth and its break-up characteristics. Rayleigh [3] proved that, for low liquid velocities, the break-up process is primarily a result of capillary instability and tend to become highly non-linear. The associated break-up generates droplets of diameter larger than the jet diameter. By extending the Rayleigh theory for Newtonian liquids, Weber [4] identified the role of aerodynamic shear at the liquid-gas interface that affects the wave behavior. In particular the influence of shear force that enhances perturbation growth, which ultimately leads to a non-linear relation between break-up length and jet velocity. By and large, the above classical studies on liquid instabilities had contributed to improve our understanding of primary break-up that dominates the initial stages of the atomization process. Although these theories have been derived from observations in laminar flows, they do represent the basis for comprehending practical flow configurations. The influence of surrounding gas jet on the liquid break-up has been studied extensively in the last century (see, for example, [5-7]) due to the recognized potential of this flow configuration in practical applications. Twin-fluid atomizers have number of advantages over pressure atomizers, which include lower requirement of fuel injection pressure and generation of finer sprays. Among various injector configurations, planar liquid sheet studies have been given particular emphasis because they allow direct examination of the air-liquid interface without the added complexity of curvature. Also, configurations of larger sheet aspect ratio are especially suited to study transverse perturbations near the nozzle exit because edge effects are limited to a small fraction of the span-wise dimension. In this context, Hagerty and Shea [8] investigated instabilities of a liquid film, which is subject to initial perturbations by an oscillating atomizer. According to their study, only two types of instabilities can exist at any given frequency. Depending on the energy transfer from external perturbation, the sheet surface either oscillates in-phase generating sinusoidal waves, or out-of-phase producing dilatational waves. Furthermore, when the energy transfer from ambient-to-liquid is high, sinusoidal waves grow faster than the dilatational waves. The existence of two independent modes of instability, that is sinuous and varicose modes were also identified by Lin et al. [9] with linear stability analysis. Their results show that, the varicose mode is always convectively unstable unless the gas-to-liquid density ratio is zero, while the stability of sinuous mode depends on the Weber number. That is, if the surface tension influence is stronger, unstable sinuous mode occurs; otherwise, it will be stable. The spatial growth-rate of varicose mode is smaller than that of the sinuous mode for the same flow parameters. Dombrowski and co-workers [10,11] identified the break-up morphology of an air-blast liquid sheet; the results signify the role of aerodynamic shear on wave generation and its subsequent breakdown into ligaments and then to drops. Mansour and Chigier [1,13] also carried-out elaborate studies on liquid sheet instabilities, and concluded that the liquid sheet break-up is controlled by two mechanisms, namely: a mechanical mode induced by the atomizing air inside the nozzle, and a shear-driven aerodynamic mode. A note-worthy aspect about their investigation is the correlation of disintegration characteristics with the primary

2 variables of the flow domain. The investigations of Crapper et al. [14], and recently by Lozano and Barreras [15] identified the dynamic effect of co-flowing air in the structural topology of the liquid sheet. Their results revealed that detachment of the boundary layer at the air-liquid interface and the subsequent growth of vortices might cause enhanced sheet flapping and liquid breakup. Theoretical analysis of liquid jet instability, specifically Kelvin-Helmholtz wave growth, is usually based on the notion that an initial disturbance propagates along the jet and increases in amplitude to a point of break-up, which is classified as convective or global instability, Weihs [16]. Spatial instability studies aims to predict the wavelength associated with the highest growth-rate of perturbations in a given flow configuration. However, a further increase in the mean air velocity causes a sudden explosion of the liquid column, so that a spectrum of structures will be generated in the breakup region. According to Huerre [17], this eruption can be attributed to the transition from convective to absolute instability; this is also