- Email: [email protected]
A numerical study on the dynamic behavior of the liquid droplet located on heterogeneous surface
Computers & Fluids 105 (2014) 294–306
Contents lists available at ScienceDirect
Computers & Fluids j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / c o m p fl u i d
A numerical study on the dynamic behavior of the liquid droplet located on heterogeneous surface Sung Wan Son a, Chan Hyeok Jeong b, ManYeong Ha a,⇑, Hyung Rak Kim a a b
School of Mechanical Engineering, Pusan National University, San 30, Jang Jeon Dong, Geum Jeong Gu, Busan 609-735, Korea Hyundai Heavy Industries, Bangeojin Ring Road 1000, Dong-gu, Ulsan 682-792, Korea
a r t i c l e
i n f o
Article history: Received 31 March 2014 Received in revised form 15 September 2014 Accepted 17 September 2014 Available online 28 September 2014 Keywords: Droplet behavior Heterogeneous surface Droplet separation Lattice Boltzmann method
a b s t r a c t The behavior of a droplet located on a heterogeneous surface is numerically analyzed using the 3-D Lattice Boltzmann Method. Predictions are made whether the droplet will be separated or not and the separation time is calculated changing the contact angles of the hydrophilic surface, hydrophobic surface and the area of the hydrophobic surface. The change in the contacting length between the droplet and the bottom surface and the change in the dynamic contact angle of the droplet on the cross section at the center of the droplet as time passes are observed, and the droplet separation mechanism is investigated by analyzing the velocity vector around the phase boundary line and the droplet which changes with time while the shape of the droplet changes. A droplet located at the center of a heterogeneous surface shows a trend of being easily separated if the difference between the contact angles of the hydrophilic and hydrophobic surfaces is bigger. The more the contact angle of the hydrophilic surface increases, the more the droplet separation time increases, and the more the contact angle of the hydrophobic surface increases, the more the droplet separation time decreases. Also, though the droplet separation time shows a decreasing trend when the width of the hydrophobic surface increases, it does not have any effect on whether the droplet will be separated or not. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction Multiphase flow is an issue which attracts people’s attention in diverse fields such as engineering, industry and environment as well as natural science including chemistry and biology. Being an important part in various fields from natural phenomena such as rainstorm, smog and snowstorm to industrial fields such as air conditioning system, refrigerating system, sea water desalination plant, nuclear power generation equipment, and various heat exchangers, it has attracted attention of many researchers. Studies on behaviors of droplets among multiphase flow issues are actively conducted experimentally or numerically with recent development of micro/nanotechnology. Tervino et al. [1], Yang and Koplik [2], and Gu and Li [3] conducted numerical studies on the phenomenon of a droplet spreading over a solid surface in a stationary state. Also, Bayer and Megaridis [4], Cossali et al. [5], Mao et al. [6], Sikalo et al. [7], and Zhang and Basaran [8] experimentally studied the behavior of a droplet crashing onto a solid surface. And Fujimoto et al. [9], and Manservisi and Scardovelli [10] conducted numerical studies on the dynamic behavior of a droplet ⇑ Corresponding author. Tel.: +82 51 510 2440; fax: +82 51 515 3101. E-mail address: [email protected] (M. Ha). http://dx.doi.org/10.1016/j.compfluid.2014.09.033 0045-7930/Ó 2014 Elsevier Ltd. All rights reserved.
colliding with a solid surface. Tanaka et al. [11] and Lunkad et al. [12] conducted numerical studies on the kinetic characteristics of a droplet moving on a slanted solid surface or on a solid surface to which a force was applied, and compared the result of the numerical analyses with the experimental result of Sikalo et al. [13]. Boltzmann Method (LBM) is a computational analysis technique which recently attracts people’s attention by reason of its fast computation speed, multiphase analysis, and easiness of parallelization. Many studies have been conducted using the Lattice Boltzmann Method, and Shan and Chen [14] in particular presented a model capable of analyzing the multiphase flow of different fluids which do not mix with each other at the same temperature using the Lattice Boltzmann Method. Swift et al. [15] introduced a free energy approach into the multiphase flow model, and Gunstensen et al. [16] proposed a method of analyzing a two-phase flow using a multiple density distribution function. Wu et al. [17] presented a model of which the pressure disturbance was corrected at the phase boundary based on the research findings of Gunstensen et al. [16]. In this study, the behavior of a hemispherical droplet located on a heterogeneous surface was studied using the 3D Lattice Boltzmann Method. A hydrophobic bottom surface exists at the
295
S.W. Son et al. / Computers & Fluids 105 (2014) 294–306
center of a hydrophilic bottom surface, and a hemispherical droplet is located at the center of the bottom surface lying from the hydrophilic bottom surface to the hydrophobic bottom surface. The droplet is separated or reaches a new equilibrium state without being separated depending on the contact angles of the hydrophilic and hydrophobic bottom surfaces. The time at which the droplet is separated differs depending on the width of the hydrophobic bottom surface, which also has an effect on whether or not the droplet will be separated. In this study, an investigation was carried out as to whether the droplet is separated or not changing the contact angle of the heterogeneous surfaces with different hydrophobic widths, and the change in the contacting length between the bottom surface and the droplet was presented. In the case the droplet was separated, the separation time of the droplet depending on the hydrophobic width and the contact angle of the bottom surface was calculated, and the dynamic contact angle between the bottom surface and the droplet which changes with time while the droplet is separated was calculated.
Fig. 1. The D3Q19 lattice model.
Accordingly, the Lattice Boltzmann Equation can be expressed as Eq. (4) below:
2. Numerical analysis method
k
k
f i ðx þ ei ; t þ dtÞ f i ðx; tÞ ¼
2.1. Lattice Boltzmann Method
1
sk
k
kðeqÞ
ðf i f i
Þðx;tÞ kðeqÞ
In this study, a 3D analysis was conducted on a droplet located on a heterogeneous surface using the Lattice Boltzmann Method. The governing equation was induced from the Boltzmann equation, and the application scope of the existing Lattice Gas Automata (LGA) governing equation was widened and its numerical stability was improved by expanding it to the real number range. The discretized Boltzmann Equation for transportation of incompressible two-phase fluid particles is as follows: k
@f i k þ ei rf i ¼ Xki @t
ð1Þ
k
Here, f i represents the density distribution function of the fluid particle, and the superscript k means the relevant phase. ei means the lattice speed, and Xki means the collision operator. The subscript i represents the direction of the lattice, which is differently defined in accordance with the lattice model. Though the collision operator contains a very complicated mathematical mechanism, it can be simply expressed using the single relaxation time proposed by Bhatnahar et al. [18]. The discretized Boltzmann equation can be expressed as follows when simplified applying the BGK model [18]: k @f i
@t
k fi
þ ei r
¼
1
s
k ðf i k
Here, the equilibrium distribution function f i
ð2Þ
Here, sk represents the single relaxation time for the kth fluid, and kðeqÞ fi represents the equilibrium distribution function. The equilibrium distribution function represents the value of the equilibrium k state f i has in the relaxation time. The D3Q19 is a model considered for the fluid particle which can have a velocity in only 19 directions in a 3D space as shown in Fig. 1, and the lattice speed ei is given as follows:
is given as follows:
3 9 3 ¼ xi qk 1 þ 2 ðei uÞ þ 4 ðei uÞ2 2 u u c 2c 2c
kðeqÞ
fi
ð5Þ
Here, xi is a weighting coefficient, and u represents the fluid velocity. In the model D3Q19, the weighting coefficient is given being divided into x0 = 1/3, xi = 1/18 (i = 1, 2, 3, 4, 5, 6), and xi = 1/36 (i = 7, ..., 18) in accordance with the direction of the lattice speed. Eq. (4) used for the Lattice Boltzmann Method is alternately calculated being divided into the collision stage and flow stage, and has an advantage that, having a form of an explicit equation, the calculation speed is faster than the analysis which uses the Navier–Stokes equation discretized in the form of an implicit equation and calculations can be done in a microscopic area. Also, as the continuity equation and Navier–Stokes equation that govern flow of fluid can be induced from the BGK model based Lattice Boltzmann Method using the Chapman-Enskog theory, it can be confirmed that the Lattice Boltzmann Method can be also applied to macroscopic flow. The macroscopic flow variables such as density (q) and pressure (p) are defined as follows:
qk ¼ kðeqÞ fi Þ
ð4Þ
X
k
fi ¼
X kðeqÞ fi
i
q¼
X
qk
k
qu ¼
X i
p¼
ð6Þ
i
f i ei ¼
ð7Þ X
eq
f i ei
ð8Þ
i
1 2 qc 3
ð9Þ
Here, qk represents the fluid density of each phase. 2.2. Two-phase flow model
e0 ¼ ð0; 0; 0Þ e1;2 ; e3;4 ; e5;6 ¼ ð1; 0; 0Þc; ð0; 1; 0Þc; ð0; 0; 1Þc e7;...;10 ; e11;...;14 ; e15;...;18 ¼ ð1; 1; 0Þc; ð1; 0; 1Þc; ð0; 1; 1Þc ð3Þ At this time, c is defined to be dx/dt(c = dx/dt), and dx and dt represent the increments of space and time respectively. The Lattice Boltzmann Equation can be obtained by discretizing Eq. (2) for the unit time and lattice space which fall under lattice speed c.
The two-phase model used in this study is the model proposed by Gunstensen et al. [16] and modified by Wu et al. [17]. This twophase flow model is suitable to solve two-phase flows which do not mix with other fluids based on the lattice BGK model, and has a feature that there is no pressure disturbance at the phase boundary. In the case two phases exist, the CSF (Continuum Surface Force) model was used to apply the surface tension at the boundary
296
S.W. Son et al. / Computers & Fluids 105 (2014) 294–306
surface and to apply the pressure gradient. Wu et al. [17] defined a new variable ueq as follows, and replaced the fluid velocity u with it.
ueq ¼ u þ
s F q
ð10Þ
Here, s is the total relaxation time and can be obtained using P k k ks q /q. The force F resulting from the surface tension is defined as follows:
s=
F ¼ rj
rC jrCj
ð11Þ
Here, r is the surface tension coefficient, and j means the curved surface of the phase. C is the ratio of the fluid, which is defined as follows:
C ¼ qred qblue n j ¼ r jnj
ð12Þ ð13Þ
Here, qred means the density of the internal fluid, and qblue means the density of the external fluid. n means the vector perpendicular to the phase surface, and can be defined using n = rC. At this time, the following calculation process should be conducted to obtain the value j.
j ¼ r
n 1 n ¼ ðr nÞ þ r jnj jnj jnj jnj
ð14Þ
2.3. Two-phase flow model In order to verify the validity of the numerical analysis method and the two-phase flow calculation model used in this study, the change in the internal and external pressures of the droplet depending on the change in the curvature radius of the droplet was calculated using the analysis code developed in this study, and the result of the calculation was compared with the value obtained using the Young–Laplace equation which is an exact solution. The Young–Laplace equation is expressed as in Eq. (19) below:
Dp ¼ pred pblue ¼
2r R
ð19Þ
Here, pred and pblue represent the dimensionless pressure of the fluids different with each other, R represents the dimensionless
Fig. 2. Change in the dimensionless internal and external pressure difference Dp of the droplet depending on the change in the inverse number of the dimensionless curvature radius 1/R.
