Energy Budget of Liquid Drop Impact at Maximum ... - ACS Publications

Subscriber access provided by GAZI UNIV

Article

Energy budget of liquid drop impact at maximum spreading: numerical simulations and experiments Jae Bong Lee, Dominique Derome, Ali Dolatabadi , and Jan Carmeliet Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.5b03848 • Publication Date (Web): 08 Jan 2016 Downloaded from http://pubs.acs.org on January 18, 2016

Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a free service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are accessible to all readers and citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.

Langmuir is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

Page 1 of 30

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

Energy budget of liquid drop impact at maximum spreading: numerical simulations and experiments Jae Bong Lee,† Dominique Derome,‡ Ali Dolatabadi,¶ and Jan Carmeliet∗,†,‡ Chair of Building Physics, ETH Zurich, Stefano-Franscini-Platz 5, CH-8093 Z¨ urich, Switzerland., Laboratory for Multiscale Studies for Building Physics, Swiss Federal ¨ Laboratories for Materials Science and Technology, EMPA, Uberlandstrasse 129, CH-8600 Dbendorf, Switzerland., and Department of Mechanical and Industrial Engineering, Concordia University, Montreal, Quebec H3G1M8, Canada. E-mail: [email protected] Phone: +41 (0)44 633 28 55. Fax: +41 (0)44 633 10 41

Abstract Maximum spreading of impinging droplet on a rigid surface is studied for low to high impact velocity, until droplet starts splashing. We investigate experimentally and numerically the role of liquid properties, such as surface tension and viscosity, on drop impact using three liquids. It is found that the use of the experimental dynamic contact angle at maximum spreading in the Kistler model, which is used as boundary condition for the CFD-VOF calculation, gives good agreement between experimental and ∗

To whom correspondence should be addressed Chair of Building Physics, ETH Zurich, Stefano-Franscini-Platz 5, CH-8093 Z¨ urich, Switzerland. ‡ Laboratory for Multiscale Studies for Building Physics, Swiss Federal Laboratories for Materials Science ¨ and Technology, EMPA, Uberlandstrasse 129, CH-8600 Dbendorf, Switzerland. ¶ Department of Mechanical and Industrial Engineering, Concordia University, Montreal, Quebec H3G1M8, Canada. †

1

ACS Paragon Plus Environment

Langmuir

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

numerical results. Analytical models commonly used for predicting the boundary layer thickness and time at maximum spreading are found to be less correct, meaning that energy balance models relying on these relations have to be considered with care. The time of maximum spreading is found to depend both on impact velocity and surface tension, both dependencies not predicted correctly in common analytical models. The relative proportion of the viscous dissipation in the total energy budget increases with impact velocity with respect to surface energy. At high impact velocity, the contribution of surface energy, even before splashing, is still substantial meaning that both surface energy and viscous dissipation have to be taken into account, and scaling laws only depending on viscous dissipation do not apply. At low impact velocity, viscous dissipation seems to play still an important role in low surface tension liquids such as ethanol.

Abbreviations IR,NMR,UV

Introduction Drop impact on solid surface is a key phenomenon in a variety of industrial and natural processes, such as thermal spray, 1 spray painting 2 and coating, 3 inkjet and 3D printing, 4,5 pesticide deposition, 6,7 rain drop impact on buildings 8 and soil erosion. 9–11 Since the first systematic study of Worthington, 12 many researchers have studied drop impact experimentally, 13–18 analytically 14,19–26 and numerically, 21,27–34 and a recent review of the drop impact can be found elsewhere. 29,35,36 One of the important characteristics of droplet impact is its maximum spreading diameter which is directly related to its final contact area and the contact time between droplet and surface. This is an imperative parameter to controlling drop deposition. The maximum spreading ratio βmax = Dmax /D0 , where Dmax is the maximum

2


Page 2 of 30

Page 3 of 30

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

spreading diameter and D0 is the initial drop diameter, is usually used to characterize the spreading of an impacting droplet. Different approaches have been employed to predict maximum spreading of drop impact including experiment, empirical scaling, energy balance methods, 19,21,22,24,26,37 computational fluid dynamics (CFD) and molecular dynamics (MD). 38 Most experiments report mainly the outer geometry of the impacting droplet, but do not offer details about the inner flow. Theoretical models are therefore needed to understand the physical problem of drop impact under different impact velocities, as well as at its asymptotic velocity conditions. 16 Most of theoretical models for droplet spreading are based on the energy balance before impact and at maximal spreading. These models are commonly formulated using two dimensionless parameters: the Weber number (We = ρD0 Vi2 /γ) describing the ratio between the kinetic and capillary energy and the Reynolds number (Re = ρD0 Vi /µ) describing the ratio between the kinetic and viscous energy. Vi is the impact velocity, ρ the liquid density, γ the surface tension and µ the viscosity. A large number of semi-empirical and energy balance models for maximum spreading has been proposed and a recent review for semi-empirical and energy balance model can be found in Refs. 39,40 Numerous studies show the possibility of CFD to simulate drop impact using different models for tracking the interface: volume-of-fluid (VOF) 33,41,42 method, SOLA-VOF, 21,27,28 CLSVOF, 43 Level Set, 44,45 and a comprehensive review can be found in Ref. 29 A most challenging task in the CFD simulation of drop impact is the correct prediction of the liquid-vapor interface at the contact line. Modeling approaches differ in how the contact angle is implemented as boundary condition at the contact line, and how this contact angle depends dynamically on the contact line velocity. Fukai et al. 30 were the first to simulate drop impact with the contact angle as boundary condition. Constant advancing and receding contact angles were applied during spreading and receding phase, respectively. Pasandideh-Fard et al. 21 compared the spreading of the droplet using constant equilibrium and dynamic contact angles, and reported that the dynamic contact angle leads to more accurate predictions of the spreading. Bussmann et al. 32 modeled the dynamic contact angle as function of velocity using a linear

