Bouncing-to-Merging Transition in Drop Impact on Liquid Film: Role of

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Bouncing-to-Merging Transition in Drop Impact on Liquid Film: Role of Liquid Viscosity Xiaoyu Tang, Abhishek Saha, Chung K. Law, and Chao Sun Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.7b03936 • Publication Date (Web): 23 Jan 2018 Downloaded from http://pubs.acs.org on January 24, 2018

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Bouncing-to-Merging Transition in Drop Impact on Liquid Film: Role of Liquid Viscosity Xiaoyu Tang, Abhishek Saha*, Chung K. Law Department of Mechanical and Aerospace Engineering, Princeton University, New Jersey 08544, USA

Chao Sun Center for Combustion Energy, Tsinghua University, Beijing 100084, China *

Corresponding Author. Email: [email protected]

Abstract: When a drop impacts on a liquid surface, it can either bounce back or merge with the surface. The outcome affects many industrial processes in that merging is preferred in spray coating to generate a uniform layer and bouncing is desired in IC engines to prevent accumulation of the fuel drop on the wall. Thus, a good understanding of how to control the impact outcome is highly demanded to optimize the performance. For a given liquid, a regime diagram of bouncing and merging outcomes can be mapped in the space of Weber number (ratio of impact inertia and surface tension) vs. film thickness. In addition, recognizing that the liquid viscosity is a fundamental fluid property that critically affects the impact outcome through viscous dissipation of the impact momentum, here we investigate liquids with a wide range of viscosity from 0.7 to 100cSt, to assess its effect on the regime diagram. Results show that while the regime diagram maintains its general structure, the merging regime becomes smaller for more viscous liquids and the retraction merging regime disappears when the viscosity is very high. The viscous effects are modeled and subsequently the mathematical relations for the transition boundaries are proposed which agree well with the experiments. The new expressions account for all the liquid properties and impact conditions, thus provides a powerful tool to predict and manipulate the outcome when a drop impacts on a liquid film.

1. Introduction Drop impact on a liquid surface is ubiquitous in many industrial applications, spanning from spray coating [1], inkjet printing [2], to internal combustion (IC) engines [3], in which the impact outcome plays a critical role in affecting the subsequent processes and the performance. For example, for thermal spray coating, the efficient merging of the drop with the pre-deposited film is highly desired to ensure a uniform and controlled coating and to minimize loss of the coating material [1]. On the other hand, in IC engines, the fuel is sprayed into the combustion chamber and is expected to burn out inside the chamber. Inevitably, however, some fuel drops, especially the larger ones and/or those within the spray core, can survive the hot environment and make 1

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their way to deposit on the engine wall. This is highly undesirable since it leads to pollutant generation due to the lower temperature at the wall [3]. The prevention of the subsequent accumulation of fuel drop on the wetted engine wall is critical for engine operation and can be facilitated by drop bouncing. Thus, it is important to recognize the conditions under which bouncing or merging occurs, and to understand the transition criteria for optimized performance. In addition, the liquid properties for the coating materials, printing materials, or fuel can exhibit large variations in the collision response, including its density, surface tension and viscosity, which should also be accounted for. There have been extensive studies on drop impact on either solid surface or deep pool, with interest on the splashing threshold, crown formation, and spreading dynamics [4-14]. However, there have been relatively few studies on the bouncing-merging transition for impact on liquid film with various thicknesses. Among them, a regime map of the bouncing-merging outcomes for tetradecane was investigated thoroughly in terms of two non-dimensional parameters: the Weber number, defined as We = 2 ρ RU 2 / σ , and the nondimensional film thickness, H * = H / R , where ρ , σ , U , R and H are the density and surface tension of the liquid, the impact velocity and radius of the drop, and the film thickness, respectively [4, 11]. These studies, however, did not look into the viscous effect, which naturally plays a key role in the impact dynamics through dissipation of the impact momentum. In particular, the importance of viscous dissipation, in addition to the impact inertia and surface tension, have been well demonstrated for drop impact on a solid surface or drop collision with another drop [5-10]. For example, for impact on a solid surface, increased viscosity reduces the maximum spreading diameter [5-7], suppresses splashing [8], levitates the spreading lamella [9], delays the merging time, and increases the merging radius [10]. Empirical and theoretical models have been developed to 2


