AIAA 04 Reno

AIAA -2004-1159 ON THE DYNAMICS OF HEAD-ON DROPLET COLLISION: EXPERIMENT AND SIMULATION K. L. Pan* and C. K. La w† Princeton University, Princeton, New Jersey 08544 Abstract The dynamics of head-on collision between two identical droplets was experimentally and computationally investigated, with particular emphasis on the transitions from merging to bouncing to merging again as the collision Weber number was increased. Experimentally the stroboscopically illuminated microphotographic images of two colliding droplet streams, generated through the ink-jet printing technique, were acquired with adequate temporal resolution of the collision event such that the instant at which the droplets merged was identified. Using this empirical information as an input, the simulated collision images agree well with the experimental observations, and allow investigation of the collision flow field including the energy budget. Detailed study of the dynamics of the colliding interfacial surfaces revealed the fundamental differences between the soft and hard collisions that led to merging, and suggested that effects of rarefied flow and gas compressibility could be responsible for the interfaces to approach to a sufficiently close distance at which van der Waals force would become effective and thereby effect merging through rupturing of the surfaces.

1. Introduction The dynamics of binary droplet collision plays a key role in various technologies such as spray combustion in liquid-fueled combustors, ink-jet printing, rain drop formation, insecticide spraying, nuclear fusion, etc. Earlier studies were primarily concerned with water in one atmosphere air1,14 because of the prevalence of water and its relevance to meteorological implications. Recent investigations involving hydrocarbons,15,23 however, revealed a richer outcome. Specifically, five distinctive regimes (Fig. 1a) of the collision outcome between two identical droplets were identified based on the collision Weber number (We) and the impact parameter (B), which are respectively defined as 2ρ l Vrel 2 R/σ and χ/(2R), with R being the droplet radius, Vrel the relative velocity, χ the projection of the separation distance between the drop centers in the direction normal to Vrel, and ρ l and σ the density and surface tension of the liquid respectively. Therefore We represents the ratio between the initial kinetic energy and surface energy of the droplet, while B = 0 and 1 respectively indicate head-on and grazing collisions. These five regimes are categorized according to: (I) permanent coalescence after minor droplet deformation, (II) bouncing, (III) permanent coalescence * PhD candidate, Department of Mechanical and Aerospace Engineering. † Robert H. Goddard Professor, Department of Mechanical and Aerospace Engineering, Fellow AIAA. Copyright © 2004 by the authors. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

after substantial droplet deformation, (IV) coalescence followed by separation for near head-on collisions, and (V) coalescence followed by separation for sufficiently off-center collisions. Daughter droplets are also produced for collisions in regimes IV and V. An interesting observation on the collision outcome is the sensitive dependence of the boundaries between bouncing and coalescence on the density of the gaseous medium. Specifically, increasing density by increasing either the pressure or molecular weight of the ambient gas promotes bouncing and widens regime II, while the opposite holds for reduced density. Consequently, by progressively reducing the ambient pressure and hence density, regime II first vanishes for B = 0 and eventually for all values of B. As such, regime II is eliminated for hydrocarbon droplets by steadily reducing the ambient pressure from 1 atmosphere. Similarly, while regime II does not exist for the head-on collision of water droplets at 1 atmosphere, it emerges as the system pressure is increased to 2.7 atmospheres. Mechanistically, the propensity for bouncing or merging is a consequence of the readiness with which the gaseous mass in the interdroplet gap can be squeezed out of the gap by the colliding interfaces such that they can make contact and hence merge. Because of the significant droplet deformation, the problem is more amenable for computational instead of analytical studies. Furthermore, because of the richness of the phenomena involved, which include the need to track the displacement of a deformable, sharp interface, computational simulation of the experimental collision images can serve to validate hydro-codes. Indeed, since

1 American Institute of Aeronautics and Astronautics

the development of the shape and phase of droplet deformation is driven through the interplay between the surface condition and the flow field, agreement in the comparison allows the extraction with confidence the flow structure such as the distributions of pressure, velocity, and energy budget. Notable efforts of this nature include those by Unverdi & Tryggvason 26 and Nobari, Jan & Tryggvason20 , who used a front tracking method to trace the interface movement in a Lagrangian manner while treated the multi-phase flow field as a unified domain, and solved the governing equations on a grid of Eulerian coordinate. The detailed evolution of the droplet surface geometry was predicted, with substantial insight into the velocity, pressure, and energy budget fields. Similar study was also conducted by Chin 7 with a level set method. A major difficulty noted in the above studies was the selection of the instant at which the colliding interfaces rupture, causing droplet merging. Since the numerical simulation was based on continuum mechanics, while the final stage of the interfacial dynamics leading to merging must necessarily involve rarefied flows as well as molecular forces, the lack of such basic information renders it fundamentally impossible to simulate the rupture of the interfaces from first principles. This difficulty was circumvented by artificially, and quite arbitrarily, removing the interfaces at a certain instant when they are sufficiently close to each other. It was however found7,2 0 that the simulated dynamics depended very sensitively on the instant at which rupture was implemented. There are several aspects of emphasis for the present study. First, in the course of our experimental investigation on droplet collision, we noticed that the instant of coalescence could be identified through a distinct change in the contour of the imaged interface. Furthermore, we also determined that such an instant could be time resolved with sufficient accuracy that, by using it as input to the computational simulation, the evolution of the experimental collision images subsequent to coalescence could be satisfactorily simulated. Thus with this experimentally supplied information, complete computational simulation of the droplet collision dynamics based on continuum description can be conducted. Investigations related to the above effort will therefore be first presented. Following this we shall study the various issues related to evolution of the inter-droplet geometry and dynamics, which provides insight into the transition between bouncing and coalescing regimes. Finally, the possible role of the van der Waals force in effecting merging is investigated by introducing an augmented intermolecular attractive force. Results suggest that compressibility and rarefied gas effects are needed for a complete description of the merging dynamics.

(a)

(b)

FIG. 1. (a) Schematic of different droplet collision regimes as function of Weber number (We) and impact number (B). (b) The computational setup of the axisymmetric domain, which is bounded by full-slip walls and resolved by a rectangular grid.