3


Langmuir

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

function between advancing, equilibrium and receding contact angles. Sikalo et al. 33 used the Kistler model 46 to describe the dynamic contact angle as function of contact line velocity, simulating the drop impact of water and glycerol onto hydrophilic and hydrophobic surfaces. They showed that, with the Kistler model for dynamic contact angle, the droplet behavior during spreading and receding phases is better predicted than when using a fixed dynamic contact angle (advancing or receding). Roisman et al. 47 applied the dynamic contact angle model of Kistler for simulating the water drop impact on stainless steel. They used, in the Kistler model, the advancing and receding contact angles instead of the equilibrium contact angle to consider contact angle hysteresis. Farhangi et al. 34 used the approach of Roisman et al. 47 to simulate the coalescence of droplets on superhydrophobic surfaces. In all these studies, the dynamic contact angle is an input to the CFD simulation as boundary condition at the contact line and the spreading is predicted and compared to the experiments. It is still not clear how this dynamic contact is best modelled and, more especially, it is not totally clear which contact angle has to be used as input to the dynamic contact model of Kistler. In this paper, we investigate the role of liquid properties, such as surface tension and viscosity, on drop impact experimentally and numerically. In contrast to most other studies, droplet spreading is measured under various conditions from low to high impact velocities for three different liquids. For the dynamic contact angle prediction by the Kistler model, we use the experimental value of the dynamic contact angle at maximum spreading and we validate this assumption comparing experimental and numerical results. Using the CFD model, we analyze the energy balance evolution during spreading and evaluate these results with respect to the analytical models commonly used for predicting maximum spreading.

4


Page 4 of 30

Page 5 of 30

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

Experimental method Liquid and surface In this study, we investigate the influence of the surface tension and viscosity of the liquid on droplet spreading: pure ethanol (ethanol), deionized water (water) and 1:1.3 glycerol-water mixture (glycerol). The liquid density ρ, viscosity µ and surface tension γ are reported in Table 1. Steel surface is used as substrate for drop impact tests. The roughness is measured by a contact profilometer (Surftest-211, Mitutoyo) equipped with a 5 µm radius diamond-tipped stylus. The arithmetic average roughness Ra of the steel surface is 0.42 µm. The wettability of steel surface is characterized with equilibrium, advancing and receding contact angles by sessile drop method (Table 1). The equilibrium contact angle is obtained with a 3 µl sessile droplet in equilibrium state (no movement). The advancing and receding contact angles are measured by quasi-statically expanding or contracting the deposited sessile droplet, respectively. Ethanol shows a total wetting on steel surface and spreads completely due to its lower surface energy (θeq ∼ 0◦ ). Water and glycerol show a wetting behavior on steel surface with an equilibrium contact angle between 52◦ to 61◦ . The advancing contact angle of water and glycerol on steel surface is quite similar to the equilibrium contact angle (difference less than 7%). Only water shows a receding contact angle higher than zero degree (θrec ∼ 7◦ ).

Drop impact measurement For the drop impact measurement, a droplet is generated from a needle and is accelerated by gravity. The impact conditions, drop size and impact velocity for the experiments and simulations are reported in Table 2. The drop impact on the steel surface is captured by a high-speed camera (NX7-S2, IDT) with a frame rate of 10 000 fps, 5 µs exposure time (LLS2, SCHOTT) and 7.38 µm spatial resolution (12x zoom lens, Navitar). The captured high-speed images are analyzed with a custom-made image analysis code to determine the 5


Langmuir

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 6 of 30

Table 1: Properties of liquids and contact angles for steel surface pure ethanol

deionized water

1:1.3 gycerol-water

(ethanol)

(water)

(glycerol)

Liquid properties ρ (kg/m3 )

789

998

1158

µ (mPa·s)

1.2

1.0

10.0

γ (mN/m)

23.2

72.8

68.3

Contact angles on steel (◦ )

θeq

θadv (◦ ) θrec

60.9 ± 1.3

52.4 ± 3.7

∼0

6.8 ± 1.0

∼0

∼0

(◦ )

