ZHANG Ren-liang, DI Qin-feng, WANG Xin-liang, DING Wei-peng, GONG Wei

Shanghai Institute of Applied Mathematics and Mechanics and Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai University, Shanghai 200072, China, E-mail: zhrleo@gmail.com

(Received December 9, 2011, Revised March 1, 2012)

NUMERICAL STUDY OF THE RELATIONSHIP BETWEEN APPARENT SLIP LENGTH AND CONTACT ANGLE BY LATTICE BOLTZMANN METHOD*

ZHANG Ren-liang, DI Qin-feng, WANG Xin-liang, DING Wei-peng, GONG Wei

Shanghai Institute of Applied Mathematics and Mechanics and Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai University, Shanghai 200072, China, E-mail: zhrleo@gmail.com

(Received December 9, 2011, Revised March 1, 2012)

The apparent slip between solid wall and liquid is studied by using the Lattice Boltzmann Method (LBM) and the Shan-Chen multiphase model in this paper. With a no-slip bounce-back scheme applied to the interface, flow regimes under different wall wettabilities are investigated. Because of the wall wettability, liquid apparent slip is observed. Slip lengths for different wall wettabilities are found to collapse nearly onto a single curve as a function of the static contact angle, and thereby a relationship between apparent slip length and contact angle is suggested. Our results also show that the wall wettability leads to the formation of a low-density layer between solid wall and liquid, which produced apparent slip in the micro-scale.

Lattice Boltzmann Method (LBM), wettability, apparent slip, contact angle, nano-particles adsorbing method

Introduction

The no-slip boundary condition, which states that the fluid velocity at a fluid-solid interface equals that of the solid surface has been proven valid at macroscopic scales, but is not fulfilled at microscopic scales[1]. A series of experiments[2,3]and numerical results[4,5]have found fluid slip at the boundaries of the flow channels in microfluidics. As the typical length scale of the micro-fluid flows gets smaller, the effect of boundary slip becomes more prominent, and has been paid more attention in engineering applications[6]. Based on this slip boundary effect, artificial super-hydrophobic surfaces have been widely used in industrial production and daily life. For example, self-cleaning paints, roof tiles, fabrics and glass windows that can be cleaned by a simple rainfall and the nanoparticles adsorbing method[2]in improving oil recovery are all in practice.

Slip length is of great importance in calculation of drag and other hydrodynamic properties of fluid flowing through micro-channels or over nano-scale patterned surfaces, but it is very difficult to directly measure the apparent slip length accurately in experiments. In 1823, Navier proposed a slip boundary condition that the fluid velocity at a point on a surface is proportional to the shear rate at the same point, but the value of slip length is hard to be determined. Chen et al.[7]investigated the Couette flows by means of a two-phase mesoscopic Lattice Boltzmann Method (LBM), and the results showed that there is a strong relationship between the magnitude of slip length and the solid-fluid interaction, but it is a very difficult or even impossible task to compute the exact slip length in dependence of interaction and it is also difficult to be applied to engineering.

In this work, we focus on investigating the effects of wall wettabilities on the slip length in order to get a slip length model governed by contact angle. Using this model, it is easy to estimate the slip length because the contact angle is a parameter that can beeasily measured.

1. Numerical model

To simulate non-ideal multiphase fluid flows, the attractive or repulsive interaction among molecules, which is referred to as the non-ideal interaction, should be included in the LBM. There are many approaches to incorporate non-ideal interactions, such as the color-fluid model, interparticle-potential model, free-energy model, mean-field theory model and so on. The interparticle-potential model proposed by Shan and Chen[8,9]is to mimic microscopic interaction forces between the fluid components. This model modified the collision operator by using an equilibrium velocity that includes an interactive force. This force guarantees phase separation and introduces surface-tension effects. This model has been applied with considerable success in measuring contact angles[10]and examining the effect of wall wettabilities, topography and micro-structure on drag reduction of fluid flow through micro-channels[11]. As an extension of the Shan-Chen model, Benzi et al.[12]first derived an analytical expression for the contact angle and the surface energy between any two of the liquid, solid and vapor phases.

The LBM, which involves a single relaxation time in the Bhatnagar-Gross-Krook (BGK) collision operator, is used here. The time evolution of this model can be written as

where fi(x,t)is the single-particle distribution function for fluid particles moving in the direction ciat (x,t), feq(x,t) the equilibrium distributionfun

i ction, Δt the time step of simulation, and parameter τ the relaxation time characterizing the collision processes by which the distribution functions relax towards their equilibrium distributions.