curvature radius of the internal fluid, and r means the dimensionless surface tension. Fig. 2 shows the change in the dimensionless internal and external pressure difference Dp of the droplet depending on the change in the inverse number of the dimensionless curvature radius 1/R. Here, the solid line represents the Young–Laplace equation shown in Eq. (19), and the rectangle represents the value obtained by numerical calculation of this study. When there is a pressure gradient on a round boundary surface of two stationary fluids resulting from the surface tension, the internal and external pressure difference can be calculated using Eq. (19). According to Eq. (19), the internal pressure at the round boundary surface of the droplet always has a value bigger than that of the external pressure. As the internal and external pressure difference Dp decreases, 1/R decreases, according to which the curvature at the boundary surface of the droplet grows bigger. Accordingly, if the internal and external pressure difference of the droplet gets very small (Dp ? 0), the radius of curvature at the boundary surface of the droplet grows infinitely, due to which the droplet approaches a flat surface. But, if the internal and external pressure difference of the droplet grows very big, the radius of curvature at the boundary surface of the droplet becomes very small. As shown in Fig. 2, the calculation result obtained using the numerical analysis method and the two-phase flow calculation model adopted in this study well corresponds with the exact solution obtained from the Young–Laplace equation given in Eq. (19). For the secondary verification of the validity of the numerical analysis method and the two-phase flow calculation model used in this study, a calculation was conducted adopting the question same as that of the study conducted by Yan and Zu [19] and the result was compared with the calculation result of Yan and Zu [19]. Yan and Zu [19] investigated the behavior of a droplet on a heterogeneous surface using the Lattice Boltzmann Method for which the D3Q15 model having 15 velocity components in a 3 dimensional space is adopted. Yan and Zu [19] conducted a calculation adopting a free-energy model to simulate a two-phase flow phenomenon. Fig. 3 shows the shape and computational domain adopted for verification of validity. Fig. 3(a) shows the shape in which a rectangular hydrophobic surface with a width of l and a longitudinal length of L exists at the center of a relatively hydrophilic square bottom surface with a side length of L. Fig. 3(b) shows the shape in which a rectangular hydrophobic surface with a width of l and a longitudinal length of L and a rectangular hydrophobic surface with a width of l and a traversal length of L intersect with each other at the center of a hydrophilic square bottom surface with a side length of L. h1 and h2 represent the contact angles of the hydrophilic and hydrophobic bottom surface, respectively. The values of h1 and h2 in Fig. 3(a) are 30° and 150° respectively, and the diameter and height of the hemispherical droplet located at the center of the bottom surface are D = 4 mm and H = 2 mm respectively. The values of h1 and h2 in Fig. 3(b) are 45° and 150° respectively, and the diameter and the height of the hemispherical droplet located at the center of the bottom surface are D = 4 mm and H = 1 mm respectively. Fig. 4 shows the dynamic behavior of the droplet at the moment when a rectangular hydrophobic surface exists at the center of a square hydrophilic bottom surface as shown in Fig. 3(a). Fig. 4(a) shows the dynamic behavior of the droplet obtained using the numerical analysis method of this study, and Fig. 4(b) shows the dynamic behavior of the droplet calculated by Yan and Zu [19]. As shown in Fig. 4, the dynamic behavior of the droplet obtained by the current numerical analysis method well reproduces the dynamic behavior of the droplet calculated by Yan and Zu [19]. Fig. 5 shows the change with time in the phase boundary surface where the droplet and the bottom surface contact each other when the two rectangular hydrophobic surfaces intersect with each other on the square hydrophilic bottom surface as shown in Fig. 3(b). The calculation result obtained
S.W. Son et al. / Computers & Fluids 105 (2014) 294–306
297
(a) Single cross hydrophobic strips
(b) Double cross hydrophobic strips Fig. 3. Initial droplet shape and computational domain adopted for verification of validity.
using the current calculation model and the numerical analysis method shown in Fig. 5(a) well reproduces the calculation result obtained by Yan and Zu [19] as shown in Fig. 5(b).
3. Result and discussion The behavior of a droplet located on a heterogeneous surface was established using the 3D Lattice Boltzmann Method after securing the validity of the numerical analysis method and two-phase flow calculation model used in this study through the verification calculations shown in Figs. 2, 4 and 5. For this, the dynamic behavior of a droplet located on such a heterogeneous surface was investigated considering a shape in which a rectangular hydrophobic surface with a width of l and a longitudinal length of L exists at the center of a hydrophilic square bottom surface with a side length of L. Here, the contact angle of the square hydrophilic bottom surface is h1, and the contact angle of the rectangular hydrophobic bottom surface is h2. The droplet is assumed to exist at the beginning in a hemispherical form with a diameter of D at the center of the bottom surface. The diameter and the height of the droplet used in this calculation are D = 3 mm and H = 1.5 mm, resulting in D/L = 0.375 and H/L = 0.1875, respectively. Fig. 6 shows the dynamic behavior of a droplet located at the center of the bottom surface when a rectangular hydrophobic surface with h2 = 150° and a width of ‘ð¼ 0:1LÞ exists at the center of the square hydrophilic surface with h1 = 30° and a side length of L.
The droplet which exists in a hemispherical shape lying across the rectangular hydrophobic surface at the beginning spreads out symmetrically with dimensionless time in the direction of the hydrophilic bottom surface which has a relatively big adhesion. The dimensionless time t in Fig. 6 is defined as follows:
sffiffiffiffiffiffiffiffi t ¼ t
r qL3
ð20Þ
where t*, r and q represent the dimensional time, surface tension and liquid phase density, respectively. The values of r and q used in this study are 0.01 N/m and 1 kg/m3, respectively. At t = 0.978, the droplet located on the hydrophobic bottom surface moves in the direction of the hydrophilic bottom surface causing a necking phenomenon where the width of the droplet located on the hydrophobic bottom surface decreases. Later, as time passes during the period of 0:978 6 t 6 21:103, the necking phenomenon continuously progresses making the width of the droplet located on the hydrophobic surface gradually decrease, and the droplet is separated into two droplets at t = 21.103. After the droplet is separated at t = 21.103, the whole droplet located on the hydrophobic surface moves to the hydrophilic area, as shown in Fig. 6 at t = 25.855. Fig. 7 shows the change with the dimensionless time in the phase boundary line of the droplet located at the center of the bottom surface and the change in the velocity vector around the droplet when a rectangular hydrophobic surface with h2 = 150° and a
298
S.W. Son et al. / Computers & Fluids 105 (2014) 294–306
t* = 0.0 s
t* = 0.015s
t* = 0.14 s
t* = 0.15s
t* = 0.152 s
t* = 0.154 s
(a) Present study
t* = 0.0 s
t* = 0.15s
t* = 0.015s
t* = 0.152 s (b) Previous research
t* = 0.14 s
t* = 0.154 s
Fig. 4. Dynamic behavior of the droplet at the moment when a rectangular hydrophobic surface exists at the center of a square hydrophilic bottom surface. (a) Present study, and (b) previous research.
width of ‘ ¼ 0:1L exists at the center of the square hydrophilic surface with h1 = 30° and a side length of L. It can be confirmed that, as the dimensionless time passes, the velocity component heading from the top part of the droplet to the bottom surface and the velocity component heading from the horizontal center of the droplet to the edges develop making the droplet spread out over the hydrophilic bottom surface. After t = 10.342, separation of the droplet starts making the phase boundary line of the horizontal center become lower than the boundary line of the surroundings. As separation of the droplet progresses, velocity vectors rotating clockwise and counterclockwise from the phase boundary of the horizontal center are generated. Fig. 8 shows the dynamic behavior of the droplet located at the center of the bottom surface when a rectangular hydrophobic surface with h2 = 130° and a width of ‘ ¼ 0:1L exists at the center of the square hydrophilic surface with h1 = 90° and a side length of L. As to the initial dynamic behavior of the droplet when h1 = 90° and h2 = 130°, the droplet which is in a hemispherical form symmetrically spreads out in the direction of the square bottom surface with h1 = 90° as in the case of h1 = 30° and h2 = 150°.
However, if h1 increases from 30° to 90°, the extent of the droplet spreading out in the direction of the square surface with h1 = 90° is considerably reduced in comparison to that of h1 = 30° due to reduction in the hydrophilic effect on the square surface. Therefore the necking phenomenon where the width of the droplet decreases as observed in the droplet located on the rectangular hydrophobic surface when h1 = 30° and h2 = 150° is not observed when h1 = 90° and h2 = 130°. For this reason, when h1 = 90° and h2 = 130°, spreading of the droplet rapidly slows down after t = 1.118 without the droplet being separated, and a new state of equilibrium is maintained after t = 5.171. Fig. 9 shows the change with the dimensionless time in the phase boundary line of the droplet located at the center of the bottom surface and the change in the velocity vector around the droplet when a rectangular hydrophobic surface with h2 = 130 and a width of ‘ ¼ 0:1L exists at the center of the square hydrophilic surface with h1 = 90° and a side length of L. As the dimensionless time passes during 0 6 t 6 2:096, the phase boundary line of the droplet symmetrically spreads out to the right and left of the hydrophilic bottom surface. However, at t P 5:171, it
S.W. Son et al. / Computers & Fluids 105 (2014) 294–306
(a) Present study
(b) Previous research Fig. 5. Change with time in the phase boundary surface for double cross hydrophobic strips.
reaches a new state of equilibrium without having almost any change in the phase boundary line of the droplet. Due to the existence of the velocity component heading from the top part of the droplet to the bottom surface and the velocity component heading from the horizontal center of the droplet to the sides, the height of the phase boundary located at the top of the droplet gradually drops as the dimensionless time passes. But, though the height of the phase boundary line at the center of the droplet has the highest value in comparison to those of the phase boundary lines of the surroundings at 0 6 t 6 2:096, the height of the phase boundary line at the center of the droplet becomes a little lower than those of the phase boundary lines of the surroundings at t P 5:171. Fig. 10 shows the change in the dimensionless separation time of the droplet depending on the contact angle of the square hydrophilic surface h1 and the contact angle of the rectangular
299
hydrophobic surface h2 when ‘ ¼ 0:1L. The shape of the droplet formed on the rectangular hydrophobic surface becomes more convex because the contact angle of the droplet contacting the surface of the hydrophobic surface grows as the contact angle of the rectangular hydrophobic surface h2 increases, due to which the dimensionless separation time of the droplet deceases. As the contact angle of the square hydrophilic surface h1 increases, the intensity of the hydrophilicity of the hydrophilic surface decreases, due to which the spreading force in the direction of the hydrophilic surface becomes weaker greatly increasing the dimensionless separation time of the droplet in comparison to the case where the contact angle of the hydrophilic surface h1 is small. Fig. 11 shows the shape of the droplet looked from the side and the top when the droplet is separated at h2 = 150° for different values of h1 = 10°, 30° and 50°. Fig. 11(a) shows the shape of the droplet looked from the side and the top of the droplet when the droplet is separated at t = 13.556 under the conditions of h1 = 10° and h2 = 150°. As the intensity of the hydrophilicity of the square hydrophilic surface is very high at h1 = 10°, the droplet widely spreads out from the center of the droplet where the hydrophobic surface is located in the right/left direction (x) and the up/down direction (y) along the surrounding hydrophilic surface, due to which the contact surface between the droplet and the bottom surface is widely formed. The phase boundary surface of the separated droplet assumes a convex shape near the rectangular hydrophobic surface. Therefore, the cross section of the droplet perpendicular to x increases at first as the point moves from x = 0 to the right and left in the direction of x and decreases after reaching the maximum value. Fig. 11(b) shows the shapes of the separated droplets looked from the side and the top when the droplet is separated at t = 21.103 under the conditions of h1 = 30° and h2 = 150°. Similarly, Fig. 11(c) shows the shapes of the separated droplets looked from the side and the top when the droplet is separated at t = 43.603 under the conditions of h1 = 50° and h2 = 150°. As to the shape of the droplet separated when h1 = 30° or h1 = 50° at h2 = 150°, the droplet widely spreads out from the center of the droplet where the rectangular hydrophobic surface is located in the right/left and the up/down directions similarly to that of h2 = 150° and h1 = 10°, due to which the cross section of the droplet perpendicular to x increases at first as the point moves from x = 0 to the right and left in the direction of x and decreases after reaching the maximum value. However, if the contact angle of the hydrophilic surface h1 increases from 10° to 30° and 50°, as the hydrophilicity of the hydrophilic surface decreases, the extent of the droplet spreading out from x = 0 in the right/left and up/down directions decreases as h1 increases. Accordingly, if h1 increases from 10° to 30° and 50° when h2 = 150°, the contact area between the droplet and the bottom surface decreases. Also, as h1 increases, the location where the droplet cross section perpendicular to x has the maximum value moves further to the right and left from x = 0, and the height of the droplet at the location increases along with h1. Fig. 12 shows the change in the dimensionless wet length between the droplet and the bottom surface at the central cross section of the droplet (y = 0) for different h1 values varying in the range from h1 = 10° to h1 = 90° when the contact angle of the hydrophobic surface is h2 = 150°. The dimensionless wet length (W/L) is defined as the contact length between the droplet and the bottom surface at the central cross section of the droplet, W, divided by the length of a square surface, L. When h1 = 10°, the dimensionless contact length between the droplet and the bottom surface rapidly increases as the droplet spreads from the hydrophobic surface to the hydrophilic surface during 0 6 t 6 10:342, and abruptly decreases in a moment at t = 10.342 as the droplet is separated. As the dimensionless time passes, the droplet spreads out continuously moving to the hydrophilic area, due to which the
300
S.W. Son et al. / Computers & Fluids 105 (2014) 294–306
t = 0.014
t = 0.419
t = 0.978
t = 2.096
t = 5.171
t = 10.342
t = 21.103
t = 25.855
Fig. 6. Dynamic behavior of a droplet located at the center of the bottom surface when a rectangular hydrophobic surface with h2 = 150° and a width of ‘ð¼ 0:1LÞ exists at the center of the square hydrophilic surface with h1 = 30° and a side length of L.