substantiated by the studies of Li and Kelly [18]. The latter analysis proved that, as the surrounding air velocity increases, the variation in temporal growth rate is higher than the spatial growth rate. In addition to aerodynamic instability, for very low gas phase Weber numbers, that is less than the ratio of gas-to-liquid density, Li and Tankin [19] reported the existence of viscosity-enhanced instability for liquid sheets. The possible scenarios of instabilities associated with twin fluid atomizers, provided by the aforementioned reviews, points to the fact that, the dominance of convective or absolute instability in the primary breakup processes of liquid sheets is still require better understanding. To accomplish this task, the present study focuses on the principal factors of initial perturbations, and its subsequent development into instabilities that ultimately leads to the breakup. The experimental investigation utilizes plane laminar liquid sheets under the influence of impinging airflows. The analysis consists of high-speed flow visualization techniques to characterize the interfacial waves for a range of air-to-liquid velocities.. EXPERIMENTAL SYSTEMS.1 Atomizer configuration Figure 1 shows a schematic layout of the liquid film generator. It comprises a two-dimensional unit to generate thin and flat liquid sheets of large aspect ratio, allowing Air t s = 0.4 mm Screens 14.4 mm Liquid Air 1 adequate control and direct examination of the air-liquid interface. To eliminate fluctuations in the flow, the test liquid water is supplied through a pressurized tank in closed circuit. The liquid emerges from a rectangular slit-orifice of aspect ratio 00:1, with an exit thickness of 0.4 mm, and is sandwiched between two impinging air-jets. The edge of the slit orifice is made sharp to prevent generation of flow disturbances and hence to ensure perfectly laminar conditions for the liquid sheet. In the present study, the liquid velocity varies from 0.8 m/s to 1.8 m/s; correspondingly the Reynolds numbers (Re l ) are 318 and 715, based on the film thickness. A calibrated rotameter continuously monitors the liquid flow-rate. The atomizing air enters through the rectangular channels on either side of the liquid slot, and impinges on the liquid sheet with a convergence angle of 30 o at the nozzle outlet. The initial velocity of both air jets was kept identical with values varying from 10 m/s to 40 m/s, and the air-channel thickness at the exit-port was 7 mm. Measurements of static pressure drop across the atomizer make it possible to control and determine precisely the air velocity at the outlet. The atomizer is mounted vertically on a traversing-table having an accuracy of 0.5 mm in all three coordinate directions.. Measurement techniques Qualitative and quantitative analysis of the flow-field were carried out by flow visualization techniques. The methodology applied in extracting the flow features will be given here, while the detailed description of the visualization set-up is available elsewhere (Santos [0]). To enhance the image contrast, a white background illumination was utilized because of the transparent nature of the object plane. A high-speed digital CCD camera acquired images at an exposure time of 100 µ s per frame. A temporal resolution of 100 µ s was chosen based on the expected maximum frequency of the flow-domain. In other words, the time-scale is predetermined from the relation that connects the liquid sheet breakup length to the respective flow velocity. The relative velocity between the liquid and the surrounding air at the atomizer outlet gives the appropriate velocity scale, while the break-up length information is extracted from front-view images of the liquid sheet (see Fig. ). Thus, the above time-resolution is sufficient to freeze the interfacial wave characteristics. The corresponding image acquisition rate was 000 frames per second, with a spatial resolution of pixels. Also, for qualitative analysis, a grabbing rate of 50 frames per second was used, which boosted the image resolution to pixels. Throughout this study, recording was done with a 5 mm focal length lens. The camera views the object plane normally, without distorting the image, in a way that allows the direct measurement of stability characteristics of the liquid sheet, such as the spatial growth of perturbations and breakup frequency. The respective scale factor for the analysis was obtained interactively using the scale function: based on the known distance between two points in the object plane, the corresponding conversion factor was calculated. To acquire reliable data, the frame sampling was carried out in such a way that the procedure identifies the growth of large amplitude waves that attains the critical state of breakup. Fig. 1. Schematic diagram of liquid sheet generator