In the two-dimensional (2-D) squ are lattice with nine velocities model[13], the equilibrium distribution function,(x,t), depends only on local density and velocity and can be chosen as the following form

where

c=Δx/Δt is the lattice velocity,Δx the lattice distance, and Δt the time step of simulation, and ρ the fluid density, which can beobtained from ρ=

The macroscopic velocity u is given by

InEq.(6), F and Fadsare the fluid-fluid interact ion force and fluid-solid interaction force, respectively. The fluid-fluid interaction is obtained by using an attractive short-range force[8]

where G is the interaction strength, which is used to co ntrol the two-phase liquid in teraction, and is negative for particle attraction,ψis the in teraction potential, which is defined as[14]

ψ0and ρ0are constants.As in Ref.[14] wiis taken as 1/9 for i=1,2,3,4,wias1/36 for i= 5,6,7,8, and 0 for i=0.

It is simple todescribe the interaction between a fluidand a wall by introducing an extra force, and this method was first used by Martys and Chen[15]. The idea is to create an analogue to the particle-particle interaction force usedto induce phase separation, and in this paper the corresponding equation is[14]

Here s=0,1 for the fluid and the solid phase, respectively, and the adhesion parameter Gadsis used to control the wettability behavior of the liqu id at solid surfaces. It can be seen that Eq.(9) incorporates an adhesive interaction between fluid and surfaces.

And the relaxation time tunes the kinematic viscosityas

The equation of state in the Shan-Chen model s[14]i

where P is the pressure.

Fig.1 Static contact angle

Fig. 2 Contact angles as a function of Gabsfor different value of G

2.Numerical simulation of contact angle

The LBM simulations were carried outin a 2-D domain. The grid mesh used is 50×200. In thesimulation, as in Ref.[11], the general non-slip bounce-back scheme[16]was employed for the solid-fluid interfaces, and periodic boundary conditions were applied at both inlet and outlet ends along the horizontal direction. A droplet with the diameter of 30, is set at the middle between two ends. After 30 000 time steps, the result tendsto be stabilized.Figure 1 shows an example of static contact angle, which is 127.6owhen G=-120, Gabs=-130and ψ(x,t)=4e-200/ρin Eqs.(7) and (9). The values of parameters G and Gabsand the simulated contact angles for each case are listed in Fig.2.

As is shown in Fig.2, Gabsand G determine the value of contact angle. Gabsrepresentsthe strength of intera ctionbetween fluid and solid surface and G the strength of interaction between fluid particles. A negative Gabsindicates attractive interaction. WhenG is gi ven, the greater the ma gnitude of, the stronger the reaction, thus resulting in smaller contact angle. The contact angle is a static parameter of measuring the wettability of a liquid on a solid surface, and it can be easily measured. In our simulation, wecanalso change the parameter Gabsto simulate arbitrary contact angle, and then easily obtain different wall wettabilities. Form Fig.2, we also see that both parameters of Gabsand G have significant impact on the simulated contact angle.

Fig.3 Velocity profiles with different contact angles

Fig.4Density profiles with different contact angles

3. Nu merical simulation of slip length

In order to investigate the effects of wallwettabilities on the slip length, we conducted numerical simulations for the 2-D Poiseuille flows. Typical density and velocity profiles of the pressure-drivenPoiseuille flows are displayed in Figs.3 and 4 (given G=-120).The ordinate in both figures represents the distance from one of the solid surface boundaryto the other. Theabscissa in Fig.3 is the normalized velocity, where v0is the max velocity measured at the center in the channel for the case of no slip. The abscissa in Fig.4 is the normalized fluid density, where0ρ is the liquid density for the case of no slip. The pressure gradient is specified as 5×10-3in lattice unit for both cases. Different contact angles (as shown in both Figs.3 and 4) can be simulated by specifying different values of the adhesion parameter (absG). All simulations were run until static equilibriumwas nearly attained, and then a pressure gradientof 5×10-3was applied in the x-direction (flow direction).