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quantify the effects of viscous dissipation, although each model predicts only a specific dynamic, such as the maximum spreading diameter or the splashing threshold [15-18]. The inhibiting role of viscosity on wetting dynamics, especially in the limits of vanishing impact speed, has also been studied for both Newtonian and non-Newtonian liquids [19-22]. On the other hand, for drop-drop collision, viscous dissipation helps maintaining the coalesced drops from separation, hence inhibit the formation of secondary and tertiary drops [23]. Higher liquid viscosity also renders merging difficult for both drop-drop [23] and jet-jet [24] collisions, thereby decrease the span of the coalesced regime. In short, while viscous effect on the impact characteristics in several impact systems as discussed has been well studied, its role on the transition of the bouncing and merging outcome in relation to drop impact on a liquid film with thickness comparable to the drop radius has not been explored. In light of the above considerations, we report herein our recent experimental results and the associated analysis on droplet impacting a liquid film with a wide range of viscosities covering the transitional boundaries of bouncing and merging. We provide physical understanding of these boundaries and the associated mathematical relations verified by the experiment. The regime diagram accounting for viscous losses provides a more comprehensive guidance in manipulating the impact outcomes, useful for many industrial applications, where change in the operating liquid is inherently associated with the corresponding change in viscosity.

Figure 1: Experimental Setup. 3


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2. Experimental Setup Our drop generation system consists of a vertically suspended needle, through which liquid is pushed using a syringe pump to generate a drop at the tip of the needle. Once the drop attains a critical size, it detached from the needle because of the gravity and landed vertically on a liquid pool contained in a cubic glass-walled chamber. The drop size is about 1.6 mm in diameter, which is controlled by the needle size and liquid properties. A schematic of the setup is shown in Fig. 1. The impact velocity of the drop is modulated by changing the distance of the needle from the liquid surface. Seven liquids with varying viscosities were used for experiments. The properties and abbreviations of these liquids are provided in Table 1.

Liquid n-Octane (C8) n-Decane (C10) n-Tetradecane (C14) n-Heptadecane (C17) Silicone Oil (S05) Silicone Oil (S20) Silicone Oil (S100)

Table 1: Properties for the liquids tested. ρ, Density σ, Surface tension (kg/m3) (mN/m) 708 21.1 730 23.8 763 26.6 777 27.5 913 20 950 20 966 20

ν, Kinematic viscosity (cSt) 0.69 2.37 3.6 4.76 5 20 100

A monochromatic high-speed camera (Phantom V7.3) along with a 50mm lens (Nikon), a 2X tele-convertor and an extension bellow were used to capture the side-view shadowgraph of the impact process with a high intensity halogen light for backlighting (Fig. 1). The imaging system was operated at 15,037 frames per second, with spatial resolutions of ~60 µm per pixel. The drop diameter, impact speed and liquid film thickness were measured by analyzing the side-view images from individual experiments.

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Figure 2: Typical impact events from the side-view. (a) Bouncing on a deep pool. (b) Impact merging at similar film thickness but higher We. (c) Bouncing on a shallow pool. (d) Retraction merging at similar We but higher film thickness.