θD (tmax )

∼0

(◦ )

61.5 ± 3.3

43.9 ± 2.5

48.5 ± 4.3

102.9 ± 3.2

121.2 ± 6.9

Table 2: Impact conditions and dynamic wettability for experiments and simulations Liquid

D0 (mm)

Vi (m/s)

We

Re

Exp. E

Ethanol

0.2 - 2.4

2.5 - 370

270 - 3 000

Sim. E

Ethanol

1.8 ± 0.04 1.8

0.2 - 2.4

2.5 - 367

247 - 2 969

2.0 ± 0.01

0.2 - 3.6

1.1 - 360

450 - 7 000

2.0

0.2 - 3.2

1.1 - 289

389 - 6 292

1.8 ± 0.01

0.2 - 3.7

1.2 - 410

42 - 760

0.2 - 3.2

1.2 - 330

42 - 686

Exp. W

Water

Sim. W

Water

Exp. G

Glycerol

Sim. G

Glycerol

1.8

initial drop diameter D0 , the impact velocity Vi , the spreading diameter D(t), the dynamic contact angle θD (t), the maximum spreading diameter Dmax and the dynamic contact angle at maximum spreading θD (tmax ) (Table 1). The dynamic contact angle θD is determined using the area angle relationship 48 in a 100 µm high region above the surface line. In this paper, we investigate experimentally the drop impact from low to high impact velocity (0.2 to 3.2 m/s) to capture the maximum spreading behavior of impinging droplets. The experimental results are used to validate our CFD simulations.

6


Page 7 of 30

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

Numerical method The numerical simulation of the droplet impact is performed using the volume-of-fluid (VOF) method for Newtonian and incompressible two-phase flow with free surface. The governing equations include the mixture, continuity and momentum equations:

∇·U=0

(1)

∂f + ∇ · (Uf ) = 0 ∂t

(2)

∂(ρU) + ∇ · (ρUU) ∂t = −∇p∗ + ∇ · (µ · ∇U) + (∇U) · ∇µ

(3)

− g · x∇ρ + γκ∇f where f is the indicator function for the phase fraction, U the velocity vector, t the time, x the coordinate vector, p∗ = p − ρg · x the modified pressure, γ the surface tension, κ the curvature of the interface. Body forces due to pressure gradient and gravity are implicitly accounted by introducing the negative gradient of the modified pressure. 49 The indicator function f represents the volume fraction of each phase and is defined as:

f=

    0,    

for points belonging to gas phase

01x105

Figure 5: Time evolution of drop impact for ethanol, water and glycerol on steel surface (Vi = 1.0 m/s). The lines show the liquid-vapor interfaces from experimental images (red) and numerical results (blue). The contour map on left side depicts kinetic energy per unit volume KE and right side shows viscous energy dissipation function. hydrophobic behavior at VCL ∼ 1.0 m/s (See Fig. 2). However, when the kinetic energy is converted into surface energy later in the spreading process, the surface tension force promotes further spreading due to the hydrophilic dynamic contact angle of ethanol on steel surface. This means that our CFD simulations are able to model the change of the dynamic contact angle from non-wetting to wetting behavior as shown in Fig. 4b, which plays an important role in the spreading process of the drop. Figure 6 shows the time evolution of the energy terms after drop impact for different liquids. Before drop impact (referred to using subscript 1), the total energy in the droplet is the sum of kinetic energy KE1 and surface energy SE1 . After impact, kinetic energy KE(t) decreases with time and is converted into surface energy SE(t) and viscous dissipation Wvis . The surface energy increases with time due to increase of surface area during the spreading phase. Simultaneously, the cumulative viscous dissipation energy Wvis increases. At maximum spreading, the remaining kinetic energy in the droplet is less than 5% compared to total energy (ethanol 0.2%, water 3.0% and glycerol 4.2% for Vi 1.0 m/s). Ethanol shows

15


Langmuir

(a)

4.0x10-6

(b)

4.0x10-6

(c)

4.0x10-6

KE1+SE1

1.0x10-6

tmax

3.0x10-6

Energy (J)

2.0x10-6

Energy (J)

3.0x10-6

2.0x10-6

1.0x10-6

0.0 0.000 0.001 0.002 0.003 0.004 0.005 0.006 0.007

Time, t (s)

0.0 0.000

tmax

KE(t) SE(t) int. W vis

tmax

3.0x10-6

Energy (J)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 16 of 30

2.0x10-6

1.0x10-6

0.001

0.002

0.003

0.004

Time, t (s)

0.0 0.000

0.001

0.002

0.003

Time, t (s)

Figure 6: Time evolution of energy terms during spreading of drop with ∼ 1.0 m/s impact velocity: (a) ethanol, (b) water and (c) glycerol. a smaller initial energy than water and glycerol before drop impact (KE1 +SE1 = 1.6 x 10−6 for ethanol and 3.4 x 10−6 J for water), but at maximum spreading, the cumulative viscous dissipation is quite similar for all liquids (Wvis = 0.9 x 10−6 J for ethanol and 1.1 x 10−6 J for water and glycerol), even though glycerol has a 10 times higher viscosity than water. This similar result can be explained by the fact that the maximum spreading time for glycerol (1.9 s) is shorter than the one of water (2.6 s). We note that the accurate prediction of the cumulative viscous dissipation at maximum spreading Wvis (tmax ) is crucial as this term is one of the inputs of energy balance models used to predict the maximum spreading.