2. Formulation and Numerical Specifications The front tracking method adopted herein was developed by Unverdi25 and discussed in Unverdi & Tryggvason.26 The actual code is an axisymmetric version of the method, described in Nobari et al.20 The physical problem and the computational domain are sketched in Fig. 1b. The droplets are initially placed at each end of the z-axis with distance z0 which is prescribed by the experiment. The Navier-Stokes equations are solved for both gas and liquid phases in a unified domain: ∂ ( ρV) + ∇ ⋅ (ρVV ) = −∇p + g ∂t (1) 1 8 +∇⋅ (∇V + ∇ V T ) − We ∫ κnδ (r − rf ) da Re ∆s where V, ρ, and p are the velocity, density, and pressure respectively normalized by the initial droplet velocity V0 , liquid density ρ l , and the dynamic pressure ρ l V02 /2, and t is time normalized by R/V0 . In Eq. (1) surface tension is added as a delta function integrated locally


over the immiscible interface within unit volume to render a singular force exerted by the droplet, which is calculated in a global way over the entire droplet surface. Furthermore, g is a body force that is turned off at a prescribed droplet separation before collision, and is merely used to provide the droplets with an initial velocity toward each other with the solved velocity field of the surrounding gas. The notations for volume and surface are for three dimensions but we have used their two-dimensional versions, i.e. surface and line. Here κ is twice the mean curvature, n the outwardly directed unit normal vector at the droplet surface, and r the space vector with the subscript f designating the interface. The conservation equations of momentum and mass are calculated on a fixed, staggered grid with a second-order central difference scheme for spatial variables and explicit second-order time integration. A symmetric boundary is imposed on the central collision plane and only half of the domain is solved. The interface is represented by separate computational points which are moved by the velocity interpolated from the neighboring grids. These markers are connected to form a front that is used to keep the density and viscosity stratifications sharp and to calculate the surface tension, which is smoothed over the droplet surface in the numerical scheme by an immersed boundary method.22 The method and the code used have been tested in several ways,20 for example by extensive grid refinement and comparison with published works (e.g. rising bubble, droplet oscillation). Figure 2 shows the droplet interface contour from a test of grid refinement for tetradecane droplet collision at We = 7.33 in 10 atmosphere nitrogen. The grid lines are unevenly spaced, with a minimum spacing of 0.001, 0.002, 0.004, and 0.008, respectively, at the center-plane. The distance between gridlines increases smoothly as it moves away from the symmetric plane. It is seen that the outer contour of the colliding droplets, away from the interfacial region, is very close for the tested resolution (Fig. 2a), which shows an insensitivity of the global dynamics to the resolution in the gap region.20 One of our goals is to study the behavior of the gas gap between the droplets. Due to the diverse scale of interest, it is critical to test the numerical convergence in the narrow zone, which is shown in Fig. 2b. It is seen that although the film is very narrow such that it consists of only a few grids, the immersed-boundary method ensures the preservation of surface sources in momentum conservation in the presence of the symmetric boundary at the collision plane. Furthermore, since the jump of fluid property is mollified throughout a thin layer which overlaps with the physical gas gap, it is the interpolated interface velocity, instead of the pressure and velocity fields, that

carries quantitative significance. The resulting accuracy in the interfacial region through the use of the immersion technique is O(h) as long as the smoothing function satisfies conservation and the velocity is continuous across the interface.4,21 Consequently convergence requires more stringent grid refinement in the interfacial region, even though the accuracy for regions away from the interface is O(h 2 ). 1.5 0.001 0.002 0.004 0.008

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FIG. 2. Droplet shape and gas gap configuration are computed with various minimum grid size near the center-plane for (a) entire droplet, (b) inter-droplet spacing. To examine the flow in the gap in more detail, we plot the velocity vector at every grid point and the gap shape for the finest resolution in Fig. 3. The results demonstrate that the shape is described well even on the coarse grid in that the velocity in the gap is essentially identical to that in the droplet over most of the film, implying that increasing resolution will not capture any additional feature. Only near the rim of the drop, where the drained gas encounters the recirculation zone outside, is there a significant difference. We further note that this plot corresponds to the state of minimum film thickness. During most of the time when the fluid is being drained from the film, the film is thicker and the film velocity is resolved by a larger number of grid


points. Since the film shape is well represented by all grids and the difference between the two finer resolutions is small, the following results focused on the evolution of the gas gap were computed on a grid of minimum size of 0.002. -3

15

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4. Global Collision Dynamics In this section a comparison between the computational and experimental sequences for tetradecane droplet collisions are presented, emphasizing on the phenomena of coalescence and bouncing around the transitional boundaries, i.e. from regime I to II (We = Wec,1 ) and from II to III (We = Wec,2 ).

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FIG. 3. Velocity vectors across the gas gap. The gas velocity inside the most part of the gap is equal to nearby liquid velocity across the interface.

3. Experimental Specifications The exp erimental apparatus and procedure were those of Jiang et al.15 and Qian and Law.23 Here two identical nozzle -generated streams of droplets of uniform size were made to impinge onto each other. Time -resolved images were taken through stroboscopy synchronized with the droplet generation circuit. A digital imaging system then accurately time-resolved the collision event, recorded the droplet image, and processed the data. This yields the fine temporal resolution of the transitional behavior between different regimes, and is critical in determining the instant of rupture of the inter-droplet gas film. The entire apparatus was housed in a chamber whose pressure could be varied between 0.1 to 20 atmospheres. Only head-on collision events were studied. We present in the following results and discussion on the various aspects of the investigation.

4.1 Comparison between experimental and simulation images Figures 4 to 6 compare several sequences of the droplet deformation obtained experimentally and computationally. In the following discussion, t is time normalized by the droplet oscillation time T = 2π ρl R3 / 8σ based on inviscid droplet oscillation with small amplitude.17 The real time is indicated in the figures as well. It is seen that (Figs. 4a and 4b), across the first transitional boundary (We = Wec,1 ), the collision outcome changes from bouncing to coalescing with decreasing We. Figure 4a shows that, for the bouncing case, the simulation agrees well with the experimental data, in terms of matching the deformed configurations and collision phases. For the coalescing case, however, since the impinging interfaces of the droplets need to be broken manually, the ensuing evolution depends sensitively on the instant at which such a rupture is applied.7,20 In conducting the experiments, it was found that, by gradually adjusting the synchronization time between droplet generation and strobe lighting, there existed a narrow range in time over which the cuspy contour between the two droplets abruptly smoothened, as shown from the instants of 1.205 and 1.385 in Fig. 4b. The physical duration was about 0.15 ms. Further refinement then narrowed the instant to 1.325. It is reasonable to identify this instant as when the coalescence takes place when the two interfaces suddenly “wets’ each other. Using this experimentally determined instant as that of surface rupturing, at which the impinging interfaces separating the liquid and gas phases are removed, the computed sequences of collision images of Fig. 4b show surprisingly close agreement with the experimental sequence. It is also of particular interest to note that the state of merging takes place subsequent to that of maximum deformation. To demonstrate the sensitivity of the computed dynamics on the timing of the rupture, Figs. 5a and 5b show the evolutions subsequent to an earlier (t = 1.271) and a later rupture (t = 1.373) of the interfaces. The comparison shows substantial differences between the computational and experimental phases in that earlier and later breaking respectively generates leading and lagging phases, as shown for t = 1.385.


t=0

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1.626 1.35 ms

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1.807 1.5 ms

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FIG. 4. Collisional sequences at first transitional boundary (We = 2.3) determined by computations (left) and experiments (right); (a) bouncing, (b) coalescence with merging at time: 1.325 (1.1 ms).