t = 0.014
t = 0.419
t = 0.978
t = 2.096
t = 5.171
t = 10.342
t = 21.103
t = 25.855
Fig. 7. Change with the dimensionless time in the phase boundary line and the velocity vector around the droplet when a rectangular hydrophobic surface with h2 = 150° and a width of ‘ ¼ 0:1L exists at the center of the square hydrophilic surface with h1 = 30° and a side length of L.
dimensionless contact length between the droplet and the bottom surface continuously increases after t = 10.342 and come to have a constant value after the droplet reaches a new equilibrium state after t = 36.196.
When h1 increases from 10° to 30°, the hydrophilicity of the hydrophilic surface decreases, due to which separation of the droplet when h1 = 30° is shown to be slower than when h1 = 10°. Therefore, as the droplet spreads out over the hydrophilic surface
301
S.W. Son et al. / Computers & Fluids 105 (2014) 294–306
t = 0.014
t = 0.14
t = 0.419
t = 0.978
t = 1.118
t = 2.096
t = 5.171
t = 10.342
Fig. 8. Dynamic behavior of a droplet located at the center of the bottom surface when a rectangular hydrophobic surface with h2 = 130° and a width of ‘ ¼ 0:1L exists at the center of the square hydrophilic surface with h1 = 90° and a side length of L.
t = 0.014
t = 0.14
t = 0.419
t = 0.978
t = 1.118
t = 2.096
t = 5.171
t = 10.342
Fig. 9. Change with the dimensionless time in the phase boundary line and the velocity vector around the droplet when a rectangular hydrophobic surface with h2 = 130° and a width of ‘ ¼ 0:1L exists at the center of the square hydrophilic surface with h1 = 90° and a side length of L.
during 0 6 t 6 16:072, the dimensionless contact length between the droplet and the bottom surface increases. However, as shown in Fig. 6, the contact length between the droplet and the bottom
surface decreases as the dimensionless time passes due to the necking phenomenon which occurs during 16:072 6 t 6 19:566 where the width of the droplet located on the hydrophobic bottom
302
S.W. Son et al. / Computers & Fluids 105 (2014) 294–306
Fig. 10. Change in the dimensionless separation time of the droplet depending on the contact angle of the square hydrophilic surface, h1, and the contact angle of the rectangular hydrophobic surface, h2.
surface decreases, and the contact length reaches the minimum value as the droplet is separated into two droplets at t = 19.566. After t = 19.566, the separated droplet continuously spreads out again in the right/left and up/down directions increasing the contact length between the droplet and the bottom surface, which comes to have a constant value after the droplet reaches a new equilibrium state after t = 43.464. When h1 increases further from 30° to 50°, the hydrophilicity of the hydrophilic surface decreases more, due to which separation of the droplet when h1 = 50° is shown to be much slower than when h1 = 30° or h1 = 10°. Therefore, as the droplet spreads out to the hydrophilic surface without being separated during 0 6 t 6 33:541, the dimensionless contact length increases. After that, as the dimensionless time passes during 33:541 6 t 6 37:175, the dimensionless contact length between the droplet and the bottom surface decreases due to necking phenomenon. The dimensionless contact length between the droplet and the bottom surface reaches the minimum value as the droplet is separated into two at t = 37.175. Similar to the case of h1 = 30°, the droplets separated
Fig. 12. Change in the dimensionless wet length between the droplet and the bottom surface at the central cross section of the droplet (y = 0) for different h1 values varying in the range from h1 = 10° to h1 = 90° when h2 = 150°.
into two after t = 37.175 continuously spread out on the hydrophilic surface in the right/left and up/down directions, increasing again the dimensionless contact length between the droplet and the bottom surface, which comes to have a constant value after the droplet reaches a new equilibrium state at t P 62:051. When h1 increases further from 70° to 90°, the hydrophilicity of the hydrophilic surface becomes very small, due to which the droplet spreads out to the hydrophilic surface without being separated and reaches a new equilibrium state in a short time. Therefore, when h1 = 70° or h1 = 90°, the dimensionless contact length between the droplet and the bottom surface increases during 0 6 t 6 3:494, and comes to have a constant value after the droplet reaches a new equilibrium state after t = 3.494. After t = 3.494, the dimensionless contact length between the droplet and the bottom surface when h1 = 70° comes to have the almost same value as that of h1 = 90°. The dimensionless contact length between the droplet and the bottom surface decreases after the droplet reaches a new equilibrium state due to the hydrophilicity of the hydrophilic surface decreasing as h1 increases.
(a) θ1 = 10°, t = 13.556
(b) θ1 = 30°, t = 21.103
(c) θ1 = 50°, t = 43.603 Fig. 11. Shape of the droplet looked from the side and the top when the droplet is separated at h2 = 150° for different values of (a) h1 = 10°, (b) h1 = 30°, and (c) h1 = 50°.
303
S.W. Son et al. / Computers & Fluids 105 (2014) 294–306
Fig. 13. Change in the contact angle between the droplet and the bottom surface as a function of the dimensionless time for different h1 values varying in the range from h1 = 10° to h1 = 90° when the contact angle of the hydrophobic surface is h2 = 150°.
Fig. 15. Change in the dimensionless separation time of the droplet depending on the contact angle of the square hydrophilic surface, h1, and the contact angle of the rectangular hydrophobic surface, h2, when ‘ ¼ 0:15L.
Fig. 13 shows the change in the contact angle between the droplet and the bottom surface as a function of the dimensionless time for different h1 values varying in the range from h1 = 10° to h1 = 90° when the contact angle of the hydrophobic surface is h2 = 150°. When h1 = 10°, the dynamic contact angle between the droplet and the bottom surface rapidly decreases as the droplet quickly spreads from the hydrophobic surface to the hydrophilic surface due to the big difference between the contact angles of the hydrophobic surface and the hydrophilic surface. As the dimensionless time passes, the rate of change in the dynamic contact angle of the droplet decreases, and the dynamic contact angle maintains a value of 10 after t = 23.898 when the droplet is divided into two and finishes its movement to the hydrophilic surface. As the dimensionless time passes, the variation pattern in the dynamic contact angle of the droplet at h1 = 30° is generally similar to that at h1 = 10°. However, as h1 increases from 10° to 30°, the hydrophilicity of the hydrophilic surface becomes smaller and, as a result, the dynamic contact angle of the droplet at h1 = 30°
changes more slowly than that at h1 = 10°, as the dimensionless time passes. The rate of change in the dynamic contact angle of the droplet decreases as the dimensionless time passes as is the case of h1 = 10°. The dynamic contact angle maintains a value of 30° after the droplet is separated into two and finishes its movement to the hydrophilic surface. When h1 = 50°, the droplet is separated into two, similar to that when h1 = 10° and h1 = 30°. Thus, when h1 = 50°, as the dimensionless time passes, the dynamic contact angle of the droplet rapidly decreases at the beginning and reaches the constant value of 30° after the droplet is separated into two and finishes its movement to the hydrophilic surface. However, when h1 = 70° and h1 = 90°, the droplet is not separated into two and reaches a new equilibrium state. Thus, as the dimensionless time passes, the dynamic contact angles for h1 = 70° and h1 = 90° decrease at the beginning and reach the new equilibrium state with constant contact angle values of 63° and 72°, respectively, which are different from the contact angles of the hydrophilic surface.
t = 0.014
t = 0.419
t = 1.118
t = 2.096
t = 5.171
t = 10.342
t = 13.975
t = 15.513
Fig. 14. Dynamic behavior of a droplet located at the center of the bottom surface when a rectangular hydrophobic surface with h2 = 150° and a width of ‘ð¼ 0:15LÞ exists at the center of the square hydrophilic surface with h1 = 30° and a side length of L.
304
S.W. Son et al. / Computers & Fluids 105 (2014) 294–306
t = 0.014
t = 0.419
t = 0.978
t = 2.096
t = 5.171
t = 10.342
t = 13.556
t = 14.954
Fig. 16. Dynamic behavior of a droplet located at the center of the bottom surface when a rectangular hydrophobic surface with h2 = 150° and a width of ‘ð¼ 0:2LÞ exists at the center of the square hydrophilic surface with h1 = 30° and a side length of L.
Fig. 14 shows the dynamic behavior of a droplet located at the center of the bottom surface when a rectangular hydrophobic surface with h2 = 150° and a width of ‘ð¼ 0:15LÞ exists at the center of the square hydrophilic surface with h1 = 30° and a side length of L. When h1 = 30° and h2 = 150°, the behavior of the droplet at ‘ ¼ 0:15L shows a similar trend to that at ‘ ¼ 0:1L, as shown in Figs. 6 and 14. At the beginning, the droplet symmetrically spreads out from the hemispherical shape in the direction of the square hydrophilic bottom surface whose contact angle is h1 = 30°. After that, a necking phenomenon appears as the droplet continues to move, and finally the whole droplet on the hydrophobic surface moves to the hydrophilic area. However, as the width of the hydrophobic surface increases from ‘ ¼ 0:1L to ‘ ¼ 0:15L, the movement and separation speeds of the droplet increase. When ‘ ¼ 0:15L, a necking phenomenon occurs at t = 0.419 where the width of the droplet located on the hydrophobic bottom surface decreases as the droplet located on the hydrophobic bottom surface moves in the direction of the hydrophilic bottom surface. After that, as the dimensionless time passes during 0:419 6 t 6 13:975, the necking phenomenon continues the progress to gradually reduce the width of the droplet located on the hydrophobic surface and separate the droplet into two at t = 13.975 which is earlier than the case of ‘ ¼ 0:1L by the dimensionless time of 7.127. The whole droplet separated at t = 13.975 then moves from the hydrophobic surface to the hydrophilic area at t = 15.513 which is earlier than the case of ‘ ¼ 0:1L by the dimensionless time of 10.342. Fig. 15 shows the change in the dimensionless separation time of the droplet depending on the contact angle of the square hydrophilic surface, h1, and the contact angle of the rectangular hydrophobic surface, h2, when ‘ ¼ 0:15L. Similar to the case of ‘ ¼ 0:1L as shown in Fig. 10, when ‘ ¼ 0:15L, the dimensionless separation time of the droplet decreases as the contact angle of the rectangular hydrophobic surface, h2, increases, while it increases as the contact angle of the square hydrophilic surface, h1, increases. When the width of the hydrophobic surface increases from ‘ ¼ 0:1L to ‘ ¼ 0:15L, the contact area between the hydrophobic surface and the droplet increases, promoting the movement of the droplet to the hydrophilic surface. As a result, the dimensionless separation time of the droplet rapidly decreases with increasing width of the hydrophobic surface from ‘ ¼ 0:1L to ‘ ¼ 0:15L.