Maximum spreading Figure 7 shows snapshots of maximum spreading for different impact velocities comparing experiments and simulation results. The boundary layer development at the surface is represented by the vector plot of the horizontal velocity Ux . The outline of the droplets at maximum spreading shows good agreement between simulation and experiment. Recall that the outline of impact droplet from the experiments (red line) shows the silhouette and the outline from the simulation (white line) is from the cutting plane in the middle of the droplet. The local boundary layer thickness δ is defined as the height where 99% of the maximal horizontal velocity is reached. To obtain a single value for the boundary layer thickness value

16


Page 17 of 30

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

(a) Ethanol

(b) Water

Vi = 0.3 m/s

tmax = 11.3 ms

Vi = 0.7 m/s

1 m/s tmax = 6.8 ms

Vi = 1.6 m/s Vi = 2.4 m/s

(c) Glycerol tmax = 4.1 ms Vi = 0.3 m/s 1 m/s tmax = 2.5 ms Vi = 0.7 m/s

tmax = 4.6 ms

tmax = 2.9 ms Vi = 0.3 m/s 1 m/s tmax = 2.0 ms Vi = 0.7 m/s

Vi =1.7 m/s

tmax = 2.2 ms Vi =1.7 m/s

tmax = 1.6 ms

Vi =3.2 m/s


tmax = 1.1 ms

tmax = 4.0 ms

Figure 7: Snapshots of the maximum spreading as function of the impact velocity. The white line shows liquid-vapor interface of simulation. The boundary layer development at the surface is represented by the blue vector plot of the horizontal velocity Ux . for each instant in time, the boundary layer thickness is spatially averaged as: 5 ¯ = δ(t)

4

R 2π R D(t)/2 0

0

δ(x, t)xdxdα πD(t)2

(15)

where is the angular coordinate. Figure 8a shows the non-dimensional boundary layer √ thickness δ¯ · Re/D0 as a function of non-dimensional time t/τ where τ is D0 /Vi in loglog plotting for three different impact velocities (Vi = 0.5, 1.0, 1.7 m/s) and three different liquids. The results are compared the prediction by Roisman 39 (dashed line with 0.5 slope). We observe that at t/τ < 1, where the inertia driven flow is dominant in the droplet, the different curves collapse for different impact velocity and liquids showing a scaling with the Reynolds number. For t/τ > 1, the curves start to deviate from the scaling slope, especially for ethanol. For glycerol and water at low impact velocity, the boundary layer thickness mainly increases until maximum spreading is reached. On the contrary for ethanol the boundary layer thickness reduces before reaching its maximum spreading, showing that the viscous dissipation becomes less important close to maximum spreading. This is explained by the fact that the ethanol droplet will still spread due to capillary forces even after inertia flow is dissipated. This time dependent boundary layer thickness may be further averaged over time to 17


Langmuir

10

1

(b)

Ethanol Water Glycerol

pe

=0

.5

(mm)

Slo

Ethanol Water Glycerol

1

Averaged

(a)

! "Re / D0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 18 of 30

0.1

1

0.1

0.01 0.1

10

1

10

Vi (m/s)

t/

Figure 8: (a) Non-dimensional boundary thickness averaged over space as function of the non-dimensional time obtained by CFD simulations. (b) Time-averaged boundary layer thickness during spreading phase as a function of impact velocity determined from CFD simulation. Dashed line is the theoretical model of Pasandideh-Fard et al. 21 for ethanol (black), water (red) and glycerol (blue). obtain a single value for the boundary layer thickness for the total spreading process. An analytical expression for this average boundary layer thickness can be attained assuming that the liquid motion in the droplet can be represented by an axisymmetric stagnation point flow: 21 D0 δ = 2√ Re

(16)

Considering Fig. 8a showing that the boundary layer thickness varies highly over time, this averaging over time to obtain one single value of the boundary thickness may be questioned. However, such a single value as given in eqn. 16 is commonly used in energy balance model because of its simplicity for predicting the cumulative viscous dissipation at maximum spreading, allowing the prediction of the maximum spreading. In Fig. 8b, the time averaged boundary layer thickness obtained by our CFD simulations is compared to the analytical prediction by eqn. 16 (dashed line) for different liquids. In general, the analytical model overpredicts the average boundary layer thickness. However, the slope of the simulated boundary layer thickness as a function of impact velocity agrees quite well with the analyti18


