research article dynamic behaviors and energy transition … · 2019. 7. 30. · using a lattice...

Research ArticleDynamic Behaviors and Energy Transition Mechanism ofDroplets Impacting on Hydrophobic Surfaces

Qiaogao Huang Ya Zhang and Guang Pan

School of Marine Science and Technology Northwestern Polytechnical University Xirsquoan Shaanxi 710072 China

Correspondence should be addressed to Qiaogao Huang huangqiaogao163com

Received 23 December 2015 Accepted 3 April 2016

Academic Editor Luisa Di Paola

Copyright copy 2016 Qiaogao Huang et alThis is an open access article distributed under the Creative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The wettability of hydrophobic surfaces and the dynamic behaviors of droplets impacting on hydrophobic surfaces are simulatedusing a lattice Boltzmann method and the condition for the rebound phenomenon of droplets impacting on solid surfaces isanalyzed The results show that there is a linear relationship between the intrinsic contact angle and the interaction strength offluid-wall particles For hydrophobic surfaces with the same intrinsic contact angle the micromorphology can increase the surfacehydrophobicity especially the hierarchical micromorphology The dynamic behaviors of droplets impacting on solid surfaces areaffected by thewettabilityThe surface hydrophobicity is stronger and the reboundphenomenonoccurs easier If the dropletrsquos kineticenergy is greater than the sumof the surface energy and theminimumconversion gravitational potential energywhen the spreadingand shrinking finish the rebound phenomenonwill occur As the hydrophobic surfacersquos viscous dissipation ismuch smaller than thehydrophilic surfacersquos the droplet still has high kinetic energy after the spreading and shrinking which is advantageous to reboundfor droplets

1 Introduction

Recently more and more researches demonstrate that hydro-phobic surface has a good applied prospect in aspects of fluiddrag reduction flow noise reduction anticontaminationanticorrosion and so forth [1ndash3] For instance it is used asdrag reduction and anticontamination coating of pipersquos innersurface and shiprsquos hull as self-cleaning coating of exteriorwalls and satellite receivers and as anti-icing and antifrost-ing coating of components of aeronautics and astronauticswhich with no exception are related to the movement ofdroplets on hydrophobic surfaces especially the dynamicbehaviors of droplets impacting on hydrophobic surfacesDavidson [4] investigated droplets impinging on solid sur-faces using a boundary integral method Pasandideh-Fard etal [5] simulated the solidification ofmoltenmetal droplets onboth horizontal and inclined stainless steel surfaces withVOF(Volume of Fluid) and continuous surface tension modelFujimoto et al [6] also employed VOF to study the effect ofthe impact angle on the deformation behavior of dropletsAlthough several computational models for studying thedynamic behaviors of droplet impacting on solid surface have

been developed by researchers the difficulty of this issue stillrelies on how to accurately track the position of the freesurface Lattice Boltzmann method as a new computationalfluid method operates on both mesoscopic and macroscopiclevels successfully In particular when things come to thenumerical simulation of multiphase and multicomponentfluid flows Shan-Chen model of lattice Boltzmann methodcan realize thermodynamic phase transition easily withan appropriate potential function [7 8] Huang et al [9]studied the movement of a droplet inside a grooved channelusing lattice Boltzmann method Shi et al [10] simulatednumerically the droplet motion driven by Marangoni effectwhich is induced by surface tension gradient on solid-liquidinterface with LBM However their main concern was themovement of droplets on solid surfaces and little attentionwas paid on the hydrodynamic behaviors and energy tran-sition mechanism of droplets impacting on solid surfacesHence this paper investigates the dynamic behaviors ofdroplets impacting on solid surfaces with different wettabilityusing LBM and discusses the occurring condition of dropletsbouncing when the solid surface gets hit from the view ofenergy

Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2016 Article ID 8517309 9 pageshttpdxdoiorg10115520168517309