(b)

FIG. 5. Collisional sequences of coalescence at first transitional boundary (We = 2.3) determined by computations (left) and experiments (right). The time sequence shown is the same as that of Fig. 4. Merging time: (a) 1.271 (1.055 ms), (b) 1.373 (1.14 ms). It is seen that earlier breakup of the colliding interfaces leads the sequential phase as compared to Fig. 4b while later breakup generates a lag.


t = 0.370 0.3 ms

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t = 0.061 0.05 ms

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When We is increased to the second transition boundary (We = Wec,2 ), the collision outcome changes from bouncing to coalescence, as shown in Figs. 6a and 6b, respectively. In Fig. 6b, the experimental time for rupture, as defined by the smoothing of the interface cusp, was found to be 0.4457 (0.368 ms). By using this time, good comparison between the computed and experimental result is again observed. It is also noted that the merging occurs as the deformed drop is flattened to a disc shape while the incoming mass at the center is still heading forward, as shown at t = 0.443 and 0.448 in Fig. 6b. Furthermore, although the forwarding motion due to the inertia of the impinging flow at the rear portion of the droplet cannot be distinguished in the experimental images as obscured by the annular ring at t = 0.545 and 0.606, the indented shape of the droplet is clearly captured by the computation. This behavior is distinct from that of regime I in which the breakup of the interfaces occurs subsequent to the attainment of the maximum deformation and the droplets have started to retract. This contrast will be demonstrated further by the evolutions of energy components and the inter-drop gas gap. 4.2 Evolution of the energy budget The total energy (TE) of the flow field consists of surface energy (SE), kinetic energy (KE), and the cumulative viscous dissipation (DE). Since the energy conservation equation is not solved in the simulation, calculations of these energy components are resorted to theoretical evaluations through the velocity and density fields as well as the surface geometry. For example, the viscous dissipation rate (VDR), given by27 2   ∂u 2  u 2  ∂w 2   ∂u ∂w  Φ = µ 2  + 2  + 2   + µ  +  r   ∂z    ∂ z ∂r  , (2)   ∂r  2

0.218 0.18 ms

0.606 0.5 ms

1.090 0.9 ms

0.363 0.3 ms

0.666 0.55 ms

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FIG. 6. Collisional sequences around second transitional boundary (Wec,2 = 12.3) determined by computations (left) and exp eriments (right); (a) bouncing (We = 9.33), (b) coalescence (We = 13.63).

2  1 ∂ ( ru ) ∂w  µ +  3  r ∂r ∂z  consists of three sources, respectively representing effects due to the normal strain, the shear strain, and volumetric dilatation. Furthermore, at the state of merging, all surfaces interior to the circumference of the cusp are removed, with the associated surface energy assumed to be dissipated thermally. The evolutions of the various energies corresponding to the bouncing and coalescent cases at Wec,1 are shown in Figs. 7a and 7b, respectively. For the bouncing case of Fig. 7a, it is seen that even though energy conservation is not imposed in the numerical scheme, the total quantity, obtained by summing the accumulated dissipation loss with the surface and kinetic energies, is conserved. This indicates that the velocity field and the surface geometry are resolved well. The state of the largest deformation, at t = 0.844, −


1

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is characterized by the largest surface energy and smallest kinetic energy. Furthermore, the viscous dissipation rate is initiated at an early phase, around the instant of contact and before significant deformation. For the coalescing case of Fig. 7b, since a certain amount of the interfacial surface area is removed at the state of merging, surface energy and hence the total energy drop off after the numerical rupture. A spike of the viscous dissipation rate is generated, which in turn contributes to a rapid rise of the total dissipation energy. This results from the perturbation of the local flow field due to the sudden connection of droplet surfaces. Since reduction of the total energy is about the same as that of surface energy, and the amount does not change for states away from the jump, the manual merging would not have distorted the energy conservation except in a short period around the discontinuity, whose influence on the ensuing evolution is trivia l because of the short action time.

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FIG. 7. Energy components versus time at Wec,1 ; (a) bouncing, (b) coalescence. Each is normalized by the initial energy of a single droplet.

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FIG. 8. Energy components versus time around Wec,2 ; (a) bouncing (We = 9.33), (b) coalescence (We = 13.63). Each is normalized by the initial energy of a single droplet. For the bouncing case with We larger than that of Fig. 7a but smaller than Wec,2 , Fig. 8a shows that the temporal variations of the various energy components are similar to those of Fig. 7a. However, since the collision is more energetic, the relative amount of kinetic energy is increased while the surface energy is decreased. Furthermore, the substantial distortion of the droplet results in relatively higher dissipation. The kinetic energy, however, is greatly reduced after collision, as compared to the low We case, because of the substantial viscous energy loss. It is also noted that, since the surrounding flow is substantially disturbed as the droplet is accelerated initially by gravity to the designated velocity, the total energy is larger than that originally carried by the droplet and so the normalized value is slightly larger than 1. For the coalescing case at higher We, Fig. 8b shows the reduction of the surface energy at the instant of coalescence, leading to a strong dissipation rate, in the same manner as that of Fig. 7b for regime I. The coalescence, however, occurs at an earlier phase of collision as compared to the soft coalescence case of


Fig. 7b, ostensibly due to the stronger impact inertia for the present situation. In summary, comparing the coalescent cases in regimes I and III (Figs. 7b and 8b), it is found that in terms of the fractional distributions of the various forms of energy, the soft coalescent case has smaller kinetic energy and larger surface energy. It also suffers less distortion and hence less dissipative loss. After merging, the hard coalescence case suffers more loss in mechanical energy which is composed of surface and kinetic energies, and hence exhibits less oscillation. It is seen (Figs. 7b and 8b) that the dissipation is almost in phase with the evolution of kinetic energy, which attains a maximum as the surface energy is minimized with the spherical shape restored.

dissipation is negligible in the gas phase for collisions with large deformation. Analyzing each component of motion, it is found that the shear strain contributes predominantly to the dissipation function, as shown in Fig. 12a. The distribution of the velocity vectors indicates that the flow is respectively turned sharply outward (Fig. 9b) and inward (Fig. 11b) in the radial direction, as the impinging interfaces squeeze the intervening gas and as they rebound such that the surrounding gas is entrained into the inner space between the receding interfaces. Therefore there exists a low pressure sink in the gas film during the receding period, as shown in Fig. 11c, and a high pressure source during the approaching period (Fig. 9c). These results demonstrate the significance of the gas inertia in the film for the soft collision, as will be further discussed in the analysis of the transition criteria.