Page 19 of 30

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

cal model. At higher impact velocity, the simulated boundary layer thickness decreases more than the one predicted by the analytical model, since, in the simulation, the boundary layer spreads over the total thickness and is therefore limited by the droplet height. Figure 9a shows the maximum spreading ratio βmax as a function of impact velocity Vi for the experiments and simulations. Overall, the numerical method captures the maximum spreading for different liquids very well, when using, for the boundary condition, the experimental dynamic contact angle measured at maximum spreading as input in eqn. 9. Ethanol shows the highest maximum spreading, while glycerol shows the lowest maximum spreading. Water shows a transition curve from the lowest spreading at lower impact velocity like glycerol to the highest spreading at higher impact velocity like ethanol. These results clearly show the roles of surface tension and viscosity of liquid on the maximum spreading. We conclude that our CFD approach, using the measured dynamic contact angle at maximum spreading in the Kistler model to define the boundary condition for numerical simulations captures correctly the experiments over the total range of impact velocities. We remark that in most publications, the maximum spreading ratio is plotted versus the Weber number to scale for the effect of the surface tension of liquid as shown in Fig. 9b. At high impact velocity, the curve for ethanol starts to collapses with the curve of water, but the trends do not change. At lower impact velocity the different curves still diverge and Weber number does not scale the data properly. Reason is that at low impact velocity also dynamic wetting plays a role as influenced by the dynamic contact angle. For this reason we will continue to plot our other results versus impact velocity and not versus We number. Finally, we also plotted in the supplementary material the non-dimensional maximum spreading, with Dmax /(D0 · Re1/5 ) as function of P = We/Re4/5 as suggested by Clanet et al. 25 This figure shows that this scaling, although predicting the behavior for water, does not predict the maximum spreading diameter for ethanol and glycerol droplets. The experimental and numerical results for the time at maximum spreading are shown in Fig. 10 for the different liquids and impact velocities. Good agreement is achieved for

19


Langmuir

(a)

6

(b)

6

5

5

4

4

3

3

Sim. Ethanol Sim. Water Sim. Glycerol Exp. Ethanol Exp. Water Exp. Glycerol

max

max

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 20 of 30

2

1 0.1

2


1

1

10

1

Vi (m/s)

10

100

1000

Weber number

Figure 9: Maximum spreading ratio of droplets of the three liquids versus impact velocity (a) and versus Weber number (b). Experimental results empty symbols, simulation results in line and full symbols. (Futher see Fig. S1 in Supplementary information for rescaled plotting proposed by Clanet et al. 25 ) ethanol and glycerol. The simulation results for water overpredict the measurement results, which is attributed to the overprediction of the maximum spreading as shown in Figs. 4a and 9. A slight overprediction in maximum spreading, as seen in Fig. 4a results in an overprediction of the time at maximum spreading. Also the simple analytical model tmax = 8D0 /3Vi as suggested by Pasandideh-Fard et al. 21 assuming the spherical drop spreading into a cylindrical disk is shown. This simple model clearly does not predict the slope correctly and also does not predict the dependence of the time at maximum spreading on fluid properties. In Fig. 11 the measured and simulated dynamic contact angles at maximum spreading θD (tmax ) as a function of impact velocity Vi are compared. The dynamic contact angle at maximum spreading is found to be quite constant as a function of impact velocity and is reported in Table 2. The dynamic contact angles at maximum spreading determined by simulation and compared with the experimental ones show a good agreement for ethanol and glycerol, but less agreement for water. The reason for this poor agreement is obviously that the time at maximum spreading for water is overpredicted, as explained above in Fig. 10. Since the dynamic contact angle is decreasing during the spreading phase, an overprediction of the

20


Page 21 of 30


10

tmax (ms)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

1

Pasandideh-Fard et al. 0.1

1

10

Vi (m/s)

Figure 10: Time at maximum spreading determined experimentally and with simulation, and comparison with model by Pasandideh-Fard et al. 21 (green dash line) time at maximum spreading results in an underprediction of the dynamic contact angle at maximum spreading. To test this explanation, we also determined the simulated dynamic contact angle at the experimental time of maximum spreading. These results are represented in Fig. 11 by the dashed red line and, as can be seen, these results coincide very well with our measured data. This means that our underestimation of the dynamic contact angle at maximum spreading can be attributed to the overestimation of time at maximum spreading for water. Spreading is governed by the balance between kinetic, capillary, viscous energy and energy loss at contact line. In energy balance approaches commonly used to predict the maximum spreading the energy budget in function of impact velocity is studied. 35 Therefore we study the energy partition at maximum spreading in more detail in Figs. 12a-c. We limit our numerical analysis to the maximum impact velocity where splashing occurs, as observed in the experiments. Figure 12d shows the relative proportion of the different energy terms. From Fig. 12d it is clear that the remaining kinetic energy at maximum spreading is almost zero

21


Langmuir

180

150

Sim. Ethanol Sim. Water Sim. Glycerol (Exp. tmax) for water D

Exp. Ethanol Exp. Water Exp. Glycerol

(tmax) (°)