3 Thus, with reasonably low levels of measurement error, the associated critical wave amplitude and breakup frequency could be extracted from the image sequence. For high velocity conditions, the analysis utilized several frames in order to achieve a statistically stable mean value. The distance between the liquid sheet centerline and the crest of the major disturbance, just before breakup, gives the corresponding critical wave amplitude (η c ). The breakup frequency (f b ) is identified as the frequency at which bursting of the high-amplitude waves of the liquid sheet occurs, and is calculated based on the acquisition frame-rate of the respective image sequence. Furthermore, the distance between the consecutive wave crest will provide the associated wavelength (λ) and its product with the respective break-up frequency will be a measure of the. wave propagation speed V w The maximum uncertainty for the averaged wave amplitude, and breakup frequency are estimated to be ± 9.5 % and ± 14 %, based on the repeatability of the measurements. A major source of uncertainty may be the enhanced activity at the liquid-air interface that broadens the breakup frequency bandwidth as well as restricting the precise detection of the breakup wave from the image sequence. 3. RESULTS AND DISCUSSION 3.1 Qualitative features A qualitative over-view of the break-up processes associated with air-assisted liquid sheets, for a range of air-to-liquid velocity ratios, can be extracted from the front-view images portrayed in Fig.. The image sequence illustrates the evolution of a liquid sheet, which is initially under the relative influence of inertia and surface tension forces, to the ultimate spray formation instigated by the enhanced momentum transfer from the surrounding gas. Depending on the initial liquid-to-gas momentum ratio, the following flow features are observed: a convergent liquid sheet bounded by thick rims that are drawn together by surface tension forces, interfacial wave generation, critical waves and its break-down, cellular-and-fiber types of liquid sheet rupturing and the eventual formation of sprays. Among these various modes of liquid sheet break-up, the present investigation confines to the high amplitude interfacial waves and their characteristics for air velocities in the range of 10-0 m/s. Figures 3, 4, and 5 depict the side-view images of liquid sheets for liquid velocities in the range of 1.0 to 1.8 m/s. Each figure distinguishes the interfacial wave structure for air velocities of 10 m/s and 0 m/s respectively. The images show distinct wave characteristics that strongly depend on the surrounding air velocity. That is, for low air velocity, waves grow rapidly, with high amplitude sinusoidal waves dominating at the liquid-air interface and reaching their critical state within one wavelength (Figs. 3a, 4a, and 5a). On the other hand, with increasing air velocity, the flow domain exhibits a system of interlaced waves (Figs. 3b, 4b, and 5b). Thus, for air-assist atomizers, the momentum-transfer from the surrounding gas to the liquid sheet controls the flow structure. To substantiate the above observation, the relative magnitude between inertial and aerodynamic forces is analyzed for the existing flow conditions. In this context, the present investigation formulated two classes of Weber number (We); the Weber number that is based on the ratio a) b) Fig. 3. Side-view images of liquid sheet disintegration [Q l = 7.4 g/sec; U l = 1.0 m/s]: (a) U a = 10 m/s; (b) U a = 0 m/s 40 m/s air velocity 0 m/s a) b) Fig. 4. Side-view images of liquid sheet disintegration [Ql = 9.1 g/sec; Ul = 1.3 m/s]: (a) Ua = 10 m/s; (b) Ua = 0 m/s 10 m/s.9 m/s 1.3 m/s liquid velocity Fig.. Front-view image sequence of liquid sheet break-up processes for a range of air and liquid velocities a) b) Fig. 5. Side-view images of liquid sheet disintegration [Ql = 1.5 g/sec; Ul = 1.8 m/s]: (a) Ua = 10 m/s; (b) Ua = 0 m/s