2 Discrete Dynamics in Nature and Society

2 Numerical Method

21 Lattice Boltzmann Method Lattice Boltzmann methodbased on mesoscopic kinetic models has developed into analternative and promising numerical scheme for simulatingfluid flows in recent years Lattice Boltzmann method pre-sented in this paper is based on the streaming and collisionof discrete particle density distribution function in the fixedlattice point and the corresponding evolution function isexpressed as [11]

119891120572(x + e

120572120575119905 119905 + 120575

119905) minus 119891120572(x 119905)

= minus1

120591[119891120572(x 119905) minus 119891(eq)

120572(x 119905)]

(1)

where 119891120572(x 119905) is the particle density distribution function

which depends on position x the particle discrete velocity e120572

and time 119905119891(eq)120572

(x 119905) is the equilibriumdistribution functionand 120591 is the dimensionless relaxation time related to thehydrodynamics viscosity

The left-side of lattice Boltzmann equation represents thestreaming of particles while the right-side represents thecollision process whichmakes119891 heading for the equilibriumdistribution 119891(eq) In D2Q9 model [11] the equilibriumdistribution 119891(eq) is given by

119891(eq)120572

(x 119905) = 120588120596120572[1 +

e120572sdot u1198882119904

+(e120572sdot u)2

21198884119904

minus1199062

21198882119904

] (2)

The discrete velocities are set as

[e0 e1 e2 e3 e4 e5 e6 e7 e8]

= [0 1 0 minus1 0 1 minus1 minus1 1

0 0 1 0 minus1 1 1 minus1 minus1] 119888

(3)

where 119888 = 120575119909120575119905is lattice velocity 120575

119909and 120575

119905are the lattice

length and the time step respectively 119888119904= 119888radic3 is the sound

speed in the discretemodel and theweighting factors dependon the link angle (120596

120572= 49 (120572 = 0) 120596

120572= 19 (120572 = 1 2 3 4)

120596120572= 136 (120572 = 5 6 7 8))The macroscopic density and velocity are defined as

follows

120588 = sum120572

119891120572

u =1

120588sum120572

119890120572119891120572

(4)

The macroscopic pressure is given by 119901 = 1205881198882119904directly

22 Shan-Chen Model To introduce intermolecular forcesbetween microscopic particles Shan and Chen proposed apseudopotential model [12 13] which easily coupled micro-forces the dominant role of microflow problems The modelreflects the physical nature of fluid dynamics and broadensthe application of lattice Boltzmann method in range ofsimulation of microflow on complex surface with wettability

In Shan-Chen model fluid particles at site of x receive forcesfrom nearest neighbor fluid particles and solid walls whichare written in the following form respectively

Fint (x 119905) = minus119866120595 (x 119905)8

sum120572=1

119908120572120595 (x + e

120572120575119905 119905) e120572 (5)

Fads (x 119905) = minus119866119904120595 (x 119905)

8

sum120572=1

119908120572swi (x + e

120572120575119905) e120572 (6)

119866 in (5) is the interaction strength and 119866119904in (6) is

called adsorption parameter representing the strength of theforce contributed by solid Varying the 119866

119904parameter allows

simulation of different solid surface wettability where swiis a ldquoswitchrdquo that takes on value one if the site at x + e

120572120575119905

is a solid and is zero otherwise 119908120572is the same direction-

dependent weighting factor used before 120595(x 119905) is the inter-action potential function which must be monotonicallyincreasing and boundedTherefore the equation used in thispaper is expressed as 120595(x 119905) = 120588

0[1 minus exp(minus120588120588

0)]

In Shan-Chen model the effect of interparticle forces isincorporated into the equilibrium distribution function feqby shifting the equilibrium velocity ulowast

120588 (x 119905) =8

sum120572=0

119891120572(x 119905)

120588 (x 119905) ulowast (x 119905) =8

sum120572=0

e120572119891120572(x 119905) + 120591 (Fint + Fads)

(7)

The equation of state is given by

119901 = 1205881198882119904+119866

61205952 (8)

3 Numerical Simulation of the Wettability ofHydrophobic Surfaces

A key parameter for characterizing the wettability ofhydrophobic surface is the apparent contact angle which ismainly determined by material properties and micromor-phology This provides a new research route first determinethe relationship between the adsorption parameter119866