120

90

D

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 22 of 30

60

30

0 0.1

1

10

Vi (m/s)

Figure 11: Dynamic contact angle at maximum spreading determined experimentally and with simulation versus impact velocity. (maximal 4% of total energy). The surface energy and viscous dissipation energy increase with impact velocity, which can be directly related to the increase in initial kinetic energy, resulting in an increase in maximum spreading. Interestingly, the relative proportion of the viscous dissipation in the total energy budget increases with impact velocity with respect to surface energy. This would justify the assumption that, in the limit of high impact velocity, surface energy does not play a role, and spreading is only governed by viscous dissipation. However, our analysis shows that the contribution of surface energy only decreases very slowly, and that just before splashing, we are still in the regime where both surface energy and viscous dissipation have to be taken into account. On the other side, our results show also that, at low velocity, viscous dissipation can still play an important role, such as for ethanol.

22


Page 23 of 30

D

[

[

E .( 6(

[

.( 6( [

6(

6(

.(

.(

: YLV

: YLV

[

(QHUJ\ -

(QHUJ\ -

[ [ [

[ [

[

[

9L P V

F

9L P V

[

.( 6( : YLV

G .( 6(

[

(WKDQRO (WKDQRO (WKDQRO

:DWHU :DWHU :DWHU

*O\FHURO *O\FHURO *O\FHURO

6(

3URSRUWLRQ RI HQHUJ\ WHUP

.( [

(QHUJ\ -

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

: YLV

[ [ [

9L P V

9L P V

Figure 12: Amount (a-c) and proportion (d) of energy terms at maximum spreading as a function of impact velocity for different liquids on a steel surface: (a) ethanol, (b) water and (c) glycerol.

Conclusions Drop impact has been experimentally studied by high-speed imaging and by CFD-VOF simulation for three liquids with different viscosity and surface tension on steel surface. The spreading of drop impact is analyzed from low to high impact velocity, thereby focusing on the spreading diameter, dynamic contact angle and time at maximum spreading. The CFD simulations are validated against experiments for ethanol, water and glycerol in all range of impact velocity. It is concluded that the dynamic contact angle at maximum spreading is a suitable choice for the input contact angle of the Kistler model for the spreading phase providing good agreement between simulations and experiments. This dynamic contact angle is found to give better agreement than the quasi-static advancing contact angle, since the former captures better the dynamic wetting behavior during spreading. Deviations

23


Langmuir

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 24 of 30

between experiments and measurements are attributed mainly to the sensitivity of the CFD model on the contact angle input to the Kistler mode used as boundary condition at the contact line. Comparison of CFD results with analytical model predictions allowed us to validate the adequacy of analytical equations for average boundary layer thickness and time at maximum spreading, as commonly used in energy balance models for predicting maximum spreading. The analytical model for the time averaged boundary layer thickness is found to predict a similar dependence on impact velocity compared to the CFD simulations, but the value of boundary thickness is overpredicted by the analytical solution. Experiments showed that the time at maximum spreading decreases with impact velocity depending on the surface tension of the liquid. The CFD results adequately predict these experimental dependencies, while the commonly used analytical model does not predict the dependence on impact velocity correctly and does not include a dependence on surface tension. It is remarked that these analytical expressions are commonly used in energy balance models, meaning the validity of the energy balance models has to be reconsidered and updated. We documented the energy budget of drop impact as a function of impact velocity, i.e. the partition between kinetic energy, surface tension energy and cumulative viscous dissipation at maximum spreading versus impact velocity. It is found that at low impact velocity viscous dissipation contributes still for a more substantial part to the energy budget and cannot neglected in the prediction of maximum spreading. At high impact velocity, it was demonstrated that just before splashing, the spreading regime is both dependent on surface energy and viscous dissipation.

Acknowledgement The authors acknowledge the support of Swiss National Science Foundation grant no. 200021 135510.

24


Page 25 of 30

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

References (1) McDonald, A.; Lamontagne, M.; Moreau, C.; Chandra, S. Impact of plasma-sprayed metal particles on hot and cold glass surfaces. Thin Solid Films 2006, 514, 212–222. (2) Aziz, S. D.; Chandra, S. Impact, recoil and splashing of molten metal droplets. Int. J. Heat Mass Transfer 2000, 43, 2841–2857. (3) Liu, H. Science and Engineering of Droplets. Noyes Publications 2000, 1–534. (4) Derby, B. Inkjet printing of functional and structural materials: fluid property requirements, feature stability, and resolution. Annu. Rev. Mater. Res. 2010, (5) Visser, C. W.; Frommhold, P. E.; Wildeman, S.; Mettin, R.; Lohse, D.; Sun, C. Dynamics of high-speed micro-drop impact: numerical simulations and experiments at frame-to-frame times below 100 ns. Soft Matter 2015, 11, 1708–1722. (6) Wirth, W.; Storp, S.; Jacobsen, W. Mechanisms controlling leaf retention of agricultural spray solutions. Pestic. Sci. 1991, 33, 411–420. (7) Bergeron, V.; Bonn, D.; Martin, J. Y.; Vovelle, L. Controlling droplet deposition with polymer additives. Nature 2000, 405, 772–775. (8) Abuku, M.; Janssen, H.; Poesen, J.; Roels, S. Impact, absorption and evaporation of raindrops on building facades. Building and Environment 2009, 44, 113–124. (9) Joung, Y. S.; Buie, C. R. Aerosol generation by raindrop impact on soil. Nat. Commun. 2015, 6, 6083. (10) Zhao, R.; Zhang, Q.; Tjugito, H.; Cheng, X. Granular impact cratering by liquid drops: Understanding raindrop imprints through an analogy to asteroid strikes. Proc. Natl. Acad. Sci. 2015, 112, 342–347.