3. Results and Discussion 3.1 Regimes of Impact Outcomes Although a wide range of characteristics of drop impact can be observed, here we focus on three types of impact outcomes, namely “bouncing”, “impact merging” (“early merging” in ref. [4]), and “retraction merging” (“late merging” in ref. [4]). Mechanically, for a given liquid, the impact conditions are controlled by the Weber number ( We ), which characterizes the ratio between kinetic energy and surface energy and film thickness nondimensionalized by the drop radius ( H * ). Figure 2a-d shows time-resolved side-view snapshots of four typical impact events for C17. At high H * and low We (Fig. 2a), the drop deforms the liquid surface and pushes the liquid away until it reaches the maximum penetration at t = 7.78 ms where the drop does not reach the bottom substrate. Then the liquid surface relaxes and pushes the drop upwards and eventually the drop bounces away. With the same H * but higher We (Fig. 2b), the drop merges readily with the liquid surface soon (t=2.59 ms) after it penetrates. On the other hand, at low H * 5


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and low We (Fig. 2c), unlike the impact shown in Fig. 2a, before reaching the maximum penetration depth, the drop reaches the bottom substrate and spreads on it, before eventually bounces back. And then with slightly higher H * and similar We (Fig. 2d), the drop merges with the liquid surface at the bottom substrate, at t=8.45 ms. The impact outcomes described above are summarized in the regime diagram of H * - We (Fig. 2e). It is seen that for We higher than the Inertial Limit and its extension the Thin Film Limit, merging always occurs; at intermediate H * , the Deformation Transition separates retraction merging from bouncing, while the Deep Pool Limit forms the upper boundary of the retraction merging regime. The rest all involves bouncing. It is noted that the similar behavior has been reported for C14 in our previous work [4,11]. 3.2 Effects of Viscosity on Regimes of Impact Outcomes Recognizing the essential role of viscosity on the impact outcomes, and that this factor has not been systematically considered as previous studies have only employed a single liquid, we have therefore substantially extended the scope of the investigation to include several liquids whose viscosities span over two orders of magnitudes (0.69-100 cSt). The transition boundaries are plotted together for all liquids tested in Fig. 3. Globally, the merging zone shrinks with increasing viscosity: the critical We for the Inertial Limit and the Thin Film Limit increases, reducing the area of impact merging; while the Deep Pool Limit shifts downward and the Deformation Transition moves upward, diminishing the retraction merging regime. It can be projected that there is a critical viscosity where the Deformation Transition will move up so much that it will be higher than the Deep Pool Limit so that the retraction merging regime will be eliminated. In fact, when viscosity is very high (S20 and S100), the retraction merging regime indeed disappears and a monotonic transition boundary between bouncing and impact merging is left behind. Phenomenologically, this behavior of diminished merging regime is expected since

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the major role of viscosity is dissipating energy, which reduces the effective kinetic energy that brings the drop and impacted liquid film surface close enough to induce merging. We now focus on the individual transition boundaries and explore their dependence on viscosities through scaling analysis.

Figure 3: Regime map of bouncing and merging. (a) Regime map of bouncing and merging for C17. Various Outcomes: B: bouncing (x), RM: retraction merging (*), IM: impact merging (+). Regime Boundaries: IL: Inertial Limit, TFL: Thin Film Limit, DT: Deformation Transition, DPL: Deep Pool Limit. (b) Bouncing-merging transition boundaries for all liquids tested with varying viscosity.

3.3 Inertial Limit The Inertial Limit identifies the transition between bouncing and impact merging, which occurs when the initial kinetic energy of the drop is large enough to break the interfacial gas layer, trapped between the drop and the impacted film surface, to induce merging. Based on the analysis of Bouwhuis et al., [12] the maximum thickness of the gas layer can be expressed as d

U p−2/3 R1/3 ρ −2/3 µ g2/3 , where Up is the penetration velocity at which the drop moves into the

liquid film, and µg the dynamic viscosity of the surrounding gas. If the gas layer thickness becomes smaller than a critical value, dcr, van der Waals force becomes stronger and induces the

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merging of the interfaces. The value of this critical gas layer thickness, dcr, depends on the molecular property of the liquid such as the Hamaker constant and can be found in the literature [9, 12, 25]. Thus, the critical penetration velocity for the bouncing to merging transition can be expressed as: U p ,cr ~ µ g R1/ 2 d cr−3/ 2 / ρ

(1)