Fig.5 Velocity profiles of simulated and fitted

As is shown in Fig.3, the fluid velocity approaches zero as y→-24.5 (the lower boundary) or y→(the upper boundary), which is consistent with the bounce-back boundary condition specified at the two boundaries. However, the fluid velocity increases dramatically in a very thin layer nearthe boundary. The layer is so thin that it is hard to see its details near the boundary, and velocity at the boundary looks like non-zero when plotted ina larger scale, as shown in Fig.3. In a macro-scale, velocity at a boundary is nearly zero, but in a micro-scale, the velocity appears non-zero (so called apparent slip). Plots shown in Fig.3 clearly indicate that the slip velocity at the boundary increases as contact angle increases. In order to understand the physical mechanism of such kind of phenomenon, the density profiles of fluid with different contact angles is drawn in Fig.4. One should note that the density of the fluid with zero contact angle is constant (as shown by the circle symbol vertical line in Fig.4). However, a sharp reduction of fluid density near the boundary is observed for a fluid with nonzero contact angle, which clearly indicates a layer of much less dense fluid (most probably gas) is induced between the dense liquid and solid surface. Fitting for the velocity data points where the density is approximately constant, we can get a parabolic of velocity profiles, and extrapolating the fitted profiles to zero velocity yields a slip length as shown in Fig.5. Given an interaction strength (G), different slip lengths can be obtained by using a series values of the adhesion parameter (absG), as shown in Fig.6. As was discussed above, the parameter ofabsG controls the interaction between the fluid and solid surface. IncreasingabsG decreases the attraction (or increases the repulsion) between the liquid and solid surface, and thus attracts more gas to the surface. The less dense of the fluid at the surface, the less viscous shear force, and the more significant slippage appears. LBM simulation[17]and observed in MDS[18]. Compa-

From Figs.3 and 4 one can see that the wetting properties of the wall have an important influence on the velocity profile, especially the slip velocity at the boundary. There is a low-density layer between the bulk liquid and the wall, and the more hydrophobic the wall is, the lower density of fluid is (see Fig.4). This result is similar to those obtained fromother red with the macroscopic flow, the ratio of the lowdensity layer region to the inner region is larger in the microscopic flow, and this is the main difference between micro-flow and macro-flow. Thus, the slip cannot be ignored in the micro-flow.

Fi g.6 Slip length as a function of Gabsfor different G

From Figs.2 and 6, we can see thatthe contact angle and slip length are associated withthe interaction strength (G) and adhesion parameter (Gabs). Numerical results of contact angles and slip lengths, represented by different symbols shown inFig.7, correspond to different values of G, respectively, represented by squares (G=-120), circle (G=-125), delta (G=-130) and gradient (G=-135). The numerical results shownin Fig.7 cover awide range of contact angles, rangingfrom18oto 150o.Results shown in Fig.7 clearly indicate that slip length is a function of contact angle. The relationship between slip length and contact angle can beeasily obtained by fitting those numerical data (see Eq.(12)) as shown by the solid curve inFig.7.

where θ is contact angle andLδ is slip length.

Fig.7 The slip length against contact angle

Form Fig.7 we can see that the slip lengthson different surfaces are found to collapse nearly onto a single curve as a function of the static contact angle characterizing the surface wettability. As was discussed above, the slip length is the result of interaction between the liquid and the wall surface, and it depends on the properties of the liquid (e.g., G) and the wall surface (e.g.,absG). So the simulation of slip length should take into account all parameters of the interaction objects, but it is a very hard task[19]to do this. In fact, the contact angle is a more comprehensive expression of interactions between liquid and solid surface, and the weaker the solid-fluid interaction is, the larger the contact angle is. So it may be a feasible and advisable way to use contact angle as control parameter to simulate the slip length.

4. Conclusion

The LBM has been successfully applied in this work to simulate contact angle of a fluid on a solid surface and slip length of a fluid flowing over a solid surface. Both the non-ideal interaction between fluid particles (controlled by a parameter of G) and the interaction between a fluid and a solid (c ontrolled by another parameter ofabsG) have significant impact on the contact angle and slip length. However, they have little influence on the relationship between the contact angle and slip length. Our numerical results confirm that there is a low-density layer between the bulk liquid, and that the wetting properties of the wall have an important influence on the velocity profile. The more hydrophobic the wall is, the lower density of fluid is. Thus, the slip cannot be ignored in the microflow.

[1] HYVALUOMA J., HARTING J. Slip flow over structured surfaces with entrapped microbubbles[J]. Physi- cal Review Letters, 2008, 100(24): 246001.