25


Langmuir

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(11) Zhao, S.-C.; de Jong, R.; van der Meer, D. Raindrop impact on sand: a dynamic explanation of crater morphologies. Soft Matter 2015, 11, 6562–6568. (12) Worthington, A. M. On the forms assumed by drops of liquids falling vertically on a horizontal plate. Proc. R. Soc. London 1876, 25, 261–272. (13) Furmidge, C. Studies at phase interfaces. I. The sliding of liquid drops on solid surfaces and a theory for spray retention. J. Colloid Sci. 1962, 17, 309–324. (14) Ford, R. E.; Furmidge, C. Impact and spreading of spray drops on foliar surfaces; Soc Chem Ind Monogr, 1967. (15) Collings, E. W.; Markworth, A. J.; McCoy, J. K.; Saunders, J. H. Splat-quench solidification of freely falling liquid-metal drops by impact on a planar substrate. J. Mater. Sci. 1990, 25, 3677–3682. ˇ ˇ Marengo, M.; Tropea, C.; Ganić, E. N. Analysis of impact of droplets on (16) Sikalo, S.; horizontal surfaces. Exp. Therm. Fluid Sci. 2002, 25, 503–510. (17) Rioboo, R.; Marengo, M.; Tropea, C. Time evolution of liquid drop impact onto solid, dry surfaces. Exp. Fluids 2002, 33, 112–124. (18) Kannan, R.; Sivakumar, D. Impact of liquid drops on a rough surface comprising microgrooves. Exp. Fluids 2008, 44, 927–938. (19) Chandra, S.; Avedisian, C. Observations of droplet impingement on a ceramic porous surface. Int. J. Heat Mass Transfer 1992, 35, 2377–2388. (20) Scheller, B. L.; Bousfield, D. W. Newtonian drop impact with a solid surface. AIChE J. 1995, 41, 1357–1367. (21) Pasandideh-Fard, M.; Qiao, Y. M.; Chandra, S.; Mostaghimi, J. Capillary effects during droplet impact on a solid surface. Phys. Fluids 1996, 8, 650. 26


Page 26 of 30

Page 27 of 30

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

(22) Mao, T.; Kuhn, D.; Tran, H. Spread and rebound of liquid droplets upon impact on flat surfaces. AIChE J. 1997, 43, 2169–2179. (23) Kim, H.; Chun, J. The recoiling of liquid droplets upon collision with solid surfaces. Phys. Fluids 2001, 13, 643–659. (24) Ukiwe, C.; Kwok, D. Y. On the Maximum Spreading Diameter of Impacting Droplets on Well-Prepared Solid Surfaces. Langmuir 2004, 21, 666–673. (25) Clanet, C.; Béguin, C.; Richard, D.; Quéré, D. Maximal deformation of an impacting drop. J. Fluid Mech. 2004, 517, 199–208. (26) Lee, J. B.; Lee, S. H. Dynamic wetting and spreading characteristics of a liquid droplet impinging on hydrophobic textured surfaces. Langmuir 2011, 27, 6565–6573. (27) Pasandideh-Fard, M.; Aziz, S. D.; Chandra, S.; Mostaghimi, J. Cooling effectiveness of a water drop impinging on a hot surface. Int. J. Heat Fluid Flow 2001, 22, 201–210. (28) Pasandideh-Fard, M.; Bhola, R.; Chandra, S. Deposition of tin droplets on a steel plate: simulations and experiments. Int. J. Heat Mass Transfer 1998, 41, 2929–2945. (29) Malgarinos, I.; Nikolopoulos, N.; Marengo, M.; Antonini, C.; Gavaises, M. VOF simulations of the contact angle dynamics during the drop spreading: Standard models and a new wetting force model. Adv. Colloid Interface Sci. 2014, 212, 1–20. (30) Fukai, J.; Shiiba, Y.; Yamamoto, T.; Miyatake, O.; Megaridis, C. M.; Zhao, Z. Wetting effects on the spreading of a liquid droplet colliding with a flat surface: Experiment and modeling. Phys. Fluids 1995, 7, 236–247. (31) Bussmann, M.; Chandra, S.; Mostaghimi, J. Modeling the splash of a droplet impacting a solid surface. Phys. Fluids 2000, 12, 3121. (32) Bussmann, M.; Mostaghimi, J.; Chandra, S. On a three-dimensional volume tracking model of droplet impact. Phys. Fluids 1999, 11, 1406. 27