Since the drop loses part of its initial kinetic energy to the liquid film and through viscous dissipation, the penetration velocity, Up is reduced from the impact velocity, U. Consequently Up can be estimated by balancing the initial drop kinetic energy ( KE ) D ,0 = 2πρU 2 R 3 / 3 with the drop kinetic energy at penetration, ( KE ) D , p = 2πρU p2 R 3 / 3 , the liquid film kinetic energy ( KE ) F , and the energy dissipated Eφ ,IL :

( KE ) D,0 = ( KE ) D , p + ( KE ) F + Eφ ,IL

(2)

It is noted that merging in this regime occurs soon after the drop impact and no significant drop and film deformations are observed (Fig. 2b). Thus, for simplicity we ignore the surface energy change for the prediction of this limit. To estimate the liquid film kinetic energy, ( KE ) F , we assume that the drop induces a spherically symmetric radial flow field in the film as shown in Fig. 4a and the velocity scales as V ≈ U p R 3 / r 3 , at distance r radially from the drop center. By integrating the kinetic energy of a thin shell with thickness dr at radius r, the kinetic energy induced in the film can be estimated as, ∞

∞

( K E ) F = ∫ 2πρ r V dr ≈ ∫ 2πρ r 2 (U p R 3 / r 3 ) 2 dr = 2πρU p2 R3 2

R

2

(3)

R

Since the drop is not substantially deformed until merging, the viscous loss is assumed to be concentrated in the boundary layer developed in the film next to the drop-film interface where

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the velocity gradient is large. The viscous loss, in general, can be estimated as

(

)

2

Eφ , IL = ∫∫ µ (∂vi / ∂x j + ∂vi / ∂x j ) 2 dx3 dt [23], which can be simplified to Eφ , IL ~ µ U p δ V%τ , where the velocity gradient in the boundary layer is approximately U p / δ and δ is the boundary layer thickness. The volume of the boundary layer can be estimated as V% ≈ 2π R 2δ , and the characteristic time can be approximated by the inertial time scale of the impact process,

τ ~ 2R / U p . The boundary layer thickness is generally expressed as a power law relation with the penetration Reynolds number, Re p = 2U p R ν , as, δ ~ R / Re np , where the value of the exponent n depends on the flow pattern. Assuming the boundary layer to be similar to the wellknown configuration of flow over a flat plate, we set n = 1/2. Thus, the dissipated energy can be expressed as, Eφ , IL ~ ρU p2 R 3 (Re p ) −1/ 2 .

(4)

It is noted that the penetration Reynolds number used here is related to the impact Reynolds number, Re = 2UR ν , through Re p = Re(U p / U ) . Now combining eqns (2) to (4) and all the expressions, we obtain

U  We U 2 = 2 ≈ 4 + CIL Re −1/2  p  We p U p U 

−1/2

(5)

where CIL is the proportionality constant. When Re is very high for low viscosity liquid, the second term on the right-hand side of eqn (5) can be neglected, which implies that the penetration velocity is half of the impact velocity, i.e. U / U p ≈ 2 , consistent with the results in references [4,10]. In Fig. 4b, we have compared the experimentally evaluated U/Up for various liquids as a function of We to show that the ratio is indeed close to 2 for liquids with low

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viscosity (C08 to S05: 0.69 – 5cSt). However, for S20 and S100, whose viscosities are high (20 and 100 cSt), we see a departure from the value 2. Finally, by substituting the critical penetration velocity in eqn (1), we arrive at an expression for the critical We which is the measure of the minimum impact inertia required for merging, defining the Inertial Limit as,

µ 2 R 2 dcr−3  U   4 + CIL Re−1/2  p  Wecr ≈ g ρσ  U 

−1/2

   

(6)

It is seen that Wecr is clearly independent of H* and thus, in the regime diagram (Fig. 2e and 3), the Inertial Limit appears as a vertical boundary. From the regime diagram (Fig. 3) we also note that Wecr shifts to a higher value as the viscosity increases, which is consistent with eqn (6). Next, we plotted (Fig. 4c) the modified functional form, eqn (5), to confirm that the power law dependence fits the experimental data well. This also validates our assumption that the boundary layer can be modeled as that of flow over a flat plate.