4 of resultant liquid inertia-to-surface tension forces (We l ), and the ratio of aerodynamic shear-to-surface tension forces ( ). Due to the inclined atomizing-air configuration, the normal air momentum can diminish the efflux momentum of the liquid sheet, while its tangential component supplements the aerodynamic shear at the interface. Consequently, We l accounts for the resultant inertia force given by (ρ l U l - ρ α (U a sinα) )xt s, and the aerodynamic force associated with becomes ρ α (U a cosα U l ) xt s. Figure 6 presents the non-dimensional parameter formed by the ratio between We l and as a function of liquid velocity U l for air velocities of 10 m/s and 0 m/s respectively. The results show that, with increasing velocity of the surrounding gas, the influence of liquid inertia substantially drops. In other words, shear force at the interface becomes significant compared to the liquid inertia, and result in the effective transfer of momentum to the liquid sheet. Wel / Wea air velocity=10 m/s air velocity=0 m/s U l (m/s) Fig. 6. Ratio of inertial to aerodynamic forces as a function of liquid velocity for different air velocities [U l: m/s ] The onset of perturbations and its augmentation to form high amplitude waves, Figs. 3(a) - 5(a), can be explained as follows: under the influence of the normal component of air-pressure, outer surface of the liquid may deprive its initial axial-momentum, and thus creates a velocity gradient through the inner-core. So, depending on the initial velocities of the liquid sheet and the surrounding air, a relative motion may instigate in the liquid mass that in-turn cause perturbation. In addition, larger perimeter associated with the rectangular slit-orifice creates enhanced shear on the liquid sheet boundary especially for low liquid velocities, as well as the sharp corners of the non-circular orifice may generate secondary flows. Combined effect of the above phenomenon makes it conducive for large-scale perturbations in the initial stages. The rapid advancement to the critical wave state, for low liquid flow-rates (Figs. 3(a) to 5(a)), can be due to the acceleration inherent in the flow field. To validate this, the wave propagation speed is analyzed for liquid velocities in the range of 1 to 1.8 m/s, Fig.7. The wave speed V w was calculated by taking the product of the average wavelength with the corresponding wave frequency extracted from the images. Figure 7 shows that, for air velocity of 10 m/s, waves are being accelerated, and the process becomes very pronounced for low liquid velocities, with wave speed attaining twice the efflux velocity of the liquid. Such an interfacial wave acceleration that attains twice the exit velocity of the liquid sheet was also observed by Lozano et al. [1]; even though, in their experiments the exit velocities of air and water were parallel. A probable cause of acceleration is the enhanced effect of gravity for low liquid flow-rates. The role of gravity in low velocity conditions can be further substantiated by calculating the Froude number at the source (Fr o ), based on the ratio of resultant liquid inertia force to gravity force at the injector outlet. In the formulation of Froude number, an equivalent diameter (d e ) is considered as the characteristic length scale instead of liquid sheet thickness (t s ) because gravitational force is a body force that acts across the entire cross-section of the outlet. Accordingly, for liquid velocities in the range of 1 to 1.8 m/s, the respective Froude number Fr o varies from 4.6 to 7.. Thus, gravitational influence may be significantly contributing the perturbation growth for low velocity conditions, without the existence of absolute instability. The latter observation is physically possible due to a feed back effect proposed by Yakubenko []. The feed back effect arises because of the change in the wave propagation speed downstream may force the perturbation upstream to distort the basic flow. This implies that, instability associated with the flow field is not activated by local phenomena. Conversely, as the air velocity increases to 0 m/s, while maintaining the liquid flow-rate variation analogous to the preceding case, the wave characteristics at the Vw (m/s) air velocity=10 m/s air velocity=0 m/s U l (m/s) Fig. 7. Wave propagation speed as a function of liquid velocity for different air velocities interface are radically altered (see Figs. 3(b) - 5(b)). The flow-domain exhibits a system of interlaced waves that become pronounced with decreasing liquid flow-rate. The observed structure can be attributed to waves originating from the rims of the liquid sheet, where the influence of atomizing air is low. In other words, except for the rim portion, the remaining part of the liquid surface will be influenced by the normal component of air momentum, which in turn reduces the efflux momentum of the liquid sheet. Therefore, rupturing of the liquid sheet may dominate, and thus the flow domain may deprive a well defined interface for the consistent development of waves. The existing rim region, which is under the predominant control of surface tension forces favors the generation of capillary waves. Subsequently, the emanated capillary waves from either side of the liquid sheet will interfere, and induce the entwine pattern, Figs. 3(b) to 5(b). Its consequent behavior may controlled by the aerodynamic shear component. 3. Stability Variables In order to understand the intrinsic properties of instability, the associated wave amplitude and break-up

