Dynamics of Droplet-Droplet and Droplet-Film Collision C. K. Law Princeton University The physical phenomena of droplet-droplet and droplet-film collision in the head-on orientation were studied experimentally and computationally, with emphasis on the transition between bouncing and merging of the liquid surfaces. Experimentally, the droplets (~300 µm diameter) were generated using the ink-jet printing technique, and imaged using stroboscopy and high-speed cine-photography for the droplet-droplet and droplet-film collision events respectively. Computationally, the collision event was simulated using the front-tracking technique. For the study of droplet-droplet collision, the instant of merging was experimentally determined and then used as an input in the computational simulation of the entire collision event. The simulation identified the differences between collision and merging at small and large Weber numbers, and satisfactorily described the dynamics of the interdroplet gap including the role of the van der Waals force in effecting surface rupture. For the study of droplet-film collision, extensive experimental mapping showed that the collision dynamics is primarily affected by the droplet Weber number (We) and the film thickness scaled by the droplet radius (H), that while droplet absorption by the film is facilitated with increasing droplet Weber number, the boundary of transition is punctuated by an absorption peninsula, in the We-H space, within which absorption is facilitated for smaller Weber numbers. Results from computation simulation revealed the essential dependence of the collision dynamics on the restraining nature of the solid surface, the energy exchange between the droplet and the film, and the coherent motion of the gas-liquid interfaces. Partial absorption with the emission of a secondary droplet of smaller size was also observed and explained. 1
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Dynamics of Droplet-Droplet and Click to edit Master title style Second level Droplet-Film Collision C. K. Law, Princeton University Fourth International level Symposium on Fifth Energy levelconversion Fundamentals June 21-25, 2004, Istanbul, Turkey 1 111 1
Practical Relevance Click to edit Master title style Droplet-droplet collision: - Spray Click processes to edit Master text styles - Rain drop formation - Model Second for nuclear level fusion Fourth chamber level - Raindrop Fifth level interaction with ocean and soil Droplet-surface collision: - Spray impingement on surface of combustion - Meteorite impaction - Ink-jet printing - Spray-painting/coating 2 222 2
Fundamental Interests Click to edit Master title style 3 Dynamics of deformable bodies Large deformation implies nonlinear dynamics Many decades of length scales: Second level - Droplet size: 10 ~ 10 3 µm - Molecular Third level force distance: 10-2 µm - Continuum flows Wide range of physical regimes: - Rarefied flows - van der Waals force Computationally & experimentally challenging Description of state of merging 333 3
Click to edit Master title style Second level 4 444 4
Computational Simulation Click to edit Master title style Front tracking method of Tryggvason & immersed Click to edit boundary Master method text styles of Peskin Second level Continuum Third levelformulation ( ρfifth V) ( level 1 T 8 + ρvv) = p + g + ( V + V ) t Re We s κnδ ( r r f ) da 5 555 5
Dynamics of Colliding Interfaces Click to edit Master title style 0.5 0-0.5 0.02 Click to edit Master t <= 0.45 text styles 0.76 t > 0.45 Second level 0.92 0.015 1.07 h 1.11 0.01 0.005 t = 0.13 0.45 0.27 0 0.002 0.004 0.006 0.008 0.01 0 0 0.2 0.4 0.6 0.8 1 1.2 t h center h rim 6 666 6