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Figure 4: Inertial Limit: (a) Schematic of the flow field induced by the drop inside a deep liquid pool. (b) Ratio between impact velocity U and penetration velocity Up as a function of We. (c) We/Wep as a function of ReUp/U. (d) Schematic of the flow field induced by the drop inside a thin liquid film.

3.4 Thin Film Limit The above discussion is based on large film thickness where the flow field in the liquid film is not affected by the bottom substrate. However, as the film thickness decreases, the flow field will be modified because the bottom solid surface would inhibit the flow in the vertical direction and the critical We for the transition to merging will be modified, which gives us the Thin Film Limit. The energy balance described for the Inertial Limit, eqn (2), is still valid, except that the kinetic energy in the liquid film and the associated viscous dissipation need to be modified:

( KE ) D,0 = ( KE ) D, p + ( KE ) F ,TFL + Eφ ,TFL

(7)

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We consider viscous dissipation up to the point when the drop reaches the bottom surface,

( (

where the radius of the displaced liquid is b = R H * 2 − H *

))

1/2

from geometry, Fig. 4d. The

flow field in the liquid film V is now cylindrically symmetric, contrary to spherical symmetry observed for Inertial Limit with large H*. Using mass balance, πρ b 2U p ≈ 2πρ bHV , we have

V ≈ bU p (2H ) . The kinetic energy of the liquid film is, thus, ( KE ) F ,TFL ≈ 1 2 πρ b 2 HV 2 . The 2 viscous dissipation can be simplified to Eφ ,TFL ~ µ (V δ ) V%τ , with V% ~ π b 2δ and τ ~ H U p .

By collecting the expressions for various modes of energies and replacing them in eqn (7), we obtain 2 We U 2 = 2 ≈ 1 + CTFL H * ( 2 − H * ) (1 + 2 Re−p1/2 ) We p U p

(8)

Combining eqn (8) with the expression for U p,cr in eqn (1), we obtain:

(

)

µ g2 R 2 d cr−3 2 1 + CTFL H * ( 2 − H * ) (1 + 2 Re −p1/ 2 ) Wecr ,TFL ≈ (9) ρσ With the decrease of the film thickness, the Wecr ,TFL decreases which serves as a tail extension of the Inertial Limit. It indicates that for the thinner film, merging can be attained with smaller We because of the reduced flow field generated in the liquid film. In addition, we also notice that both the Inertial Limit and the Thin Film Limit have similar dependence on viscosity, through Re −1/ 2 , which indicates that for higher viscosity liquids, the Thin Film Limit shifts to higher We,

similar to the Inertial Limit. We note in passing that while the global trend in this limit has been experimentally verified, i.e. the critical We attains a smaller value with decrease in H* and viscosity (Fig. 3), a quantitative validation is difficult due to the large uncertainty in measuring the film thickness.

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3.5 Deformation Transition Limit As identified earlier, deformation transition occurs when the increase in the film thickness causes a “delayed” merging, even We H *p ,max , the time taken by the drop to reach the maximum penetration depth for a given liquid is independent of the impact inertia, suggesting a capillary controlled process. This is evidenced in Fig. 7a, where we compare the penetration time normalized by the capillary time scale, for all the liquids tested, to show that it is indeed close to unity except for S100. The evolution of the penetration depths for all the liquids are shown in Fig. 7b, where both the penetration and rebounding stages collapse except S100. The peculiar behavior of S100 suggests the possibility of overdamping in drop impact. In general, viscous dissipation serves as a damping mechanism in capillary wave propagation and as in any damped oscillating system, there exists a critical damping beyond which the capillary wave loses its oscillatory nature. Such critical damping occurs when the damping ratio, defined as ε = η / ωcap equals to unity ( ε cr = 1 ), where the damping coefficient η = 2ν k 2 and the