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FIG. 9. Distributions of flow properties at t = 0.361 for the bouncing case at Wec,1 ; (a) contours of viscous dissipation rate, (b) velocity vectors, and (c) pressure mesh surface with delineated droplet shape and contours projected at the bottom. 4.3 Spatia l distributions of pressure, velocity, and viscous dissipation The profiles of flow properties for the bouncing case at Wec,1 (regime II) are plotted in Figs. 9 to 12. It is seen that there is strong dissipation in the gas film at early and late phases when the drop is about to deform (t = 0.361, Fig. 9a) and recover its spherical shape (t = 1.505, Fig. 11a). Particularly noteworthy is the result that these peaks are situated in the gas medium, in contrast to the previous suggestion15,23 that viscous

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FIG. 10. Distributions of flow properties at t = 0.722 for the bouncing case at Wec,1 ; (a) contours of viscous dissipation rate, (b) velocity vectors, and (c) pressure mesh surface with delineated droplet shape and contours projected at the bottom. In addition to the gas phase, dissipation also occurs within the droplet, as manifested by the contours at the rear of the drop when it is deformed and expands laterally (t = 0.722, Fig. 10a). The dissipation is caused by the compression due to the forward motion of the rear part of the droplet, as indicated by the substantial contribution from the normal strain rate of the axial component (F zz), as shown in Fig. 12b. Another source


of dissipation is generated around the waist of the deformed droplet, which is due to the recirculation of the surrounding gas, as illustrated by the velocity vectors in Fig. 10b. These sources of dissipation, however, are significant only for a short time (e.g. t = 0.722 to 1.083), which is to be contrasted with the main accumulation period of total dissipation, from t = 0.361 to 1.384. Consequently, viscous loss is mostly through the gas medium. This is physically realistic because collision with small We leads to small and slow deformation, resulting in weak strain rates in the liquid while the influence on the intervening gas lasts throughout the soft collision with long collision time. In the lack of substantial energy dissipation within the droplet, the dissipative loss is small relative to the total energy (Fig. 7a).

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FIG. 12. Components of viscous dissipation rate for the bouncing case at Wec,1 ; (a) t = 0.361, (b) t = 0.722. Each subfigure includes Frr (top panel) that is caused by normal strain in radial direction, F zz (middle panel) that is caused by normal strain in axial direction, and Frz (bottom panel) that is caused by shear strain rate. It is seen that the dissipation rate at the opening of the gas film is mainly caused by shear strain while that in the liquid droplet is significantly contributed by the normal stress in z direction due to compression.

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FIG. 11. Distributions of flow properties at t = 1.505 for the bouncing case at Wec,1 ; (a) contours of viscous dissipation rate, (b) velocity vectors, and (c) pressure mesh surface with delineated droplet shape and contours projected at the bottom.

If the approaching interface is ruptured as We is decreased slightly from Wec,1 , the development before the onset of merging is similar to that of bouncing while the subsequent dynamics is quite different, as shown in Figs. 13 to 15. It is seen that before coalescence, the surrounding gas is already entrained into the intervening film (Fig. 13b) and dissipation is generated at its opening to the ambience (Fig. 13a). Immediately after coalescence, the connected interfaces with negative curvature contract rapidly due to surface tension and a strong recirculation is produced as depicted by the velocity vectors in Fig. 14b. This then leads to a region of high dissipation rate (Fig. 14a) around the connected cusp, and consequently a substantial increase in the global dissipation rate, as shown in Fig. 7b. This zone expands outward along the


droplet surface, being pulled by the surface tension exerting at the expanding rim of merging (Fig. 15). The subsequent evolution is governed by the velocity and pressure fields, resulting in prolonged and contracted oscillations whose periodicity is of the order of the natural oscillation time as shown in Fig. 7b and to be discussed shortly. If the approaching interface is ruptured as We is decreased slightly from Wec,1 , the development before the onset of merging is similar to that of bouncing while the subsequent dynamics is quite different, as shown in Figs. 13 to 15. It is seen that before coalescence, the surrounding gas is already entrained into the intervening film (Fig. 13b) and dissipation is generated at its opening to the ambience (Fig. 13a). Immediately after the coalescence, the connected interfaces with negative curvature contract rapidly due to surface tension and a strong recirculation is produced as depicted by the velocity vectors in Fig. 14b. This then leads to a region of high dissipation rate (Fig. 14a) around the connected cusp, and consequently a substantial increase in the global dissipation rate, as shown in Fig. 7b. This zone expands outward along the droplet surface, being pulled by the surface tension exerting at the expanding rim of merging (Fig. 15). The subsequent evolution is governed by the velocity and pressure fields, resulting in prolonged and contracted oscillations whose periodicity is of the order of the natural oscillation time as shown in Fig. 7b and to be discussed shortly.

FIG. 13. Distributions of flow properties at t = 1.265 for the coalescent case at Wec,1 ; (a) contours of viscous dissipation rate, (b) velocity vectors, and (c) pressure mesh surface with delineated droplet shape and contours projected at the bottom.

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FIG. 14. Distributions of flow properties at t = 1.385 for the coalescent case at Wec,1 ; (a) contours of viscous dissipation rate, (b) velocity vectors, and (c) pressure mesh surface with delineated droplet shape and contours projected at the bottom.

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For bouncing at higher We close to Wec,2 , the droplet spreads out into a disc shape and the indentation is larger due to the larger inertia of the impacting liquid. The dissipation loss is caused not only by the squeezed gas at the approaching and receding moments as demonstrated for the soft collision with low We, but also by the strained liquid as the droplet is deformed (Fig. 16). We note that while dissipation in the liquid in soft bouncing is much smaller than that in the gas due to the smaller droplet deformation, it becomes a major source in hard collisions. This can be demonstrated by the fact that substantial dissipation is present in the liquid for a long time while in the gas it exists only for a short period in a narrow region where the interstitial gap is largely squeezed. It therefore does not constitute much of the total dissipation energy, as shown in Fig. 8a. Consequently the dissipation loss is substantial and dominated by that through the liquid phase, in contrast


to the small dis sipation contributed mostly by the gas phase in the soft collision. The region of high dissipation rate starts from the waist when the impinging liquid flow is turned outward in the radial direction (t = 0.580). It is then swept along the droplet surface toward the indentation at the rear portion where the incoming flow motion is halted (when t is somewhat smaller than 1.049 in the sequence shown). The curvature there becomes negative due to the balance between surface tension and local pressure, as interpreted by the negative pressure shown in Fig. 17. At the receding moment (t = 1.585), significant dissipation is induced by the entrained gas associated with greater shear strain and lower pressure.