Fig. 17. Change in the dimensionless separation time of the droplet depending on the contact angle of the square hydrophilic surface, h1, and the contact angle of the rectangular hydrophobic surface, h2, when ‘ ¼ 0:2L.
Fig. 16 shows the dynamic behavior of a droplet located at the center of the bottom surface when a rectangular hydrophobic surface with h2 = 150° and a width of ‘ð¼ 0:2LÞ exists at the center of the square hydrophilic surface with h1 = 30° and a side length of L. When h1 = 30° and h2 = 150°, the behavior of the droplet at ‘ ¼ 0:2L is generally similar to that at ‘ ¼ 0:1L and ‘ ¼ 0:1L, as shown in Figs. 6, 14 and 16. However, the movement and separation speed of the droplet further increases as the width of the hydrophobic surface increases, causing a necking phenomenon to occur at t = 0.419 when ‘ ¼ 0:2L. After that, as the dimensionless time passes during 0:419 6 t 6 13:556, the necking phenomenon continues the progress to gradually reduce the width of the droplet located on the hydrophobic surface and separate the droplet into two at t = 13.556 which is earlier than the case of ‘ ¼ 0:1L by the dimensionless time of 7.127. The whole droplet separated at t = 13.556 then moves from the hydrophobic surface to the
S.W. Son et al. / Computers & Fluids 105 (2014) 294–306
305
increasing value of h1. Thus, the ranges of the contact angles of h1 and h2, in which the droplet is separated, are 10 6 h1 6 30 at h2 = 110°, 10 6 h1 6 40 at h2 = 120°, 10 6 h1 6 50 at 130 6 h2 6 140 , and 10 6 h1 6 60 at 150 6 h2 6 170 , as shown in Fig. 18.
4. Conclusion
Fig. 18. Map of droplet separation (denoted by the gray color) depending on the changes in the contact angle of the square hydrophilic surface, h1, and the contact angle of the rectangular hydrophobic surface, h2.
hydrophilic area at t = 14.954 which is earlier than the case of ‘ ¼ 0:1L by the dimensionless time of 9.783. Fig. 17 shows the change in the separation time of the droplet depending on the contact angle of the square hydrophilic surface, h1, and the contact angle of the rectangular hydrophobic surface, h2, when ‘ ¼ 0:2L As the contact angle of the rectangular hydrophobic surface, h2, increases, the separation time of the droplet decreases, and the separation time of the droplet increases as the contact angle of the square hydrophilic surface increases as in the cases of ‘ ¼ 0:1L and ‘ ¼ 0:15L shown in Figs. 10 and 15, respectively. When the length increases from ‘ ¼ 0:15L to ‘ ¼ 0:2L, the contact area between the hydrophobic surface and the droplet increases promoting movement of the droplet. On the other hand, the distance between the hydrophilic surface and the droplet also increases along with it increasing the time it takes for the droplet to move to the hydrophilic area. Accordingly, the decrease in the dimensionless separation time of the droplet caused by the increase from ‘ ¼ 0:15L to ‘ ¼ 0:2L is minor in comparison to that caused by the increase from ‘ ¼ 0:1L to ‘ ¼ 0:15L. Fig. 18 shows whether the droplet separation phenomenon appears or not depending on the changes in the contact angle of the square hydrophilic surface, h1, and the contact angle of the rectangular hydrophobic surface, h2. The droplet separation does not depend on the width of the hydrophobic surface, ‘, when ‘ has the value of 0.1L, 0.15L and 0.2L considered in this study, but depends on the values of h1 and h2. The gray-colored zone in Fig. 18 represents the contact angles, h1 and h2, at which the droplet is separated as time passes. When the contact angle of the rectangular hydrophobic surface, h2, is h2 = 100°, the droplet is not separated in the region of 10 6 h1 6 90 , because the hydrophobicity of the hydrophobic surface is not large. However, if the hydrophobicity of the hydrophobic surface increases slightly as the contact angle of the hydrophobic surface, h2, increases to h2 = 110°, the droplet is separated in the region of h1 6 30 where the hydrophilicity of the hydrophilic surface is strong. If the hydrophobicity of the hydrophobic surface increases further with h2 P 120 , the range of the square hydrophilic surface contact angle, h1, in which the droplet is separated, increases, even when the hydrophilicity of the hydrophilic surface decreases with
A numerical analysis was conducted for the behavior of a droplet located on a heterogeneous surface using the 3D Lattice Boltzmann Method. The contact angle range of the hydrophilic surface considered in this study was 10 6 h1 6 90 , and that of the hydro phobic surface was 100 6 h2 6 170 . Also, the change in the behavior of the droplet depending on the area of the hydrophobic surface was observed by changing the width of the hydrophobic surface as ‘ ¼ 0:1L, ‘ ¼ 0:15L and ‘ ¼ 0:2L. The diameter and the height of the droplet used in this calculation are D = 3 mm and H = 1.5 mm, resulting in D/L = 0.375 and H/L = 0.1875, respectively. In the case of a droplet located on a heterogeneous surface where a rectangular hydrophobic surface with a width of ‘ and a longitudinal length of L exists at the center of a square hydrophilic bottom surface with a side length of L, whether or not the droplet is separated and the dimensionless separation time are determined by the sizes of the contact angle of the hydrophilic surface, h1, and the contact angle of the hydrophobic surface, h2. The droplet is easily separated if the difference between the contact angle of the hydrophilic surface h1 and the contact angle of the hydrophobic surface h2 is big, and is not easily separated if the difference between the contact angles of the hydrophilic surface and the hydrophobic surface, h2 – h1, is smaller than 80°. The droplet located on a heterogeneous surface symmetrically spreads out from its initial hemispherical shape in the direction of the hydrophilic surface which has a relatively big adhesion. At this time, if the difference between the contact angles of the hydrophilic surface and the hydrophobic surface is not big enough, the droplet is not separated but reaches a new equilibrium state lying across the heterogeneous surfaces. But, if the difference between the contact angles of the two heterogeneous surfaces is big enough, the droplet located on the hydrophobic surface moves in the direction of the hydrophilic bottom surface causing a necking phenomenon where the width of the droplet located on the hydrophobic surface reduces. After that, the necking phenomenon continues the progress to gradually reduce the width of the droplet located on the hydrophobic surface and separate the droplet into two. When the separation of the droplet is observed, the more the contact angle of the hydrophilic surface, h1, increases, the more the dimensionless time it takes the droplet to be separated increases, while, the more the contact angle of the hydrophilic surface, h2, increases, the more the dimensionless separation time of the droplet decreases. Also, though the movement of the droplet is accelerated if the width of the hydrophobic surface ‘ increases and, as a result, the dimensionless separation time of the droplet shows a decreasing trend, the width of the hydrophobic surface ‘ does not have any effect on whether or not the droplet is separated in the conditions considered in this study. Acknowledgment This research was supported by Leading Foreign Research Institute Recruiment Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT & Future Planning (No. 2009-00495), and also by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (No. 2013R1A2A2A01067251).
306
S.W. Son et al. / Computers & Fluids 105 (2014) 294–306
Appendix A The two terms induced in Eq. (14) are obtained through the following process:
@nx @ny @nz @x i;j;k @y i;j;k @z i;j;k 1 ¼ : nx;iþ1=2;jþ1=2;kþ1=2 þ nx;iþ1=2;j1=2;kþ1=2 2 Dx
ðr nÞ ¼ @nx @x i;j;k
þnx;iþ1=2;jþ1=2;k1=2 þ nx;iþ1=2;j1=2;k1=2 nx;i1=2;jþ1=2;kþ1=2 nx;i1=2;j1=2;kþ1=2 nx;i1=2;jþ1=2;k1=2 nx;i1=2;j1=2;k1=2 ni;j;k nx @jnj ny @jnj þ r jnj ¼ jni;j;k j jnj i;j;k @x i;j;k jnj i;j;k @y i;j;k nz @jnj þ jnj i;j;k @z i;j;k i niþ1=2;jþ1=2;kþ1=2 ¼ C iþ1;j;k þ C iþ1;jþ1;k þ C iþ1;j;kþ1 2Dx
þC iþ1;jþ1;kþ1 C i;j;k C i;jþ1;k C i;j;kþ1 C i;jþ1;kþ1 j þ C i;jþ1;k þ C iþ1;jþ1;k þ C i;jþ1;kþ1 þ C iþ1;jþ1;kþ1 C i;j;k 2Dy
k C iþ1;j;k C i;j;kþ1 C iþ1;j;kþ1 þ C i;j;kþ1 þ C iþ1;j;kþ1 2 Dz
þC i;jþ1;kþ1 þ C iþ1;jþ1;kþ1 C i;j;k C iþ1;j;k C i;jþ1;k C iþ1;jþ1;k
ðA1Þ
ðA2Þ
ðA3Þ
ðA4Þ
The phase boundary was established through the process described above, and an additional force was set along the boundary surface so that the two fluids do not mix with each other. References [1] Trevino C, Forro-Fontan C, Mendez F. Asymptotic analysis of axisymmetric drop spreading. Phys. Rev. E 1998;58:4478–84.
[2] Yang J, Koplik J. Molecular dynamics of drop spreading on a solid surface. Phys. Rev. Lett. 1991;67:3539–42. [3] Gu Y, Li D. Model for a liquid drop spreading on a solid surface. Colloids Surf. A. Physicochem. Eng. Aspects 1998;142:243–56. [4] Bayer IS, Megaridis CM. Contact angle dynamics in droplets impacting on flat surfaces with different wetting characteristics. J. Fluid Mech. 2006;558:415–49. [5] Cossali GE, Coghe A, Marengo M. The impact of a single drop on a wetted solid surface. Exp. Fluids 1997;22:463–72. [6] Mao T, Kuhn DCS, Tran H. Spread and rebound of liquid droplet upon impact on flat surfaces. AIChE. J. 1997;43:2169–79. [7] Sikalo S, Marengo M, Tropea C, Ganic EN. Analysis of impact of droplets on horizontal surfaces. Exp. Therm. Fluid Sci. 2002;25:503–10. [8] Zhang X, Basaran OA. Dynamic surface tension effects in impact of a drop with solid surface. J. Colloid Interf. Sci. 1997;187:166–78. [9] Fujimoto H, Ogino T, Takuda H, Hatta N. Collision of a droplet with a hemispherical static droplet on a solid. Int. J. Multiphase Flow 2001;27:1127–245. [10] Manservisi S, Scardovelli R. A variational approach to the contact angle dynamics of spreading droplets. Comput. Fluids 2009;38:406–24. [11] Tanaka Y, Washio Y, Yoshino M, Hirata T. Numerical simulation of dynamic behavior of droplet on solid surface by the two-phase Lattice Boltzmann Method. Comput. Fluids 2011;40:68–78. [12] Lunkad SF, Buwa VV, Nigam KDP. Numerical simulations of drop impact and spreading on horizontal and inclined surfaces. Chem. Eng. Sci. 2007;62:7214–24. [13] Sikalo S, Tropea C, Ganic EN. Impact of droplets onto inclined surface. J. Colloid Interf. Sci. 2005;286:661–9. [14] Shan X, Chen H. Lattice Boltzmann model of simulating flows with multiple phases and components. Phy. Rev. E 1993;47:1815–9. [15] Swift MR, Osborn WR, Yeomans JM. Lattice Boltzmann Simulation of nonideal fluids. Phy. Rev. Lett. 1995;75:830–3. [16] Gunstensen AK, Rothman DH, Zaleski S, Zanetti G. Lattice Boltzmann model of immiscible fluids. Phys. Rev. A 1991;43:4320–7. [17] Wu L, Tsutahara M, Kim LS, Ha MY. Three-dimensional lattice Boltzmann simulation of droplet formulation in a cross-junction microchannel. Int. J. Multiphase Flow 2008;34:852–64. [18] Bhatnagar PL, Gross EP, Krook M. A model for collision processes in gases. I: small amplitude process in charged and neutral one-component system. Phy. Rev 1954;94:511. [19] Yan YY, Zu YQ. A lattice Boltzmann method for incompressible two-phase flow in particle wetting surface with large density ratio. J. Comp. Phys. 2007;227:763–75.