Matched Instant of Merging Click to edit Master title style Identify experimental instant of curvature reversal of interface 0.298 (0.124 ms) cusp Second as state level of surface Third level rupture 0.622 (0.258 ms) simulation Fourth level 0.916 (0.380 ms) Use this instant in Close agreement with experimental images t = 0 0.953 (0.395 ms) 0.984 (0.408 ms) 1.076 (0.446 ms) 1.362 (0.565 ms) 1.637 (0.679 ms) 1.944 (0.806 ms) 2.390 (0.990 ms) 2.923 (1.212 ms) 7 1.013 (0.420 ms) 3.280 (1.360 ms) 777 7
Mis-Matched Instants of Merging Click to edit Master title style Second level t = 0 1.076 0.298 0.622 0.916 0.953 1.362 1.637 1.944 2.390 0.984 2.923 8 1.013 3.280 Advanced rupture Delayed rupture 888 8
Surface Rupture Induced by Click Molecular to edit Master Force title (1/3) style To effect surface rupture, add van der 0.01 Waals force in formulation: Second level A/(6πδ 3 0.008 ) h 0.006 divergence of 0.004 interface Fourth for level certain A 0.002 A* = 1.6 10-5 Simulation shows Adjust A to agree with empirical instant of rupture 0.012 h rim (A* 0) h center (A* = 0) h rim (A* = 0) A* = 2.4 10 - Rupture 0 5 0.4 0.6 0.8 1 1.2 1.4 1.6 t A* = 0.8 10-5 9 999 9
Surface Rupture Induced by Click Molecular to edit Master Force title (2/3) style 0.012 0.012 h 0.01 0.008 0.006 0.004 0.002 0.01 Click to hedit rim (A* 0) Master text styles 0.008 Second h center (A* = 0) level 0.006 h rim (A* = 0) 0.004 A* = 0.8 10-5 A* = 1.6 10-5 0.002 A* = 2.4 10-5 Rupture 0 0 0.4 0.6 0.8 1 1.2 1.4 1.6 Soft merging occurs late in collision, at rim t h center (A* = 0) h rim (A* 0) Rupture A* = 0.8 10-6 A* = 2.4 10-6 A* = 2.0 10-6 0.2 0.4 0.6 0.8 1 1.2 Hard merging occurs early in collision, at rim t h rim (A* = 0) 10 10 10
Surface Rupture Induced by Click Molecular to edit Master Force title (3/3) style Augmented Hamaker constant is 10 3 to 10Click 6 larger to edit than Master the real text one styles Implying Second the level interfacial gap width from simulation Third level is one to two orders too large Consistent Fourth level with: Simulated Fifth levelgap width ~ 10-1 µm van der Waals force range ~ 10-3 -10-2 µm 11 11 11
Missing Physics Click to edit Master title style Additional physics needed so that the interface Click to can edit reach Master the text range styles of the molecular Second level force Need Third rarefied level flow treatment Need Fourth variable level density treatment Computationally Fifth level challenging Possibly described through analysis 12 12 12
Dynamics of Droplet-Film Collision Click to edit (Thin Master film) title style Second level (a) (a) Bouncing (b) Partial absorption (c) Total absorption (b) 13 (c) 13 13
Dynamics of Droplet-Film Collision Click to edit (Thick Master film) title style 14 Second level (a) (b) (c) (a) Bouncing (b) Partial absorption (c) Total absorption 14 14
Regime Diagram of Click Droplet-Film to edit Master Collision title style Controlling parameters: Weber 2 number & film thickness Second (H=h/R) level 1.5 Bouncing 1 peninsula around H 1 0.5 Unquantified boundary Existence of merging Existence of triple reversal in absorption & bouncing with increasing H H = h /R 2.5 0 0 5 10 15 20 25 30 We = 2R ρ V 2 /σ Merging R = 0.165 mm R = 0.250 mm R = 0.305 mm 15 15 15
Factors Affecting Click Bouncing to edit Master vs Absorption title style Solid surface restrains droplet motion Impact inertia narrows interfacial gap Droplet Second flattening level increases resistance Droplet deformation increases viscous loss Fourth level Energy Fifth level transfer to film Oscillatory resonance around H 1 16 16 16
Summary Click to edit Master title style Instant of droplet merging can be empirically determined and used to simulate complete droplet collision sequence Second level augmented Third level van der Waals force Surface rupture can be induced through an Proper description of interfacial dynamics, allowing for rarefied flow and variable density, is essential For droplet-film collision, existence of merging peninsula around H 1 17 17 17