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E r

z

(a)

r

r

z z (a)

(b)

FIG. 16. Distributions of viscous dissipation rate for the bouncing case around Wec,2 (We = 9.33); (a) mesh surface with delineated droplet shape, (b) filled contours. It is noted that the range has been magnified at t = 0.580, 1.357, and 1.585.

(b)

p

r

t = 0.580

1.116

0.740

1.234

0.864

1.357

1.049

1.585

z (c)

FIG. 15. Distributions of flow properties at t = 1.506 for the coalescent case at Wec,1 ; (a) contours of viscous dissipation rate, (b) velocity vectors, and (c) pressure mesh surface with delineated droplet shape and contours projected at the bottom. p

r

z

FIG. 17. Pressure distributions for the bouncing case around Wec,2 (We = 9.33). The plotted is pressure mesh surface with delineated droplet shape and contours projected at the bottom.


The flow structure for the coalescent collision with We slightly larger than Wec,2 is shown in Figs. 18 and 19. Analogous to that of regime I, immediately after rupture of the interfaces, the ensuing mixing of the flow induces a swirl due to the outward motion of the connected surface segment, which leads to a strong strain rate (t = 0.485) and a period of substantial viscous loss, and thus generates the first spike in Fig. 8b. The second spike of the dissipation rate occurs at t ˜ 0.635. It is caused by the oscillating dimple of the deformed droplet, which lasts about one cycle as referred to the sequential images in Fig. 6b, from t = 0.448 when the liquid flow is impinging until t = 0.703 when the flow is returned and a bulge is formed. The contracted and protruding phases of the oscillating surface correspond to two local maxima of the surface energy as shown in the evolution of the energy budget (Fig. 8b, t ˜ 0.585 and 0.715, where a local minimum between them represents the neutral position of the oscillation). This is a consequence of the balance between the impinging inertia and surface tension, which then induces high strain rate and hence substantial dissipation. Specifically, the local pressure minimum at the surface indentation (Fig. 19, t = 0.606) indicates a sink caused by the substantial deformation and negative curvature, which then entrains the liquid at the sides. Consequently a growing bulge is generated at the center, similar to that of the bouncing case. Because of merging, however, this bulge is now pulled back, as shown at t = 0.703, and moves outward in the radial direction due to expansion of the merged interface. It ends in an oscillatory motion that is analogous to the natural oscillation of a spherical droplet. 4.4 Collision time It has been observed numerically 9,20 that the droplet collision time is close to the natural oscillation time T of the droplet. A direct comparison between experimental and computational results, however, has not been conducted. We have therefore performed a series of experiments and the corresponding numerical simulations for the collision time of water and tetradecane droplets with different Weber numbers in 14.6 atmosphere argon. The experimental collision time, τ, is measured as the time period during which the droplets remain visually in contact. The numerical collision time is defined as the period between the state when the slope of the gap clearance at center, h center, starts to level off and the state at which it starts to increase sharply. Figure 20a shows that these two states are indeed sharply defined in terms of h center. Figures 21a and 21b show that the collision times determined experimentally and numerically indeed agree well.

t = 0.363

0.606

0.443

0.703

0.485

0.818

z 0.545

1.090

E r

(a)

r z

(b)

FIG. 18. Distributions of viscous dissipation rate for the coalescent case around Wec,2 (We = 13.63); (a) mesh surface with delineated droplet shape, (b) filled contours.

t = 0.363

0.606

0.443

0.703

0.485

0.818

0.545

1.090

p

r

z

FIG. 19. Pressure distributions for the coalescent case around Wec,2 (We = 13.63). The plotted is pressure mesh surface with delineated droplet shape and contours projected at the bottom.


surprising that both τ1 and τ2 are close and each is about half of the oscillation time, as shown in Figs. 22a and 22b respectively for water and tetradecane in 14.6 atmosphere argon. While the deformation rate is set by the droplet impact velocity, the time needed to convert the kinetic energy to surface energy and then from surface energy back to kinetic energy is close to the oscillation time, i.e. τm,c ˜ 1, modified slightly by Weber number effects. If the collision was completely loss free, the drop would recover its initial velocity when it rebounds and we would expect the second half of the collision to be more or less equal to the initial impact and τ1 = τ2 . Due to losses, this is generally not the case as seen in Fig. 22. 1.40 τ (experimental) τ (numerical) τ m,c (experimental) τm,c (numerical)

1.30

Collision Time

Figure 20b plots h m,c, which is the calculated half distance between the mass centers of the droplets (for We = 6.96), as a function of time when the droplets collide and bounce away. It is seen that the variation of h m,c can be divided into two stages corresponding to two time intervals, τ1 and τ2 . The first stage is between the state when the droplet surface starts to deform at h m,c = 1 until the state when it has achieved maximum deformation and h m,c is minimum. The second stage is between the time of maximum deformation and the time when the droplets separate again at h m,c = 1. The sum of τ1 and τ2 , τm,c = τ1 + τ2 , is another way to describe the collision time as shown in Fig. 21. Here τ (the time the drops are visually in contact) is larger than τm,c (the time between when the drop centers are one diameter apart) because the droplets are deformed into an elongated shape in the direction of motion when they rebound.

1.20

1.10

1.00

Water

0.90 0

2

4

6

8

We

(a)

1.40 τ (experimental) τ (numerical) τ m,c (experimental) τ m,c (numerical)

1.30

Collision Time

(a) 1.20

1.10 1.00 0.90

Tetradecane 0.80 0

2

4

6

We

(b)

FIG. 20. (a) Calculated inter-center distance, h center, as a function of time, which defines the collision time t. (b) The variation of mass center of droplet, h m,c, with time during tetradecane droplet collision with We = 6.96 in 14.6 atmosphere argon.

8

10

(b)

FIG. 21. Collision times, t and τm,c, as a function of in 14.6 atmosphere argon (a) for water, (b) tetradecane. Here t is larger than τm,c because droplets are deformed into elongated shape in direction of motion.

The collision process is very similar to a droplet oscillation in that a drop is deformed and then recovers back to achieve a spherical shape. Therefore it is not 13 American Institute of Aeronautics and Astronautics

We for the the

1.40

Water

τ 1, τ 2 and τm,c

1.20

τm,c 1.00

0.80

τ2

0.60

τ1 0.40 0

1

2

3

5

4

6

7

(a)

We 1.40

Tetradecane

τ 1, τ 2 and τm,c

1.20

τm,c

centers. Such “dimples” are frequently observed during the drainage of axisymmetric films .19 As the droplet surface approaches the maximum deformation with increasing flattening, the indentation in the center increases while the size of the minimum clearance ring expands. After the droplets have reached their maximum deformation at t = 0.60, most of the initial kinetic energy of the droplet is transformed into surface tension energy, as identified by the evolution of energy components shown earlier. At this stage, the droplet starts to contract due to surface tension, and rebound by the contracting flow inside. This is depicted by ht e dashed lines, which show the gap evolution for t > 0.60 after the droplets have reached the maximum deformation. It is seen that, as the droplet bounces away, the radius of the clearance ring decreases and the gap thickness increases.