Contents lists available at ScienceDirect
Computers & Fluids j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / c o m p fl u i d
A numerical study on the dynamic behavior of the liquid droplet located on heterogeneous surface Sung Wan Son a, Chan Hyeok Jeong b, ManYeong Ha a,⇑, Hyung Rak Kim a a b
School of Mechanical Engineering, Pusan National University, San 30, Jang Jeon Dong, Geum Jeong Gu, Busan 609-735, Korea Hyundai Heavy Industries, Bangeojin Ring Road 1000, Dong-gu, Ulsan 682-792, Korea
a r t i c l e
i n f o
Article history: Received 31 March 2014 Received in revised form 15 September 2014 Accepted 17 September 2014 Available online 28 September 2014 Keywords: Droplet behavior Heterogeneous surface Droplet separation Lattice Boltzmann method
a b s t r a c t The behavior of a droplet located on a heterogeneous surface is numerically analyzed using the 3-D Lattice Boltzmann Method. Predictions are made whether the droplet will be separated or not and the separation time is calculated changing the contact angles of the hydrophilic surface, hydrophobic surface and the area of the hydrophobic surface. The change in the contacting length between the droplet and the bottom surface and the change in the dynamic contact angle of the droplet on the cross section at the center of the droplet as time passes are observed, and the droplet separation mechanism is investigated by analyzing the velocity vector around the phase boundary line and the droplet which changes with time while the shape of the droplet changes. A droplet located at the center of a heterogeneous surface shows a trend of being easily separated if the difference between the contact angles of the hydrophilic and hydrophobic surfaces is bigger. The more the contact angle of the hydrophilic surface increases, the more the droplet separation time increases, and the more the contact angle of the hydrophobic surface increases, the more the droplet separation time decreases. Also, though the droplet separation time shows a decreasing trend when the width of the hydrophobic surface increases, it does not have any effect on whether the droplet will be separated or not. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction Multiphase flow is an issue which attracts people’s attention in diverse fields such as engineering, industry and environment as well as natural science including chemistry and biology. Being an important part in various fields from natural phenomena such as rainstorm, smog and snowstorm to industrial fields such as air conditioning system, refrigerating system, sea water desalination plant, nuclear power generation equipment, and various heat exchangers, it has attracted attention of many researchers. Studies on behaviors of droplets among multiphase flow issues are actively conducted experimentally or numerically with recent development of micro/nanotechnology. Tervino et al. [1], Yang and Koplik [2], and Gu and Li [3] conducted numerical studies on the phenomenon of a droplet spreading over a solid surface in a stationary state. Also, Bayer and Megaridis [4], Cossali et al. [5], Mao et al. [6], Sikalo et al. [7], and Zhang and Basaran [8] experimentally studied the behavior of a droplet crashing onto a solid surface. And Fujimoto et al. [9], and Manservisi and Scardovelli [10] conducted numerical studies on the dynamic behavior of a droplet ⇑ Corresponding author. Tel.: +82 51 510 2440; fax: +82 51 515 3101. E-mail address: [email protected] (M. Ha). http://dx.doi.org/10.1016/j.compfluid.2014.09.033 0045-7930/Ó 2014 Elsevier Ltd. All rights reserved.
colliding with a solid surface. Tanaka et al. [11] and Lunkad et al. [12] conducted numerical studies on the kinetic characteristics of a droplet moving on a slanted solid surface or on a solid surface to which a force was applied, and compared the result of the numerical analyses with the experimental result of Sikalo et al. [13]. Boltzmann Method (LBM) is a computational analysis technique which recently attracts people’s attention by reason of its fast computation speed, multiphase analysis, and easiness of parallelization. Many studies have been conducted using the Lattice Boltzmann Method, and Shan and Chen [14] in particular presented a model capable of analyzing the multiphase flow of different fluids which do not mix with each other at the same temperature using the Lattice Boltzmann Method. Swift et al. [15] introduced a free energy approach into the multiphase flow model, and Gunstensen et al. [16] proposed a method of analyzing a two-phase flow using a multiple density distribution function. Wu et al. [17] presented a model of which the pressure disturbance was corrected at the phase boundary based on the research findings of Gunstensen et al. [16]. In this study, the behavior of a hemispherical droplet located on a heterogeneous surface was studied using the 3D Lattice Boltzmann Method. A hydrophobic bottom surface exists at the
295
S.W. Son et al. / Computers & Fluids 105 (2014) 294–306
center of a hydrophilic bottom surface, and a hemispherical droplet is located at the center of the bottom surface lying from the hydrophilic bottom surface to the hydrophobic bottom surface. The droplet is separated or reaches a new equilibrium state without being separated depending on the contact angles of the hydrophilic and hydrophobic bottom surfaces. The time at which the droplet is separated differs depending on the width of the hydrophobic bottom surface, which also has an effect on whether or not the droplet will be separated. In this study, an investigation was carried out as to whether the droplet is separated or not changing the contact angle of the heterogeneous surfaces with different hydrophobic widths, and the change in the contacting length between the bottom surface and the droplet was presented. In the case the droplet was separated, the separation time of the droplet depending on the hydrophobic width and the contact angle of the bottom surface was calculated, and the dynamic contact angle between the bottom surface and the droplet which changes with time while the droplet is separated was calculated.
Fig. 1. The D3Q19 lattice model.
Accordingly, the Lattice Boltzmann Equation can be expressed as Eq. (4) below:
2. Numerical analysis method
k
k
f i ðx þ ei ; t þ dtÞ f i ðx; tÞ ¼
2.1. Lattice Boltzmann Method
1
sk
k
kðeqÞ
ðf i f i
Þðx;tÞ kðeqÞ
In this study, a 3D analysis was conducted on a droplet located on a heterogeneous surface using the Lattice Boltzmann Method. The governing equation was induced from the Boltzmann equation, and the application scope of the existing Lattice Gas Automata (LGA) governing equation was widened and its numerical stability was improved by expanding it to the real number range. The discretized Boltzmann Equation for transportation of incompressible two-phase fluid particles is as follows: k
@f i k þ ei rf i ¼ Xki @t
ð1Þ
k
Here, f i represents the density distribution function of the fluid particle, and the superscript k means the relevant phase. ei means the lattice speed, and Xki means the collision operator. The subscript i represents the direction of the lattice, which is differently defined in accordance with the lattice model. Though the collision operator contains a very complicated mathematical mechanism, it can be simply expressed using the single relaxation time proposed by Bhatnahar et al. [18]. The discretized Boltzmann equation can be expressed as follows when simplified applying the BGK model [18]: k @f i
@t
k fi
þ ei r
¼
1
s
k ðf i k
Here, the equilibrium distribution function f i
ð2Þ
Here, sk represents the single relaxation time for the kth fluid, and kðeqÞ fi represents the equilibrium distribution function. The equilibrium distribution function represents the value of the equilibrium k state f i has in the relaxation time. The D3Q19 is a model considered for the fluid particle which can have a velocity in only 19 directions in a 3D space as shown in Fig. 1, and the lattice speed ei is given as follows:
is given as follows:
3 9 3 ¼ xi qk 1 þ 2 ðei uÞ þ 4 ðei uÞ2 2 u u c 2c 2c
kðeqÞ
fi
ð5Þ
Here, xi is a weighting coefficient, and u represents the fluid velocity. In the model D3Q19, the weighting coefficient is given being divided into x0 = 1/3, xi = 1/18 (i = 1, 2, 3, 4, 5, 6), and xi = 1/36 (i = 7, ..., 18) in accordance with the direction of the lattice speed. Eq. (4) used for the Lattice Boltzmann Method is alternately calculated being divided into the collision stage and flow stage, and has an advantage that, having a form of an explicit equation, the calculation speed is faster than the analysis which uses the Navier–Stokes equation discretized in the form of an implicit equation and calculations can be done in a microscopic area. Also, as the continuity equation and Navier–Stokes equation that govern flow of fluid can be induced from the BGK model based Lattice Boltzmann Method using the Chapman-Enskog theory, it can be confirmed that the Lattice Boltzmann Method can be also applied to macroscopic flow. The macroscopic flow variables such as density (q) and pressure (p) are defined as follows:
qk ¼ kðeqÞ fi Þ
ð4Þ
X
k
fi ¼
X kðeqÞ fi
i
q¼
X
qk
k
qu ¼
X i
p¼
ð6Þ
i
f i ei ¼
ð7Þ X
eq
f i ei
ð8Þ
i
1 2 qc 3
ð9Þ
Here, qk represents the fluid density of each phase. 2.2. Two-phase flow model
e0 ¼ ð0; 0; 0Þ e1;2 ; e3;4 ; e5;6 ¼ ð1; 0; 0Þc; ð0; 1; 0Þc; ð0; 0; 1Þc e7;...;10 ; e11;...;14 ; e15;...;18 ¼ ð1; 1; 0Þc; ð1; 0; 1Þc; ð0; 1; 1Þc ð3Þ At this time, c is defined to be dx/dt(c = dx/dt), and dx and dt represent the increments of space and time respectively. The Lattice Boltzmann Equation can be obtained by discretizing Eq. (2) for the unit time and lattice space which fall under lattice speed c.
The two-phase model used in this study is the model proposed by Gunstensen et al. [16] and modified by Wu et al. [17]. This twophase flow model is suitable to solve two-phase flows which do not mix with other fluids based on the lattice BGK model, and has a feature that there is no pressure disturbance at the phase boundary. In the case two phases exist, the CSF (Continuum Surface Force) model was used to apply the surface tension at the boundary
296
S.W. Son et al. / Computers & Fluids 105 (2014) 294–306
surface and to apply the pressure gradient. Wu et al. [17] defined a new variable ueq as follows, and replaced the fluid velocity u with it.
ueq ¼ u þ
s F q
ð10Þ
Here, s is the total relaxation time and can be obtained using P k k ks q /q. The force F resulting from the surface tension is defined as follows:
s=
F ¼ rj
rC jrCj
ð11Þ
Here, r is the surface tension coefficient, and j means the curved surface of the phase. C is the ratio of the fluid, which is defined as follows:
C ¼ qred qblue n j ¼ r jnj
ð12Þ ð13Þ
Here, qred means the density of the internal fluid, and qblue means the density of the external fluid. n means the vector perpendicular to the phase surface, and can be defined using n = rC. At this time, the following calculation process should be conducted to obtain the value j.
j ¼ r
n 1 n ¼ ðr nÞ þ r jnj jnj jnj jnj
ð14Þ
2.3. Two-phase flow model In order to verify the validity of the numerical analysis method and the two-phase flow calculation model used in this study, the change in the internal and external pressures of the droplet depending on the change in the curvature radius of the droplet was calculated using the analysis code developed in this study, and the result of the calculation was compared with the value obtained using the Young–Laplace equation which is an exact solution. The Young–Laplace equation is expressed as in Eq. (19) below:
Dp ¼ pred pblue ¼
2r R
ð19Þ
Here, pred and pblue represent the dimensionless pressure of the fluids different with each other, R represents the dimensionless
Fig. 2. Change in the dimensionless internal and external pressure difference Dp of the droplet depending on the change in the inverse number of the dimensionless curvature radius 1/R.