5 frequency are analyzed. Figures 8 and 9 depict the non-dimensional critical wave amplitude η c as a function of Weber number based on the aerodynamic shear, for air velocities of 10 m/s and 0 m/s respectively. The critical wave amplitude is non-dimensionalized by the initial liquid sheet thickness t s. The liquid sheet thickness has been considered because the shorter side of the injector limits the initial perturbation amplitude or energy. Now, the half-value of t s is utilized for non-dimensionalization since wave amplitude is defined by the distance between the liquid sheet centerline to the crest of major disturbances. ηc / (0.5ts) Fig. 8. Critical wave amplitude as a function of Weber number [U a = 10 m/s]; the error-bar represent the standard deviation, with maximum deviation of ± 9 % except the rim region, interfacial wave development may hamper due to the enhanced normal component of air momentum that might rupture the liquid sheet. Therefore, interface becomes delicate, and hence the critical state of breakup may reach rapidly at low wave amplitudes itself. Under these conditions, the remnant capillary waves of the rim region are nurtured progressively by the aerodynamic shear force acting at the interface (see Fig. 9). To substantiate the aforementioned observations, the associated break-up frequency of the waves is analyzed. Figures 10 and 11 show the break-up frequency f b variation with Weber number, for the particular flow conditions as in the preceding case. f b is non-dimensionalized by the combination of initial liquid sheet thickness, s t, and liquid velocity, U l. The results demonstrate that, liquid sheet under low aerodynamic influence exhibit cyclic behavior in their break-up frequency, Fig. 10, in a way analogous to the behavior of critical wave amplitude of the corresponding case. The non-linear scenario justifies that, at low velocity of the surrounding air, convective instability may be the source of primary break-up. That is, liquid sheet breakup is controlled by upstream non-linear events, such as the effect of injector geometry on initial flow conditions and the gravitational influence in accelerating the interfacial waves. In contrast, with increasing air velocity, the break-up frequency tends to become quasi-linear due to the enhanced aerodynamic shear at the liquid-air interface, Fig. 11. This is analogous to the critical wave amplitude behavior (see Fig. 9). Moreover, under such conditions, the break-up process attains appreciable increase, with frequency values as high as 15 Hz. η c / (0.5t s) Fig. 9. Critical wave amplitude as a function of Weber number [U a = 0 m/s]; the error-bar represent the standard deviation, with maximum deviation of ± 9.5 % f bts / U l The results demonstrate that a non-linear state of wave development persist for low air velocity conditions (Fig. 8). As the surrounding air velocity increases, the magnitude of wave amplitude sharply diminishes as well exhibit a quasi-linear variation with aerodynamic shear force (Fig. 9). The observed trend may be due to the influence of atomizer geometry effecting the initial perturbations as discussed in the preceding section of qualitative features. That is, the genesis and sustenance of perturbations are controlled by non-linear mechanisms, such as the faster rate of shearing of the liquid sheet caused by its larger perimeter and also the effect of secondary flows that may occur at the sharp corners of the slit orifice. Besides, acceleration of waves caused by gravity under low liquid flow rate conditions (see Fig. 7) may also contribute the non-linear wave development. However, as the air velocity increases, the non-linearity reduces and the variation of amplitude becomes predictive with lower values, Fig. 9. In this case, Fig. 10. Break-up frequency as a function of Weber number [U a = 10 m/s]; the error-bar represent the standard deviation, with maximum deviation of ± 10.5 % f bts / U l Fig. 11. Break-up frequency as a function of Weber number [U a = 0 m/s]; the error-bar represent the standard deviation, with maximum deviation of ± 14 %