[2] DI Qin-feng, SHEN Chen and WANG Zhang-hong et al. Experimental research on drag reduction of flow in micro-channels of rocks using nano-particle adsorption method[J]. Acta Petrolei Sinica, 2009, 30(1): 125- 128(in Chinese).

[3] BURTON Z., BHUSHAN B. Hydrophobicity, adhesion, and friction properties of nanopatterned polymers and scale dependence for micro-and nanoelectromechanical systems[J]. Nano Letters, 2005, 5(8): 1607-1613.

[4]ZHANG J. Lattice Boltzmann method for microfluidics: Models and applications[J]. Microfluidics and Nano- fluidics, 2010, 10(1): 1-28.

[5] KHAN M., HAYAT T. and WANG Y. Slip effects on shearing flows in a porous medium[J]. Acta Mechannic Sinica, 2008, 24(1): 51-59.

[6]WANG Xin-liang, DI Qin-feng and ZHANG Ren-liang et al. Progress in theories of super-hydrophobic surface slip effect and its application to drag reduction technology[J]. Advances in Mechanics, 2010, 40(3): 241- 249(in Chinese).

[7] CHEN Yan-yan, YI Hou-hui and LI Hua-bing Boundary slip and surface interaction: A lattice Boltzmann simulation[J]. Chinese Physics Letters,2008, 25(1): 184-187.

[8]SHAN X., CHEN H. Lattice Boltzmann model for simulating flows with multiple phases and components[J]. Physical Review E, 1993, 47(3): 1815-1819.

[9]SHAN X., CHEN H. Simulation of nonideal gases and liquid-gas phase transitions by the lattice Boltzmann equation[J]. Physical Review E, 1994, 49(4): 2941- 2948.

[10] HUANG H., THORNE D. T. Jr and SCHAAP M. G. et al. Proposed approximation for contact angles in Shanand-Chen-type multicomponent multiphase lattice Boltzmann models[J]. Physical Review E, 2007, 76(6): 1-6.

[11] ZHANG Ren-liang, DI Qin-feng and WANG Xin-liang, et al. Numerical study of wall wettabilities and topographyon drag reduction effect in micro-channel flow by lattice Boltzmann method[J]. Journal of Hydrodynamics, 2010, 22(3): 366-372.

[12]BENZI R., BIFERALE L. and SBRAGAGLIA M. et al. Mesoscopic modeling of a two-phase flow in the presenceof boundaries: The contact angle[J]. Physical Review E, 2006, 74(2): 021509.

[13] QIAN Y. H., D?HUMIERES D. and LALLEMAND P. Lattice BGK models for Navier-Stokes equations[J]. Europhysics Letters, 1992, 17(6): 479-484.

[14]SUKOP M. C., THORNE D. T. Lattice Boltzmann modeling: An introduction for geoscientists and engineers[M]. Berlin, Heidelberg, Germany: Springer- Verlag, 2006.

[15] MARTYS N. S., CHEN H. Simulation of multicomponent fluids in complex three-dimensional geometries by the lattice Boltzmann method[J]. Physical Review E, 1996, 53(1): 743-750.

[16] GUO Zhao-li, ZHENG Chu-guang. The principle and application of the method[M]. Beijing: Science Press, 2009(in Chinese).

[17]ZHANG J., KWOK D. Y. Apparent slip over a solidliquid interface with a no-slip boundary condition[J]. Physical Review E, 2004, 70(5): 1-4.

[18] CAO Bing-Yang, CHEN Min and GUO Zeng-Yuan. Velocity slip of liquid flow in nanochannels[J]. Acta Physica Sinica, 2006, 55(10): 5305-5310(in Chinese).

[19] HARTING J., KUNERT C. and HERRMANN H. J. Lattice Boltzmann simulations of apparent slip in hydrophobic microchannels[J]. Europhysics Letters, 2006, 75(2): 328-334.

10.1016/S1001-6058(11)60275-8

* Project supported by the National Natural Science Foundation of China (Grant No. 50874071), the National High Technology Research and Development of China (863 Program, Grant No. 2008AA06Z201), the Key Program of Science and Technology Commission of Shanghai Municipality (Grant No. 071605102) and the Leading Talent Funding of Shanghai.

Biography: ZHANG Ren-liang (1982-), Male, Ph. D. Candidate

DI Qin-feng,

E-mail: qinfengd@sina.com