curvature radius of the internal fluid, and r means the dimensionless surface tension. Fig. 2 shows the change in the dimensionless internal and external pressure difference Dp of the droplet depending on the change in the inverse number of the dimensionless curvature radius 1/R. Here, the solid line represents the Young–Laplace equation shown in Eq. (19), and the rectangle represents the value obtained by numerical calculation of this study. When there is a pressure gradient on a round boundary surface of two stationary fluids resulting from the surface tension, the internal and external pressure difference can be calculated using Eq. (19). According to Eq. (19), the internal pressure at the round boundary surface of the droplet always has a value bigger than that of the external pressure. As the internal and external pressure difference Dp decreases, 1/R decreases, according to which the curvature at the boundary surface of the droplet grows bigger. Accordingly, if the internal and external pressure difference of the droplet gets very small (Dp ? 0), the radius of curvature at the boundary surface of the droplet grows infinitely, due to which the droplet approaches a flat surface. But, if the internal and external pressure difference of the droplet grows very big, the radius of curvature at the boundary surface of the droplet becomes very small. As shown in Fig. 2, the calculation result obtained using the numerical analysis method and the two-phase flow calculation model adopted in this study well corresponds with the exact solution obtained from the Young–Laplace equation given in Eq. (19). For the secondary verification of the validity of the numerical analysis method and the two-phase flow calculation model used in this study, a calculation was conducted adopting the question same as that of the study conducted by Yan and Zu [19] and the result was compared with the calculation result of Yan and Zu [19]. Yan and Zu [19] investigated the behavior of a droplet on a heterogeneous surface using the Lattice Boltzmann Method for which the D3Q15 model having 15 velocity components in a 3 dimensional space is adopted. Yan and Zu [19] conducted a calculation adopting a free-energy model to simulate a two-phase flow phenomenon. Fig. 3 shows the shape and computational domain adopted for verification of validity. Fig. 3(a) shows the shape in which a rectangular hydrophobic surface with a width of l and a longitudinal length of L exists at the center of a relatively hydrophilic square bottom surface with a side length of L. Fig. 3(b) shows the shape in which a rectangular hydrophobic surface with a width of l and a longitudinal length of L and a rectangular hydrophobic surface with a width of l and a traversal length of L intersect with each other at the center of a hydrophilic square bottom surface with a side length of L. h1 and h2 represent the contact angles of the hydrophilic and hydrophobic bottom surface, respectively. The values of h1 and h2 in Fig. 3(a) are 30° and 150° respectively, and the diameter and height of the hemispherical droplet located at the center of the bottom surface are D = 4 mm and H = 2 mm respectively. The values of h1 and h2 in Fig. 3(b) are 45° and 150° respectively, and the diameter and the height of the hemispherical droplet located at the center of the bottom surface are D = 4 mm and H = 1 mm respectively. Fig. 4 shows the dynamic behavior of the droplet at the moment when a rectangular hydrophobic surface exists at the center of a square hydrophilic bottom surface as shown in Fig. 3(a). Fig. 4(a) shows the dynamic behavior of the droplet obtained using the numerical analysis method of this study, and Fig. 4(b) shows the dynamic behavior of the droplet calculated by Yan and Zu [19]. As shown in Fig. 4, the dynamic behavior of the droplet obtained by the current numerical analysis method well reproduces the dynamic behavior of the droplet calculated by Yan and Zu [19]. Fig. 5 shows the change with time in the phase boundary surface where the droplet and the bottom surface contact each other when the two rectangular hydrophobic surfaces intersect with each other on the square hydrophilic bottom surface as shown in Fig. 3(b). The calculation result obtained
S.W. Son et al. / Computers & Fluids 105 (2014) 294–306
297
(a) Single cross hydrophobic strips
(b) Double cross hydrophobic strips Fig. 3. Initial droplet shape and computational domain adopted for verification of validity.
using the current calculation model and the numerical analysis method shown in Fig. 5(a) well reproduces the calculation result obtained by Yan and Zu [19] as shown in Fig. 5(b).
3. Result and discussion The behavior of a droplet located on a heterogeneous surface was established using the 3D Lattice Boltzmann Method after securing the validity of the numerical analysis method and two-phase flow calculation model used in this study through the verification calculations shown in Figs. 2, 4 and 5. For this, the dynamic behavior of a droplet located on such a heterogeneous surface was investigated considering a shape in which a rectangular hydrophobic surface with a width of l and a longitudinal length of L exists at the center of a hydrophilic square bottom surface with a side length of L. Here, the contact angle of the square hydrophilic bottom surface is h1, and the contact angle of the rectangular hydrophobic bottom surface is h2. The droplet is assumed to exist at the beginning in a hemispherical form with a diameter of D at the center of the bottom surface. The diameter and the height of the droplet used in this calculation are D = 3 mm and H = 1.5 mm, resulting in D/L = 0.375 and H/L = 0.1875, respectively. Fig. 6 shows the dynamic behavior of a droplet located at the center of the bottom surface when a rectangular hydrophobic surface with h2 = 150° and a width of ‘ð¼ 0:1LÞ exists at the center of the square hydrophilic surface with h1 = 30° and a side length of L.
The droplet which exists in a hemispherical shape lying across the rectangular hydrophobic surface at the beginning spreads out symmetrically with dimensionless time in the direction of the hydrophilic bottom surface which has a relatively big adhesion. The dimensionless time t in Fig. 6 is defined as follows:
sffiffiffiffiffiffiffiffi t ¼ t
r qL3
ð20Þ
where t*, r and q represent the dimensional time, surface tension and liquid phase density, respectively. The values of r and q used in this study are 0.01 N/m and 1 kg/m3, respectively. At t = 0.978, the droplet located on the hydrophobic bottom surface moves in the direction of the hydrophilic bottom surface causing a necking phenomenon where the width of the droplet located on the hydrophobic bottom surface decreases. Later, as time passes during the period of 0:978 6 t 6 21:103, the necking phenomenon continuously progresses making the width of the droplet located on the hydrophobic surface gradually decrease, and the droplet is separated into two droplets at t = 21.103. After the droplet is separated at t = 21.103, the whole droplet located on the hydrophobic surface moves to the hydrophilic area, as shown in Fig. 6 at t = 25.855. Fig. 7 shows the change with the dimensionless time in the phase boundary line of the droplet located at the center of the bottom surface and the change in the velocity vector around the droplet when a rectangular hydrophobic surface with h2 = 150° and a
298
S.W. Son et al. / Computers & Fluids 105 (2014) 294–306
t* = 0.0 s
t* = 0.015s
t* = 0.14 s
t* = 0.15s
t* = 0.152 s
t* = 0.154 s
(a) Present study
t* = 0.0 s
t* = 0.15s
t* = 0.015s
t* = 0.152 s (b) Previous research
t* = 0.14 s
t* = 0.154 s
Fig. 4. Dynamic behavior of the droplet at the moment when a rectangular hydrophobic surface exists at the center of a square hydrophilic bottom surface. (a) Present study, and (b) previous research.
width of ‘ ¼ 0:1L exists at the center of the square hydrophilic surface with h1 = 30° and a side length of L. It can be confirmed that, as the dimensionless time passes, the velocity component heading from the top part of the droplet to the bottom surface and the velocity component heading from the horizontal center of the droplet to the edges develop making the droplet spread out over the hydrophilic bottom surface. After t = 10.342, separation of the droplet starts making the phase boundary line of the horizontal center become lower than the boundary line of the surroundings. As separation of the droplet progresses, velocity vectors rotating clockwise and counterclockwise from the phase boundary of the horizontal center are generated. Fig. 8 shows the dynamic behavior of the droplet located at the center of the bottom surface when a rectangular hydrophobic surface with h2 = 130° and a width of ‘ ¼ 0:1L exists at the center of the square hydrophilic surface with h1 = 90° and a side length of L. As to the initial dynamic behavior of the droplet when h1 = 90° and h2 = 130°, the droplet which is in a hemispherical form symmetrically spreads out in the direction of the square bottom surface with h1 = 90° as in the case of h1 = 30° and h2 = 150°.
However, if h1 increases from 30° to 90°, the extent of the droplet spreading out in the direction of the square surface with h1 = 90° is considerably reduced in comparison to that of h1 = 30° due to reduction in the hydrophilic effect on the square surface. Therefore the necking phenomenon where the width of the droplet decreases as observed in the droplet located on the rectangular hydrophobic surface when h1 = 30° and h2 = 150° is not observed when h1 = 90° and h2 = 130°. For this reason, when h1 = 90° and h2 = 130°, spreading of the droplet rapidly slows down after t = 1.118 without the droplet being separated, and a new state of equilibrium is maintained after t = 5.171. Fig. 9 shows the change with the dimensionless time in the phase boundary line of the droplet located at the center of the bottom surface and the change in the velocity vector around the droplet when a rectangular hydrophobic surface with h2 = 130 and a width of ‘ ¼ 0:1L exists at the center of the square hydrophilic surface with h1 = 90° and a side length of L. As the dimensionless time passes during 0 6 t 6 2:096, the phase boundary line of the droplet symmetrically spreads out to the right and left of the hydrophilic bottom surface. However, at t P 5:171, it
S.W. Son et al. / Computers & Fluids 105 (2014) 294–306
(a) Present study
(b) Previous research Fig. 5. Change with time in the phase boundary surface for double cross hydrophobic strips.
reaches a new state of equilibrium without having almost any change in the phase boundary line of the droplet. Due to the existence of the velocity component heading from the top part of the droplet to the bottom surface and the velocity component heading from the horizontal center of the droplet to the sides, the height of the phase boundary located at the top of the droplet gradually drops as the dimensionless time passes. But, though the height of the phase boundary line at the center of the droplet has the highest value in comparison to those of the phase boundary lines of the surroundings at 0 6 t 6 2:096, the height of the phase boundary line at the center of the droplet becomes a little lower than those of the phase boundary lines of the surroundings at t P 5:171. Fig. 10 shows the change in the dimensionless separation time of the droplet depending on the contact angle of the square hydrophilic surface h1 and the contact angle of the rectangular
299
hydrophobic surface h2 when ‘ ¼ 0:1L. The shape of the droplet formed on the rectangular hydrophobic surface becomes more convex because the contact angle of the droplet contacting the surface of the hydrophobic surface grows as the contact angle of the rectangular hydrophobic surface h2 increases, due to which the dimensionless separation time of the droplet deceases. As the contact angle of the square hydrophilic surface h1 increases, the intensity of the hydrophilicity of the hydrophilic surface decreases, due to which the spreading force in the direction of the hydrophilic surface becomes weaker greatly increasing the dimensionless separation time of the droplet in comparison to the case where the contact angle of the hydrophilic surface h1 is small. Fig. 11 shows the shape of the droplet looked from the side and the top when the droplet is separated at h2 = 150° for different values of h1 = 10°, 30° and 50°. Fig. 11(a) shows the shape of the droplet looked from the side and the top of the droplet when the droplet is separated at t = 13.556 under the conditions of h1 = 10° and h2 = 150°. As the intensity of the hydrophilicity of the square hydrophilic surface is very high at h1 = 10°, the droplet widely spreads out from the center of the droplet where the hydrophobic surface is located in the right/left direction (x) and the up/down direction (y) along the surrounding hydrophilic surface, due to which the contact surface between the droplet and the bottom surface is widely formed. The phase boundary surface of the separated droplet assumes a convex shape near the rectangular hydrophobic surface. Therefore, the cross section of the droplet perpendicular to x increases at first as the point moves from x = 0 to the right and left in the direction of x and decreases after reaching the maximum value. Fig. 11(b) shows the shapes of the separated droplets looked from the side and the top when the droplet is separated at t = 21.103 under the conditions of h1 = 30° and h2 = 150°. Similarly, Fig. 11(c) shows the shapes of the separated droplets looked from the side and the top when the droplet is separated at t = 43.603 under the conditions of h1 = 50° and h2 = 150°. As to the shape of the droplet separated when h1 = 30° or h1 = 50° at h2 = 150°, the droplet widely spreads out from the center of the droplet where the rectangular hydrophobic surface is located in the right/left and the up/down directions similarly to that of h2 = 150° and h1 = 10°, due to which the cross section of the droplet perpendicular to x increases at first as the point moves from x = 0 to the right and left in the direction of x and decreases after reaching the maximum value. However, if the contact angle of the hydrophilic surface h1 increases from 10° to 30° and 50°, as the hydrophilicity of the hydrophilic surface decreases, the extent of the droplet spreading out from x = 0 in the right/left and up/down directions decreases as h1 increases. Accordingly, if h1 increases from 10° to 30° and 50° when h2 = 150°, the contact area between the droplet and the bottom surface decreases. Also, as h1 increases, the location where the droplet cross section perpendicular to x has the maximum value moves further to the right and left from x = 0, and the height of the droplet at the location increases along with h1. Fig. 12 shows the change in the dimensionless wet length between the droplet and the bottom surface at the central cross section of the droplet (y = 0) for different h1 values varying in the range from h1 = 10° to h1 = 90° when the contact angle of the hydrophobic surface is h2 = 150°. The dimensionless wet length (W/L) is defined as the contact length between the droplet and the bottom surface at the central cross section of the droplet, W, divided by the length of a square surface, L. When h1 = 10°, the dimensionless contact length between the droplet and the bottom surface rapidly increases as the droplet spreads from the hydrophobic surface to the hydrophilic surface during 0 6 t 6 10:342, and abruptly decreases in a moment at t = 10.342 as the droplet is separated. As the dimensionless time passes, the droplet spreads out continuously moving to the hydrophilic area, due to which the
300
S.W. Son et al. / Computers & Fluids 105 (2014) 294–306
t = 0.014
t = 0.419
t = 0.978
t = 2.096
t = 5.171
t = 10.342
t = 21.103
t = 25.855
Fig. 6. Dynamic behavior of a droplet located at the center of the bottom surface when a rectangular hydrophobic surface with h2 = 150° and a width of ‘ð¼ 0:1LÞ exists at the center of the square hydrophilic surface with h1 = 30° and a side length of L.