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undamped frequency ωcap = (σ k 3 / ρ ) tanh(kH ) depend on liquid properties, film thickness, H

and the wave number ( k ≈ 2 / R ) [26]. For a deep pool where H * = H / R > 1 , we can simplify tanh( kH ) ≈ 1 such that ε ≈ 2 2Oh and thus Ohcr ,OD ≈ 1/ 2 2 ≈ 0.35 . Clearly, for overdamping

conditions with Oh > Ohcr ,OD , the capillary wave of the film loses its oscillatory nature and we expect to observe a different behavior in the penetration and rebounding process. Interestingly, if we extrapolate the best fit curve for eqn (18), we see that the slope (∂H *p ,max ) / ∂We of penetration depth vs We curve becomes zero at Oh=0.33 (Fig. 6d), which is very close to 0.35. In our experiment, only S100 (Oh =0.83) satisfies the overdamping condition, which agrees with the previous observations in Fig. 7 with shorter time to reach the maximum penetration and drastically different penetration behavior. Moreover, the slope (∂H *p ,max ) / ∂We for S100 in Fig. 6d attains a value close to zero, suggesting weak dependence on We of the penetration process. Next, we note that through the influences on the Deep Pool Limit and the Deformation Transition Limit, viscosity has a diminishing effect on the “late” merging regime (Fig. 3). With increasing viscosity, the Deformation Transition Limit is shifted towards higher H* (eqn (14)) and the slope of Deep Pool Limit is reduced, eqn (19), at a certain large viscosity the two limits will eventually merge to engulf the “late” merging regime. Specifically, this condition will occur when the intersecting H* of the Deformation Transition Limit and Inertial Limit (Point H*DT_IL in Fig. 2e) is larger than the intersecting H* of the Deep Pool Limit and Inertial Limit (Point H*DP_IL in Fig. 2e). Based on the experimental data of C08, C10, C14, C17 and S05, we find that such criticality would occur at Ohcr,LM =0.046, which is just slightly higher than (Oh=0.041) for S05. From the regime diagram we see that S05 indeed exhibits very small retraction merging regime

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and for S20 (Oh=0.16) and S100 (Oh=0.83), the retraction merging behavior completely disappears and as such the non-monotonicity of impact outcome with H* is lost.

4. Conclusion: In summary, we have investigated the bouncing to merging transition for drop impact on a liquid film using liquids with a wide range of viscosities. In general, we found that larger viscosity renders merging more difficult through dissipation of the impact kinetic energy and as such shrinks the merging regimes. Specifically, with increasing viscosity, the Inertial Limit which separates the impact merging and bouncing regime shifts to higher We, reducing the impact merging regime. The retraction merging regime bounded by the Deformation Transition Limit and the Deep Pool Limit also diminishes with increasing viscosity as the former shifts to larger film thickness and the latter moves to the lower film thickness with reduced sensitivity with We. Continuous increase in viscosity minimizes the span of retraction merging, which eventually disappears at a critical viscosity (Ohcr,LM =0.046), and as such larger viscosity liquids do not exhibit the retraction merging behavior as substantiated by the experiments. We have also identified that the penetration process is strongly coupled with the capillary oscillation of the liquid film, which becomes overdamped beyond larger viscosity (Ohcr,OD =0.35), changing the penetration characteristics completely as it becomes insensitive to impact inertia. The detailed viscous effect on each transition boundary is analyzed and quantified with mathematical expressions, and agree well with the experiments. With the understanding of the viscous effect and the resultant new mathematical expressions, the bouncing-merging transition when a drop impacts on a liquid film can now be predicted comprehensively for a wide range of liquids and impact conditions.

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Acknowledgement The work at Princeton University was supported by the Army Research Office under Grant #W911NF-16-0449 and by the Xerox Corporation under a UAC grant.

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Bouncing-to-Merging Transition in Drop Impact on Liquid Film: Role of

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