1.00

0.6 0.80

t 0.60

0.95

0.4

τ2

0.60

τ1

r

0.40 0

2

4

1.15

0.2 6

8

1.24 1.25

0

(b)

We

-0.2

FIG. 22. t1 , t2 , and τm,c , as a function of We for (a) water and (b) tetradecane in 14.6 atmosphere argon.

0.29 -0.6 0

5. Dynamics of inter-droplet gas gap To investigate the characteristics of the intervening film, experiments were conducted at high pressures under which the bouncing regime is broadened23 such that the behavior around the transition boundaries, such as the relation between inter-drop spacing and We, can be studied without concern for the various numerical issues related to merging. The discussion is based on the shape and minimum separation between the droplets. We have avoided extracting the detailed flow structure in the channel since this narrow region may overlap with the numerically immersed boundary layer. The motion of the interface, however, can still be tracked through interpolation of the conserved quantities at neighboring grid points. 5.1 Evolution of the gap contour Figure 23a shows the shape of half of the inter-droplet gap, h(r), during water droplet collision in 14.6 atmosphere argon, with We = 2.22, at different times. It is seen that as the two interfaces approach each other, high pressure is built up in the direction of the lines connecting the centers (Fig. 9c), causing indentation on the droplet surface such that the minimum clearance takes the form of a ring concentric along the line of

t = 0.16

-0.4

0.60 0.002

0.004

0.006

0.008

z

(a) t 0.45

0.76 0.5

0.01

0.92 1.07

r

1.11

0

-0.5

t = 0.13 0.45

0

0.002

0.27 0.004

0.006

0.008

0.01

(b)

FIG. 23. Evolution of inter-droplet gap geometry; (a) water in 14.6 atmosphere argon with We = 2.22, (b) tetradecane in 10 atmosphere nitrogen with We = 7.33. An interesting phenomenon observed from the gap evolution is the deformation of the droplet surface in the center region during collision. It is seen that the center indentation develops gradually until t = 0.60. As the droplet bounces away, this center region initially remains largely stationary. However, between t = 1.15


to 1.24, the larger pressure difference between the higher pressure within the liquid and the lower one in the gas film (refer to Fig. 17, t = 1.585) “pumps” the indented surface out rapidly. This causes the minimum clearance position to switch from the ring to the center, and the minimum gap size first decreases and then increases again. The development of the gas gap during tetradecane droplet collision at 10 atmosphere nitrogen with We = 7.33 is shown in Figs. 23b, which shows similar characteristics observed for water droplet collision e xcept the film thickness is larger. Specifically, the expanding and then shrinking of the minimum clearance ring during the collision and the “pumping” of the center indentation at the final collision stage are again exhibited. 5.2 The role of the minimum gap width Figure 24 plots the center clearance, h center, and minimum gap clearance at the rim, h rim, as functions of time for water and tetradecane droplet collision. It is seen that during most of the collision time the indentation is developed causing h center > h rim, while during the initial and final stages h center represents the minimum distance when the indentation is absent.

increases, the collision outcome changes from coalescence to bouncing, and then back to coalescence again, the variation of the gap size as the droplet kinetic energy changes has been calculated. Figure 25 plots the minimum gap width h m as a function of We for decane, dodecane, and tetradecane in 10 atmosphere nitrogen, where for each liquid We has been changed by varying the initial droplet velocity while keeping the droplet size fixed. Here h m is the minimum value of h rim, which represents the minimum distance between the droplet surface and the symmetric plane during the entire collis ion process. It is seen that h m initially increases and then decreases as We increases. In other words, with small relative velocity or drop size, or large surface tension, the droplets can approach each other relatively easy since the gas resistance due to pressure buildup between the droplet surfaces is small. However, as the droplet kinetic energy increases, the trapped gas and hence pressure buildup inside the gap increase, resulting in a larger flattening of the interface. The enhancement effect of the droplet inertia is smaller than the increase of resistance and thus the droplets bounce away at larger h m. With further increase of kinetic energy, however, the droplets can overcome the gas resistance and again attain a closer approach. 0.0017

0.02 Tetradecane

0.0015

0.015

0.0013

hm

h

Dodecane 0.0011

0.01 0.0009

0.005

0 0

Tetradecane

0.2

0.4

0.6

0.8

Decane

0.0007

1

Water 1.2

t

FIG. 24. The evolution of h center and h rim for the water and tetradecane droplet collisional sequences of Fig. 23. The study of the gap evolution brings a critical question about the specific mechanism that controls the collision outcome to be either coalescence or bouncing. It is the gas pressure buildup that retards the intersurface movement and reverses the incoming motion by transforming the original kinetic energy to the surface energy and then releasing it back to the fluid, which makes the droplet bounce away after the collision. It is suggested that coalescence occurs when the gap size reaches the molecular interaction range of a few hundred Angstroms ,5,18 while bouncing occurs when the minimum gap is still larger than that range. Recognizing (Fig. 1) that as the droplet kinetic energy

0.0005 0

2

4

6

8

10

We

FIG. 25. Minimu m gap distance during the entire collision sequence as a function of We for decane, dodecane, and tetradecane in 10 atmosphere nitrogen. Experimentally determined Weber numbers at transition between regimes I and II, and II and III, are marked off. A series of experiments has been performed for decane, dodecane, and tetradecane in 10 atmosphere nitrogen to acquire the critical Weber numbers, Wec,1 and Wec,2 , for the transitions between regimes I and II, and II and III respectively, as marked in Fig. 25. It is seen that, for a given fuel, the two critical Weber numbers yield values of h m that lie on opposite sides of the maximum h m, indicating a change of mechanism for bouncing and coalescence. Furthermore, the critical values of h m are not equal. Thus the non-monotonic


transition between the collision outcomes of coalescence and bouncing corresponds to a nonmonotonic variation of the minimum gap width h m, and the state of transition does not depend on h m alone. To further investigate the propensity of bouncing versus merging with respect to the width of the gas gap, we have tracked the evolution of the interface boundary for the coalescent states around the first (Fig. 26a) and second (26b) transitional boundaries. The instants of merging were noted from the experiments, but were not implemented in the simulation so that hrim and h center are shown to provide an indication of their evolvement. Using the experimental rupture times as markers, it is thus seen that breakup of the impinging surfaces occurs at an early phase of collision for We around Wec,2 when the gap of the rim approaches its minimum value (Fig. 26b), while it occurs at a late phase for We around Wec,1 after the interfaces have remained stagnant at the shortest separation until they start to recede (Fig. 26a). Thus merging occurs at the rim for both soft and hard collisions, although it takes place during the receding and approaching phases of the interface motion respectively. 0.012 hcenter hrim hcenter (A* = 4.0×10-6) hrim (A* ? 0)

0.01 0.008 h

0.006 0.004 0.002 0 0.4

0.6

0.8

1

t

1.2

1.4

1.6

(a)

0.012 0.01

hcenter hrim hrim (A* ? 0)

0.008 0.006 0.004 0.002 0

0.2

0.4

0.6

0.8

1

1.2 (b)

FIG. 26. The evolution of h center and h rim for tetradecane droplet collisions with coalescence in 1 atmosphere air; (a) We = 2.3, (b) We = 13.63. The h rim‘s associated with imposed merging mechanism that mimics van der Waals force are also presented, of which A* ? 0.