t = 0.014
t = 0.419
t = 0.978
t = 2.096
t = 5.171
t = 10.342
t = 21.103
t = 25.855
Fig. 7. Change with the dimensionless time in the phase boundary line and the velocity vector around the droplet when a rectangular hydrophobic surface with h2 = 150° and a width of ‘ ¼ 0:1L exists at the center of the square hydrophilic surface with h1 = 30° and a side length of L.
dimensionless contact length between the droplet and the bottom surface continuously increases after t = 10.342 and come to have a constant value after the droplet reaches a new equilibrium state after t = 36.196.
When h1 increases from 10° to 30°, the hydrophilicity of the hydrophilic surface decreases, due to which separation of the droplet when h1 = 30° is shown to be slower than when h1 = 10°. Therefore, as the droplet spreads out over the hydrophilic surface
301
S.W. Son et al. / Computers & Fluids 105 (2014) 294–306
t = 0.014
t = 0.14
t = 0.419
t = 0.978
t = 1.118
t = 2.096
t = 5.171
t = 10.342
Fig. 8. Dynamic behavior of a droplet located at the center of the bottom surface when a rectangular hydrophobic surface with h2 = 130° and a width of ‘ ¼ 0:1L exists at the center of the square hydrophilic surface with h1 = 90° and a side length of L.
t = 0.014
t = 0.14
t = 0.419
t = 0.978
t = 1.118
t = 2.096
t = 5.171
t = 10.342
Fig. 9. Change with the dimensionless time in the phase boundary line and the velocity vector around the droplet when a rectangular hydrophobic surface with h2 = 130° and a width of ‘ ¼ 0:1L exists at the center of the square hydrophilic surface with h1 = 90° and a side length of L.
during 0 6 t 6 16:072, the dimensionless contact length between the droplet and the bottom surface increases. However, as shown in Fig. 6, the contact length between the droplet and the bottom
surface decreases as the dimensionless time passes due to the necking phenomenon which occurs during 16:072 6 t 6 19:566 where the width of the droplet located on the hydrophobic bottom
302
S.W. Son et al. / Computers & Fluids 105 (2014) 294–306
Fig. 10. Change in the dimensionless separation time of the droplet depending on the contact angle of the square hydrophilic surface, h1, and the contact angle of the rectangular hydrophobic surface, h2.
surface decreases, and the contact length reaches the minimum value as the droplet is separated into two droplets at t = 19.566. After t = 19.566, the separated droplet continuously spreads out again in the right/left and up/down directions increasing the contact length between the droplet and the bottom surface, which comes to have a constant value after the droplet reaches a new equilibrium state after t = 43.464. When h1 increases further from 30° to 50°, the hydrophilicity of the hydrophilic surface decreases more, due to which separation of the droplet when h1 = 50° is shown to be much slower than when h1 = 30° or h1 = 10°. Therefore, as the droplet spreads out to the hydrophilic surface without being separated during 0 6 t 6 33:541, the dimensionless contact length increases. After that, as the dimensionless time passes during 33:541 6 t 6 37:175, the dimensionless contact length between the droplet and the bottom surface decreases due to necking phenomenon. The dimensionless contact length between the droplet and the bottom surface reaches the minimum value as the droplet is separated into two at t = 37.175. Similar to the case of h1 = 30°, the droplets separated
Fig. 12. Change in the dimensionless wet length between the droplet and the bottom surface at the central cross section of the droplet (y = 0) for different h1 values varying in the range from h1 = 10° to h1 = 90° when h2 = 150°.
into two after t = 37.175 continuously spread out on the hydrophilic surface in the right/left and up/down directions, increasing again the dimensionless contact length between the droplet and the bottom surface, which comes to have a constant value after the droplet reaches a new equilibrium state at t P 62:051. When h1 increases further from 70° to 90°, the hydrophilicity of the hydrophilic surface becomes very small, due to which the droplet spreads out to the hydrophilic surface without being separated and reaches a new equilibrium state in a short time. Therefore, when h1 = 70° or h1 = 90°, the dimensionless contact length between the droplet and the bottom surface increases during 0 6 t 6 3:494, and comes to have a constant value after the droplet reaches a new equilibrium state after t = 3.494. After t = 3.494, the dimensionless contact length between the droplet and the bottom surface when h1 = 70° comes to have the almost same value as that of h1 = 90°. The dimensionless contact length between the droplet and the bottom surface decreases after the droplet reaches a new equilibrium state due to the hydrophilicity of the hydrophilic surface decreasing as h1 increases.
(a) θ1 = 10°, t = 13.556
(b) θ1 = 30°, t = 21.103
(c) θ1 = 50°, t = 43.603 Fig. 11. Shape of the droplet looked from the side and the top when the droplet is separated at h2 = 150° for different values of (a) h1 = 10°, (b) h1 = 30°, and (c) h1 = 50°.
303
S.W. Son et al. / Computers & Fluids 105 (2014) 294–306
Fig. 13. Change in the contact angle between the droplet and the bottom surface as a function of the dimensionless time for different h1 values varying in the range from h1 = 10° to h1 = 90° when the contact angle of the hydrophobic surface is h2 = 150°.
Fig. 15. Change in the dimensionless separation time of the droplet depending on the contact angle of the square hydrophilic surface, h1, and the contact angle of the rectangular hydrophobic surface, h2, when ‘ ¼ 0:15L.
Fig. 13 shows the change in the contact angle between the droplet and the bottom surface as a function of the dimensionless time for different h1 values varying in the range from h1 = 10° to h1 = 90° when the contact angle of the hydrophobic surface is h2 = 150°. When h1 = 10°, the dynamic contact angle between the droplet and the bottom surface rapidly decreases as the droplet quickly spreads from the hydrophobic surface to the hydrophilic surface due to the big difference between the contact angles of the hydrophobic surface and the hydrophilic surface. As the dimensionless time passes, the rate of change in the dynamic contact angle of the droplet decreases, and the dynamic contact angle maintains a value of 10 after t = 23.898 when the droplet is divided into two and finishes its movement to the hydrophilic surface. As the dimensionless time passes, the variation pattern in the dynamic contact angle of the droplet at h1 = 30° is generally similar to that at h1 = 10°. However, as h1 increases from 10° to 30°, the hydrophilicity of the hydrophilic surface becomes smaller and, as a result, the dynamic contact angle of the droplet at h1 = 30°
changes more slowly than that at h1 = 10°, as the dimensionless time passes. The rate of change in the dynamic contact angle of the droplet decreases as the dimensionless time passes as is the case of h1 = 10°. The dynamic contact angle maintains a value of 30° after the droplet is separated into two and finishes its movement to the hydrophilic surface. When h1 = 50°, the droplet is separated into two, similar to that when h1 = 10° and h1 = 30°. Thus, when h1 = 50°, as the dimensionless time passes, the dynamic contact angle of the droplet rapidly decreases at the beginning and reaches the constant value of 30° after the droplet is separated into two and finishes its movement to the hydrophilic surface. However, when h1 = 70° and h1 = 90°, the droplet is not separated into two and reaches a new equilibrium state. Thus, as the dimensionless time passes, the dynamic contact angles for h1 = 70° and h1 = 90° decrease at the beginning and reach the new equilibrium state with constant contact angle values of 63° and 72°, respectively, which are different from the contact angles of the hydrophilic surface.
t = 0.014
t = 0.419
t = 1.118
t = 2.096
t = 5.171
t = 10.342
t = 13.975
t = 15.513
Fig. 14. Dynamic behavior of a droplet located at the center of the bottom surface when a rectangular hydrophobic surface with h2 = 150° and a width of ‘ð¼ 0:15LÞ exists at the center of the square hydrophilic surface with h1 = 30° and a side length of L.
304
S.W. Son et al. / Computers & Fluids 105 (2014) 294–306
t = 0.014
t = 0.419
t = 0.978
t = 2.096
t = 5.171
t = 10.342
t = 13.556
t = 14.954
Fig. 16. Dynamic behavior of a droplet located at the center of the bottom surface when a rectangular hydrophobic surface with h2 = 150° and a width of ‘ð¼ 0:2LÞ exists at the center of the square hydrophilic surface with h1 = 30° and a side length of L.
Fig. 14 shows the dynamic behavior of a droplet located at the center of the bottom surface when a rectangular hydrophobic surface with h2 = 150° and a width of ‘ð¼ 0:15LÞ exists at the center of the square hydrophilic surface with h1 = 30° and a side length of L. When h1 = 30° and h2 = 150°, the behavior of the droplet at ‘ ¼ 0:15L shows a similar trend to that at ‘ ¼ 0:1L, as shown in Figs. 6 and 14. At the beginning, the droplet symmetrically spreads out from the hemispherical shape in the direction of the square hydrophilic bottom surface whose contact angle is h1 = 30°. After that, a necking phenomenon appears as the droplet continues to move, and finally the whole droplet on the hydrophobic surface moves to the hydrophilic area. However, as the width of the hydrophobic surface increases from ‘ ¼ 0:1L to ‘ ¼ 0:15L, the movement and separation speeds of the droplet increase. When ‘ ¼ 0:15L, a necking phenomenon occurs at t = 0.419 where the width of the droplet located on the hydrophobic bottom surface decreases as the droplet located on the hydrophobic bottom surface moves in the direction of the hydrophilic bottom surface. After that, as the dimensionless time passes during 0:419 6 t 6 13:975, the necking phenomenon continues the progress to gradually reduce the width of the droplet located on the hydrophobic surface and separate the droplet into two at t = 13.975 which is earlier than the case of ‘ ¼ 0:1L by the dimensionless time of 7.127. The whole droplet separated at t = 13.975 then moves from the hydrophobic surface to the hydrophilic area at t = 15.513 which is earlier than the case of ‘ ¼ 0:1L by the dimensionless time of 10.342. Fig. 15 shows the change in the dimensionless separation time of the droplet depending on the contact angle of the square hydrophilic surface, h1, and the contact angle of the rectangular hydrophobic surface, h2, when ‘ ¼ 0:15L. Similar to the case of ‘ ¼ 0:1L as shown in Fig. 10, when ‘ ¼ 0:15L, the dimensionless separation time of the droplet decreases as the contact angle of the rectangular hydrophobic surface, h2, increases, while it increases as the contact angle of the square hydrophilic surface, h1, increases. When the width of the hydrophobic surface increases from ‘ ¼ 0:1L to ‘ ¼ 0:15L, the contact area between the hydrophobic surface and the droplet increases, promoting the movement of the droplet to the hydrophilic surface. As a result, the dimensionless separation time of the droplet rapidly decreases with increasing width of the hydrophobic surface from ‘ ¼ 0:1L to ‘ ¼ 0:15L.
Fig. 17. Change in the dimensionless separation time of the droplet depending on the contact angle of the square hydrophilic surface, h1, and the contact angle of the rectangular hydrophobic surface, h2, when ‘ ¼ 0:2L.