Langmuir

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

ˇ ˇ Wilhelm, H. D.; Roisman, I. V.; Jakirlić, S.; Tropea, C. Dynamic contact (33) Sikalo, S.; angle of spreading droplets: Experiments and simulations. Phys. Fluids 2005, 17, 1–13. (34) Farhangi, M. M.; Graham, P. J.; Choudhury, N. R.; Dolatabadi, A. Induced Detachment of Coalescing Droplets on Superhydrophobic Surfaces. Langmuir 2012, 28, 1290–1303. (35) Yarin, A. L. Drop impact dynamics: splashing, spreading, receding, bouncing. . . . Annu. Rev. Fluid Mech. 2006, 38, 159–192. (36) Marengo, M.; Antonini, C.; Roisman, I. V.; Tropea, C. Drop collisions with simple and complex surfaces. Curr. Opin. Colloid Interface Sci. 2011, 16, 292–302. (37) Vadillo, D. C.; Soucemarianadin, A.; Delattre, C.; Roux, D. C. D. Dynamic contact angle effects onto the maximum drop impact spreading on solid surfaces. Phys. Fluids 2009, 21, 1–8. (38) Li, X.-H.; Zhang, X.-X.; Chen, M. Estimation of viscous dissipation in nanodroplet impact and spreading. Phys. Fluids 2015, 27, 052007. (39) Roisman, I. V.; Berberović, E.; Tropea, C. Inertia dominated drop collisions. I. On the universal flow in the lamella. Phys. Fluids 2009, 21, 052103. (40) Attané, P.; Girard, F.; Morin, V. An energy balance approach of the dynamics of drop impact on a solid surface. Phys. Fluids 2007, 19, 012101. (41) Nikolopoulos, N.; Theodorakakos, A.; Bergeles, G. A numerical investigation of the evaporation process of a liquid droplet impinging onto a hot substrate. Int. J. Heat Mass Transfer 2007, 50, 303–319. (42) Strotos, G.; Aleksis, G.; Gavaises, M.; Nikas, K. S.; Nikolopoulos, N.; Theodorakakos, A. Non-dimensionalisation parameters for predicting the cooling effectiveness of droplets impinging on moderate temperature solid surfaces. Int. J. Therm. Sci. 2011, 50, 698– 711. 28


Page 28 of 30

Page 29 of 30

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

(43) Yokoi, K.; Vadillo, D.; Hinch, J.; Hutchings, I. Numerical studies of the influence of the dynamic contact angle on a droplet impacting on a dry surface. Phys. Fluids 2009, 21, 072102. (44) Caviezel, D. .; Narayanan, C. .; Lakehal, D. . Adherence and bouncing of liquid droplets impacting on dry surfaces. Microfluid. Nanofluid. 2008, 5, 469–478. (45) Griebel, M.; Klitz, M. Singular Phenomena and Scaling in Mathematical Models; Springer International Publishing: Cham, 2014; pp 297–325. (46) Kistler, S. F. Hydrodynamics of wetting; In Wettability: Marcel Dekker: New York, 1993; pp 311–429. (47) Roisman, I. V.; Opfer, L.; Tropea, C.; Raessi, M.; Mostaghimi, J.; Chandra, S. Drop impact onto a dry surface: Role of the dynamic contact angle. Colloids Surf., A 2008, 322, 183–191. (48) Biolè, D.; Bertola, V. A goniometric mask to measure contact angles from digital images of liquid drops. Colloids Surf., A 2015, 467, 149–156. (49) Rusche, H. Computational fluid dynamics of dispersed two-phase flows at high phase fractions. Ph.D. thesis, Imperial College London (University of London), 2003. (50) Qian, T.; Wang, X.-P.; Sheng, P. A variational approach to moving contact line hydrodynamics. J. Fluid Mech. 2006, 564, 333–360. (51) Chandra, S.; Avedisian, C. T. On the collision of a droplet with a solid surface. Proc. R. Soc. Lond. A 1991, 432, 13–41.

29


Langmuir

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 30 of 30

TOC Pure ethanol

Deionized water

Vi = 0.3 m/s

tmax = 11.3 ms

Vi = 0.7 m/s

1 m/s tmax = 6.8 ms

Vi = 1.6 m/s Vi = 2.4 m/s

Exp.

1:1.3 glycerol-water mixture


CFD sim. tmax = 4.6 ms


Vi =1.7 m/s


tmax = 1.6 ms

Vi =3.2 m/s


tmax = 1.1 ms

tmax = 4.0 ms

30


Energy Budget of Liquid Drop Impact at Maximum ... - ACS Publications

Recommend Documents