Several questions can be raised regarding the nonmonotonic relation between the gap width and We. First, the minimum distance that the droplet can attain, which is about 0.2 µm, is an order larger than the reported range for van der Waals force to be effective. In addition, if coalescence is purely dominated by the intermolecular force which depends on the inter-droplet spacing, the smallest h‘s at the two transitional boundaries should be close to each other and the merging at Wec,1 should have occurred at an early phase of the stagnant stage, immediately after the gap has attained its minimum value. Furthermore, if the delay is due to gas drainage, why is the gap not gradually narrowed? The considerable period of near stagnation before soft merging takes place indicates that the computational simulation could have overlooked certain mechanisms, particularly those which could exert an attractive force that is strengthened as the droplet interfaces approach each other. We now consider such a mechanis m. 5.3 A merging mechanism through augmented van der Waals force To explore the above issues, we introduce a merging mechanism that mimics the van der Waals force by adding a “negative” pressure at the droplet surface through the gas medium, i.e. A/(6ph i 3 ), where A is a coefficient and h i the inter-droplet separation. It is treated as a surface force and is smoothed out by means of the immersed technique in the same manner as that for the surface tension. This attractive force, which is inversely proportional to the cubic power of the separation of the interface, was derived based on two flat plates with semi -infinite mass.12 We thus note that, for the present situation the surface at the closest approach of the rim does locally approach a flat shape, as can be justified by the small curvature in the region where the axial variation of the front position is about two orders smaller than that in the radial direction, as shown in Fig. 27a. Since the attraction is extremely sensitive to the inter-droplet separation, its influence prevails only around the trough which is located at the shortest distance. We are interested to assess if force of this nature can provide sufficient attraction to effect merging. Using A* = A/(ρ l V0 2 R3 ) = 2×10-6 , Fig. 27b shows that, under the influence of the attractive force, the interfacial surfaces, specifically the troughs, now approach each other in an accelerative manner. Asymptotic theories that assume certain limit conditions such as small We have also predicted similar evolution of droplet surfaces before rupturing.6 In our numerical simulation, if the interfaces were not manually connected when they become very close, the computation would eventually diverge, indicating the onset of merging. It was thus found that the


dimensionless force coefficient, A*, falls within the range of 10-6 ~ 10-5 , with corresponding A being about 10-15 joule, and is sensitively related to the occurrence of the attractive movement. It is seen in Fig. 26a, for We = Wec,1 , that a somewhat smaller A* (4×10-6 ) fails to bring the surfaces close enough for merging to occur, while a slightly larger value (7.2×10-6 ) leads to an earlier merging time than that captured experimentally. Adopting A* as 5.3×10-6 (A = 1.2×10-15 J) then generates coalescence at the appropriate instant. For the strong merging situation of Fig. 26b, a value of 2.0×10-6 for A* (A = 2.65×10-15 J) was found to reproduce the experimentally observed merging time. 0.53 0.52 0.51 0.5 0.49

r 0.48 0.47 0.46 0.45 0.44 2

2.2

2.4

z

x 10

-3

(a)

0.6 0.4

1.144 1.265

0.2 r

0

-0.2 1.023

-0.4

t = 0.783 0.903

-0.6 0

0.002

0.004 z 0.006

0.008

0.01 (b)

FIG. 27. The inter-droplet (tetradecane) gap geometry in atmospheric air condition with We = 2.3; (a) magnified trough region (t = 0.783), (b) surface movement due to the merging mechanism that is analogous to van der Waals force with A* = 5.3×10-6 . In reality, A = π2 q 2 λ is the Hamaker constant and is determined by the atom densities (q) and London-van der Waals constant of the interacting materials (λ), with modified effects of the immersion fluid.12 It is always positive for particles of the same material and thus generates attractive interaction. The magnitude of A is

generally between 10-21 ~ 10-18 J, and is about 5.0×10-20 J for liquid tetradecane in vacuum. Since the force coefficients used in effecting merging in Figs. 26 are of the order of 10-15 J, which is much larger than the Hamaker constant of 10-21 ~ 10-18 J, the approach distance is too large for the van der Waals force to be effective. However, if the interface separation distance is one order smaller, e.g. changing from 0.2 to 0.01 µm, and since the attractive force varies inversely with the cube of the separation distance, our calculated coefficient A would fall in the range for the intermolecular force to take substantial effect. This implies that some other factors, to be discussed next, could be present that would have allowed the interfaces to approach each other closer than calculated. Since the characteristic dimension of the intervening film is close to that of the rarefied gas regime, which is about 0.1 µm at atmospheric condition, it is reasonable to suggest that a proper rarefied gas treatment would have resulted in a lower pressure within the gap, which would have driven the interfaces closer to each other and subsequently triggered the molecular attractive force. Indeed, investigations have been conducted analytically for solid particles with small deformation in collision by Hocking13 , Barnocky & Davis 2 , and Sundararajakumar & Koch.24 By implementing the Maxwell slip boundary condition or solving the linearized Boltzmann equation, it was found that when the inter-particle gas gap becomes as small as the mean free path (l) or less, the mass flux drained out is modified and the resistant force in the lubrication layer is reduced as compared to that predicted based on continuum mechanics. Consequently the singularity of the lubrication force as the film thickness approaches zero, which results from the inverse proportionality between the repulsive force and gap thickness, is avoided and the particles are able to make contact. Another possible factor that restrains the interfaces from moving toward each is the neglect of compressibility during collision. This effect has been studied in particle collisions by asymptotic analyses .3,10,1 6 Since the gas is squeezed as the gap becomes small, the pressure, as well as density, can be elevated substantially. Since the Reynolds number of the gas flow in the gap is O(1),11,23 the inertial and viscous terms are of equal significance in balancing the driving pressure. By equating (?p/?r) ~ 2µg (?2 u/?z2 ) and using mass conservation, V0 Rh ~ uh , based on characteristic dimensions of (Rh)1/2 and h for r and z directions, respectively, in an approximation assuming parabolic surface shape at the trough,8,1 6 the order of the pressure difference across the channel is found to be 2µg V0 R / h 2 . If this term is comparable to the pressure of


the ambient gas p 0 , i.e. h cp ~ (2µg V0 R / p 0 )1/2, compressibility will play a significant role in the drainage such that the density change needs to be considered in mass conservation.10 Previous analytical studies showed that compressible effects could reduce the lubrication force in the gas gap as compared to incompressible results and hence promote the propensity for particles to come into contact. In the current condition, h cp is of the order 0.1 µm and thus compressibility could also have lowered the pressure in the gap.