Fig. 16 shows the dynamic behavior of a droplet located at the center of the bottom surface when a rectangular hydrophobic surface with h2 = 150° and a width of ‘ð¼ 0:2LÞ exists at the center of the square hydrophilic surface with h1 = 30° and a side length of L. When h1 = 30° and h2 = 150°, the behavior of the droplet at ‘ ¼ 0:2L is generally similar to that at ‘ ¼ 0:1L and ‘ ¼ 0:1L, as shown in Figs. 6, 14 and 16. However, the movement and separation speed of the droplet further increases as the width of the hydrophobic surface increases, causing a necking phenomenon to occur at t = 0.419 when ‘ ¼ 0:2L. After that, as the dimensionless time passes during 0:419 6 t 6 13:556, the necking phenomenon continues the progress to gradually reduce the width of the droplet located on the hydrophobic surface and separate the droplet into two at t = 13.556 which is earlier than the case of ‘ ¼ 0:1L by the dimensionless time of 7.127. The whole droplet separated at t = 13.556 then moves from the hydrophobic surface to the
S.W. Son et al. / Computers & Fluids 105 (2014) 294–306
305
increasing value of h1. Thus, the ranges of the contact angles of h1 and h2, in which the droplet is separated, are 10 6 h1 6 30 at h2 = 110°, 10 6 h1 6 40 at h2 = 120°, 10 6 h1 6 50 at 130 6 h2 6 140 , and 10 6 h1 6 60 at 150 6 h2 6 170 , as shown in Fig. 18.
4. Conclusion
Fig. 18. Map of droplet separation (denoted by the gray color) depending on the changes in the contact angle of the square hydrophilic surface, h1, and the contact angle of the rectangular hydrophobic surface, h2.
hydrophilic area at t = 14.954 which is earlier than the case of ‘ ¼ 0:1L by the dimensionless time of 9.783. Fig. 17 shows the change in the separation time of the droplet depending on the contact angle of the square hydrophilic surface, h1, and the contact angle of the rectangular hydrophobic surface, h2, when ‘ ¼ 0:2L As the contact angle of the rectangular hydrophobic surface, h2, increases, the separation time of the droplet decreases, and the separation time of the droplet increases as the contact angle of the square hydrophilic surface increases as in the cases of ‘ ¼ 0:1L and ‘ ¼ 0:15L shown in Figs. 10 and 15, respectively. When the length increases from ‘ ¼ 0:15L to ‘ ¼ 0:2L, the contact area between the hydrophobic surface and the droplet increases promoting movement of the droplet. On the other hand, the distance between the hydrophilic surface and the droplet also increases along with it increasing the time it takes for the droplet to move to the hydrophilic area. Accordingly, the decrease in the dimensionless separation time of the droplet caused by the increase from ‘ ¼ 0:15L to ‘ ¼ 0:2L is minor in comparison to that caused by the increase from ‘ ¼ 0:1L to ‘ ¼ 0:15L. Fig. 18 shows whether the droplet separation phenomenon appears or not depending on the changes in the contact angle of the square hydrophilic surface, h1, and the contact angle of the rectangular hydrophobic surface, h2. The droplet separation does not depend on the width of the hydrophobic surface, ‘, when ‘ has the value of 0.1L, 0.15L and 0.2L considered in this study, but depends on the values of h1 and h2. The gray-colored zone in Fig. 18 represents the contact angles, h1 and h2, at which the droplet is separated as time passes. When the contact angle of the rectangular hydrophobic surface, h2, is h2 = 100°, the droplet is not separated in the region of 10 6 h1 6 90 , because the hydrophobicity of the hydrophobic surface is not large. However, if the hydrophobicity of the hydrophobic surface increases slightly as the contact angle of the hydrophobic surface, h2, increases to h2 = 110°, the droplet is separated in the region of h1 6 30 where the hydrophilicity of the hydrophilic surface is strong. If the hydrophobicity of the hydrophobic surface increases further with h2 P 120 , the range of the square hydrophilic surface contact angle, h1, in which the droplet is separated, increases, even when the hydrophilicity of the hydrophilic surface decreases with
A numerical analysis was conducted for the behavior of a droplet located on a heterogeneous surface using the 3D Lattice Boltzmann Method. The contact angle range of the hydrophilic surface considered in this study was 10 6 h1 6 90 , and that of the hydro phobic surface was 100 6 h2 6 170 . Also, the change in the behavior of the droplet depending on the area of the hydrophobic surface was observed by changing the width of the hydrophobic surface as ‘ ¼ 0:1L, ‘ ¼ 0:15L and ‘ ¼ 0:2L. The diameter and the height of the droplet used in this calculation are D = 3 mm and H = 1.5 mm, resulting in D/L = 0.375 and H/L = 0.1875, respectively. In the case of a droplet located on a heterogeneous surface where a rectangular hydrophobic surface with a width of ‘ and a longitudinal length of L exists at the center of a square hydrophilic bottom surface with a side length of L, whether or not the droplet is separated and the dimensionless separation time are determined by the sizes of the contact angle of the hydrophilic surface, h1, and the contact angle of the hydrophobic surface, h2. The droplet is easily separated if the difference between the contact angle of the hydrophilic surface h1 and the contact angle of the hydrophobic surface h2 is big, and is not easily separated if the difference between the contact angles of the hydrophilic surface and the hydrophobic surface, h2 – h1, is smaller than 80°. The droplet located on a heterogeneous surface symmetrically spreads out from its initial hemispherical shape in the direction of the hydrophilic surface which has a relatively big adhesion. At this time, if the difference between the contact angles of the hydrophilic surface and the hydrophobic surface is not big enough, the droplet is not separated but reaches a new equilibrium state lying across the heterogeneous surfaces. But, if the difference between the contact angles of the two heterogeneous surfaces is big enough, the droplet located on the hydrophobic surface moves in the direction of the hydrophilic bottom surface causing a necking phenomenon where the width of the droplet located on the hydrophobic surface reduces. After that, the necking phenomenon continues the progress to gradually reduce the width of the droplet located on the hydrophobic surface and separate the droplet into two. When the separation of the droplet is observed, the more the contact angle of the hydrophilic surface, h1, increases, the more the dimensionless time it takes the droplet to be separated increases, while, the more the contact angle of the hydrophilic surface, h2, increases, the more the dimensionless separation time of the droplet decreases. Also, though the movement of the droplet is accelerated if the width of the hydrophobic surface ‘ increases and, as a result, the dimensionless separation time of the droplet shows a decreasing trend, the width of the hydrophobic surface ‘ does not have any effect on whether or not the droplet is separated in the conditions considered in this study. Acknowledgment This research was supported by Leading Foreign Research Institute Recruiment Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT & Future Planning (No. 2009-00495), and also by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (No. 2013R1A2A2A01067251).
306
S.W. Son et al. / Computers & Fluids 105 (2014) 294–306
Appendix A The two terms induced in Eq. (14) are obtained through the following process:
@nx @ny @nz @x i;j;k @y i;j;k @z i;j;k 1 ¼ : nx;iþ1=2;jþ1=2;kþ1=2 þ nx;iþ1=2;j1=2;kþ1=2 2 Dx
ðr nÞ ¼ @nx @x i;j;k
þnx;iþ1=2;jþ1=2;k1=2 þ nx;iþ1=2;j1=2;k1=2 nx;i1=2;jþ1=2;kþ1=2 nx;i1=2;j1=2;kþ1=2 nx;i1=2;jþ1=2;k1=2 nx;i1=2;j1=2;k1=2 ni;j;k nx @jnj ny @jnj þ r jnj ¼ jni;j;k j jnj i;j;k @x i;j;k jnj i;j;k @y i;j;k nz @jnj þ jnj i;j;k @z i;j;k i niþ1=2;jþ1=2;kþ1=2 ¼ C iþ1;j;k þ C iþ1;jþ1;k þ C iþ1;j;kþ1 2Dx
þC iþ1;jþ1;kþ1 C i;j;k C i;jþ1;k C i;j;kþ1 C i;jþ1;kþ1 j þ C i;jþ1;k þ C iþ1;jþ1;k þ C i;jþ1;kþ1 þ C iþ1;jþ1;kþ1 C i;j;k 2Dy
k C iþ1;j;k C i;j;kþ1 C iþ1;j;kþ1 þ C i;j;kþ1 þ C iþ1;j;kþ1 2 Dz
þC i;jþ1;kþ1 þ C iþ1;jþ1;kþ1 C i;j;k C iþ1;j;k C i;jþ1;k C iþ1;jþ1;k
ðA1Þ
ðA2Þ
ðA3Þ
ðA4Þ
The phase boundary was established through the process described above, and an additional force was set along the boundary surface so that the two fluids do not mix with each other. References [1] Trevino C, Forro-Fontan C, Mendez F. Asymptotic analysis of axisymmetric drop spreading. Phys. Rev. E 1998;58:4478–84.
[2] Yang J, Koplik J. Molecular dynamics of drop spreading on a solid surface. Phys. Rev. Lett. 1991;67:3539–42. [3] Gu Y, Li D. Model for a liquid drop spreading on a solid surface. Colloids Surf. A. Physicochem. Eng. Aspects 1998;142:243–56. [4] Bayer IS, Megaridis CM. Contact angle dynamics in droplets impacting on flat surfaces with different wetting characteristics. J. Fluid Mech. 2006;558:415–49. [5] Cossali GE, Coghe A, Marengo M. The impact of a single drop on a wetted solid surface. Exp. Fluids 1997;22:463–72. [6] Mao T, Kuhn DCS, Tran H. Spread and rebound of liquid droplet upon impact on flat surfaces. AIChE. J. 1997;43:2169–79. [7] Sikalo S, Marengo M, Tropea C, Ganic EN. Analysis of impact of droplets on horizontal surfaces. Exp. Therm. Fluid Sci. 2002;25:503–10. [8] Zhang X, Basaran OA. Dynamic surface tension effects in impact of a drop with solid surface. J. Colloid Interf. Sci. 1997;187:166–78. [9] Fujimoto H, Ogino T, Takuda H, Hatta N. Collision of a droplet with a hemispherical static droplet on a solid. Int. J. Multiphase Flow 2001;27:1127–245. [10] Manservisi S, Scardovelli R. A variational approach to the contact angle dynamics of spreading droplets. Comput. Fluids 2009;38:406–24. [11] Tanaka Y, Washio Y, Yoshino M, Hirata T. Numerical simulation of dynamic behavior of droplet on solid surface by the two-phase Lattice Boltzmann Method. Comput. Fluids 2011;40:68–78. [12] Lunkad SF, Buwa VV, Nigam KDP. Numerical simulations of drop impact and spreading on horizontal and inclined surfaces. Chem. Eng. Sci. 2007;62:7214–24. [13] Sikalo S, Tropea C, Ganic EN. Impact of droplets onto inclined surface. J. Colloid Interf. Sci. 2005;286:661–9. [14] Shan X, Chen H. Lattice Boltzmann model of simulating flows with multiple phases and components. Phy. Rev. E 1993;47:1815–9. [15] Swift MR, Osborn WR, Yeomans JM. Lattice Boltzmann Simulation of nonideal fluids. Phy. Rev. Lett. 1995;75:830–3. [16] Gunstensen AK, Rothman DH, Zaleski S, Zanetti G. Lattice Boltzmann model of immiscible fluids. Phys. Rev. A 1991;43:4320–7. [17] Wu L, Tsutahara M, Kim LS, Ha MY. Three-dimensional lattice Boltzmann simulation of droplet formulation in a cross-junction microchannel. Int. J. Multiphase Flow 2008;34:852–64. [18] Bhatnagar PL, Gross EP, Krook M. A model for collision processes in gases. I: small amplitude process in charged and neutral one-component system. Phy. Rev 1954;94:511. [19] Yan YY, Zu YQ. A lattice Boltzmann method for incompressible two-phase flow in particle wetting surface with large density ratio. J. Comp. Phys. 2007;227:763–75.