6 4. CONCLUSIONS The wave characteristics of a laminar liquid sheet exhibit structurally different patterns with respect to the dynamic state of its ambient. For low velocity of the surrounding air, the interface is dominated by sinusoidal waves, and the physical processes of their generation and subsequent development depends upon the outlet geometry, the inertial force and the gravitational force. The analysis has identified the prevailing role of gravity in augmenting the wave amplitude, and its subsequent non-linear mode of breakup. In addition, the Weber number classifications based on liquid inertia force and aerodynamic shear force shows that, with increasing air velocity, shear force at the liquid-air interface becomes equally significant with inertia force and give rise to tangled waves. As a consequence of enhanced aerodynamic influence, the break-up processes tends to become quasi-linear. The gravitational affect and the imprints of initial disturbances at the downstream imply that convective instability may be the source of primary break-up for low Weber number liquid sheets. In other words, it can be concluded that the low-frequency events at the two-phase boundary are sustained by the initial dynamic state of the high-density fluid, and the process may less dependent on absolute or local instability characterized by the surrounding air flow. 5. NOMENCLATURE A d e f b physical area of the liquid sheet at the injector outlet Equivalent diameter of the liquid sheet at the source = 4A π break-up frequency of the liquid sheet F ro = ( ρlu l ρau a sin α ) ρl gde Q l liquid flow-rate at the nozzle outlet t s initial liquid sheet thickness U a mean air velocity at the nozzle outlet U l liquid velocity at the nozzle outlet U n component of air velocity normal to the liquid sheet axis (U a sinα) U r relative velocity between air and liquid (U a cosα U l ) wave propagation speed V w wave propagation speed Weber number based on aerodynamic shear ρ a U r t s /σ We l Weber number based on liquid inertia [ρ l U l -ρ a U n ]xt s /σ ρ a density of air ρ l density of liquid σ surface tension of water at 0 o C η c critical wave amplitude α impingement angle of air with the liquid sheet 6. REFERENCES 1. L. Kelvin, Hydrokinetic Solutions and Observations, Philosophical Magazine, vol. 4, pp , G. I. Taylor, Generation of Ripples by Wind Blowing Over a Viscous Liquid, The Scientific Papers of G.I. Taylor, vol. 3, Cambridge University Press, Cambridge, L. Rayleigh, On the Instability of Jets, Proceedings of the London Mathematical Society, vol. 10, pp. 4 13, C. Weber, Disintegration of Liquid Jets, Zeitschrift Angew. Math. Mech., vol. 11, pp , N. Chigier, Energy Combustion and the Environment, McGraw-Hill, New York, R. D. Reitz and F. V. Bracco, Mechanisms of Breakup of Round Liquid Jets, N. Cheremisnoff, ed., The Encyclopedia of Fluid Mechanics, Gulf Publishing, Houston, Texas, pp , A. H. Lefebvre, Atomization and Sprays, Hemisphere Publishing Corp., New York, pp , W. Hagerty and J. F. Shea, A Study of Stability of Plane Fluid Sheets, Journal of Applied Mechanics, vol., pp , S. P. Lin, Z.W. Lian and B. J. Creighton, Absolute and Convective Instability of a Liquid Sheet, Journal of Fluid Mechanics, vol. 0, pp , N. Dombrowski and R.P. Fraser, A Photographic Investigation into the Disintegration of Liquid Sheets, Philosophical Transactions of the Royal Society of London, vol. 47, pp , N. Dombrowski and W.R. Johns, The Aerodynamic Instability and Disintegration of Viscous Liquid Sheets, Chemical Engineering Science, vol. 18, pp , A. Mansour and N. Chigier, Disintegration of Liquid Sheets, Physics of Fluids, vol., pp , A. Mansour and N. Chigier, Dynamic Behavior of Liquid Sheets, Physics of Fluids, vol. 3, pp , G. D. Crapper, N. Dombrowski, W.P. Jepson and G.A. Pyott, A Note on the Growth of Kelvin-Helmholtz Waves on Thin Liquid Sheets, Journal of Fluid Mechanics, vol. 57, pp , A. Lozano and F. Barreras, Experimental Study of the Gas Flow in an Air-Blasted Liquid Sheet, Experiments in Fluids, vol. 31, pp , D. Weihs, Stability of Thin, Radially Moving Liquid Sheets, Journal of Fluid Mechanics, vol. 87, pp , P. Huerre, Local and Global Instabilities in Spatially Developing Flows, Annual Review of Fluid Mechanics, vol., pp , H. S. Li and R. E. Kelly, The Instability of a Liquid Jet in a Compressible Air Stream, Physics of Fluids A, vol. 4, pp , X. Li and R.S. Tankin, On the Temporal Instability of a -D Viscous Liquid Sheet, Journal of Fluid Mechanics, vol. 6, pp , D. Santos, Liquid Film Disintegration, MSc thesis (In Portuguese), Technical University of Lisbon, Lisbon, A. Lozano, F. Barreras, G. Hauke and C. Dopazo, Longitudinal Instabilities in an Air-Blasted Liquid Sheet, Journal of Fluid Mechanics, vol. 437, pp , P. A. Yakubenko, Global Capillary Instability of an Inclined Jet, Journal of Fluid Mechanics, vol. 346, pp , 1997.