6. Concluding Remarks In the present investigation we have studied the collision dynamics of two impinging droplets experimentally as well as through computational simulation. The crucial factor responsible for the success of the present study is the ability to experimentally identify and resolve the instant of merging, at which the cuspy curvature of un-merged interfaces suddenly smoothens. Using this empirically supplied information in the computational simulation we were able to identify with realism the various aspects of the droplet collision dynamics especially those associated with merging. In general, the experimental collision images were satisfactorily simulated, given the instant of merging. Analysis of the evolution of the energy budget shows that, for soft collisions, with We close to Wec,1 , viscous dissipation is generated mostly through the gas phase near the opening of the inter-droplet gap, especially during the impacting and rebounding stages at which the intervening gas is respectively squeezed out and entrained into the gap. For the hard collision with high We near Wec,2 , however, dissipation occurs mainly inside the droplet and the total viscous loss is relatively larger than that of the soft collision. This justifies the previous assumption15 of negligible gas-phase dissipation situations involving hard collisions. It is further shown that merging in regime I around Wec,1 occurs at the end of collision, following a near stagnant stage of minimum film thickness, while in regime III close to Wec,2 merging occurs early when the interfaces readily attain the smallest separation distance. The minimum film thickness was found to vary non-monotonically with We, and bouncing is favored for larger gaps, as expected. The role of the intermolecular force was investigated by using a merging mechanism that employs an augmented van der Waals force. Results show that the present simulation based on continuum mechanics fails to bring the colliding interfaces to within the range of molecular attraction to effect merging. It is therefore suggested that effects of

rarefied gas flow and compressibility are needed in the simulation to properly induce merging. Acknowledgment The work was supported by the Air Force Office of Scientific Research under the technical monitoring of Dr. M. Birkan. We thank Professors Wei Shyy and G. Tryggavason for helpful discussions. References 1. J. R. Adam, N. R Lindblad,. and C. D. Hendricks, The collision, coalescence, and disruption of water droplets, J. Appl. Phys. 39, 5173 (1968). 2. G. Barnocky and R. H. Davis, The effect of Maxwell slip on the aerodynamic collision and rebound of spherical particles, J. Colloid Interface Sci. 121, 226 (1988). 3. G. Barnocky and R. H. Davis, The influence of pressure-dependent density and viscosity on the elastohydrodynamic collision and rebound of two spheres, J. Fluid Mech. 209, 501 (1989). 4. R. P. Beyer and R. J. Leveque, Analysis of a onedimensional model for the immersed boundary method, SIAM J. Numer. Anal. 29, 332 (1992). 5. S. G. Bradley and C. D. Stow, Collisions between liquid drops, Phil. Trans. R. Soc. Lond. A 287, 635 (1978). 6. A. K. Chesters, The modeling of coalescence processes in fluid-liquid dispersions: a review of current understanding, Trans. Inst. Chem. Engrs 69, 259 (1991). 7. L. P. Chin, A numerical study on the collision behavior of droplets, Central States Section Meeting of the Combustion Institute, St. Louis, MO (1996). 8. R. H. Davis, J. A. Schonberg, and J. M. Rallison, The lubrication force between two viscous drops, Phys. Fluids A 1, 77 (1989). 9. G. B. Foote, The water drop rebound problem: dynamics of collision, J. Atmos. Sci. 32, 390 (1975). 10. A. Gopinath, S. B. Chen, and D. L. Koch, Lubrication flows between spherical particles colliding in a compressible non-continuum gas, J. Fluid Mech. 344, 245 (1997). 11. A. Gopinath and D. L. Koch, Collision and rebound of small droplets in an incompressible continuum gas, J. Fluid Mech. 454, 145 (2002). 12. H. C. Hamaker, The London-van der Waals attraction between spherical particles, Physica IV 10, 1058 (1937). 13. L. M. Hocking, The effect of slip on the motion of a sphere close to a wall and of two adjacent spheres , J. Eng. Math. 7, 207 (1973).


14. O. W. Jayaratne and B. J. Mason, The coalescence and bouncing of water drops at an air/water interface, Proc. R. Soc. Lond. A 280, 545 (1964). 15. Y. J. Jiang, A. Umemura, and C. K. Law, An experimental investigation on the collision behaviour of hydrocarbon droplets, J. Fluid Mech. 234, 171 (1992). 16. H. K. Kytömaa and R. J. Schmid, On the collision of rigid spheres in a weakly compressible fluid, Phys. Fluids A 4, 2683 (1992). 17. H. Lamb, Hydrodynamics, Dover, New York (1932). 18. G. D. M. MacKay and S. G. Mason, The gravity approach and coalescence of fluid drops at liquid interfaces, Can. J. Chem. Engng 41, 203 (1963). 19. S. Middleman, Modeling axisymmetric flows: dynamics of films, jets, and drops, Academic Press, New York (1995). 20. M. R. Nobari, Y.-J. Jan, and G. Tryggvason, Headon Collision of drops–A numerical investigation, Phys. Fluids 8, 29 (1996).

21. K. L. Pan, W. Shyy, and C. K. Law, An immersedboundary.method for the dynamics of premixed flames, Int. J. Heat Mass Transfer 45, 3503 (2002). 22. C. S. Peskin, Numerical analysis of blood flow in the heart, J. Comput. Phys. 25, 220 (1977). 23. J. Qian and C. K. Law, Regimes of coalescence and separation in droplet collision, J. Fluid Mech. 331, 59 (1997). 24. R. R. Sundararajakumar and D. L. Koch, Noncontinuum lubrication flows between particles colliding in a gas, J. Fluid Mech. 313, 283 (1996). 25. S. O. Unverdi, Numerical simulations of multi-fluid flows, Ph.D. thesis, The Unversity of Michigan (1990). 26. S. O. Unverdi and G. Tryggvason, A front-tracking method for viscous, incompressible, multi-fluid flows, J. Comput. Phys. 100, 25 (1992). 27. F. M. White, Viscous Fluid Flow (2nd ed.), McGraw-Hill Inc. (1991).