119904and the

intrinsic contact angle at a specific interaction strength 119866 bysimulating the contact angle of a smooth solid surface thenconstruct microtopography on the surface and simulate theapparent contact angle of real solid surface

31 Determining the Intrinsic Contact Angle The intrinsiccontact angle of the smooth solid surface is adjusted bychanging the adsorption parameter 119866

119904for a specific inter-

action strength 119866 = minus58 The computational domain is arectangular space of 500 times 500 lattice units and a latticeunits is approximately 12120583minphysical scale Dimensionlesstime 120591 = 10 and gravity acceleration 119892 = 05 times 10minus6Periodic scheme is used in the left and right boundaries whileno-slip bounce back scheme is used in the top and bottomboundaries A circular liquid droplet with a diameter of 150is placed in the middle of the bottom wall

Discrete Dynamics in Nature and Society 3

Gs = minus08

(a) CA = 1593∘

Gs = minus12

(b) CA = 1454∘

Gs = minus16

(c) CA = 1327∘

Gs = minus20

(d) CA = 1205∘

Gs = minus24

(e) CA = 1089∘

Gs = minus28

(f) CA = 968∘

Gs = minus32

(g) CA = 816∘

Gs = minus36

(h) CA = 681∘

Gs = minus40

(i) CA = 562∘

Figure 1 Droplet shape and intrinsic contact angle of smooth surfaces under different adsorption parameter

By the different forces from the solid wall the fluid dropsshow different shapes when the system reaches equilibriumShapes and contact angles of droplets for different absorptioncoefficients are shown in Figure 1 The result of fitting 120579 and|119866119904| in Figure 2 illustrates that there is a linear relationship

between 120579 and |119866119904| and 120579 decreases with the increase of |119866

119904|

The fitting result is expressed as the linear equation

120579 = minus32091003816100381610038161003816119866119904

1003816100381610038161003816 + 18474 (9)

According to (9) for 2952 lt |119866119904| lt 5757 the solid

material is hydrophilic with 0∘ lt 120579 lt 90∘ for 0148 lt |119866119904| lt

2952 the solid material is hydrophobic with 90∘ lt 120579 lt 180∘

32 Determining the Apparent Contact Angle In order tosimulate the apparent contact angle of real solid surface weneed to build microtopography on the smooth solid surfacesthat have specific material properties In the simulation the2D square cylinders represent the rough elements on thehydrophobic surfaceThe cylinders are defined by parametersof the column width 119886 spacing 119887 and height ℎ Whencomputations are converged we obtain the droplet shape andthe apparent contact angle of different hydrophobic surfaceswhich are shown in Figure 3 Simulations in Figure 3 arebased on the same absorption parameter which means thesesurfaces share the same material properties According tothe figure the contact angle of the droplet on the smoothhydrophobic surface is 1089∘ indicating a relatively weakhydrophobicity However after adding a rough structure onthe surface its hydrophobicity is improved significantly

For droplets placed on composite surfaces shown inFigure 3 when the gas-liquid interfacial fraction 119891V = 119887(119886 +

120579(∘)

|Gs|

180

150

120

90

60

30

000 10 20 30 40 50

SimulationFitting

Figure 2 The relationship between intrinsic contact angle andadsorption parameter

119887) = 0333 the apparent contact angle is 1231∘ when 119891V =0667 the contact angle is 1231∘ when the surface is deco-rated by hierarchical micromorphology the contact angle is1518∘ superhydrophobicity The results validate the previoustheory obtained by experimental studies that surfaces with


Gs = minus24 smooth surfaces

(a) CA = 1089∘

Gs = minus24 a = 10 b = 5 h = 20

(b) CA = 1231∘

Gs = minus24 a = 5 b = 10 h = 20

(c) CA = 1397∘

hierarchical micromorphologyGs = minus24 a = 5 b = 10 h = 20

(d) CA = 1518∘

Figure 3 Droplet shape and apparent contact angle of different hydrophobic surfaces

Table 1 The property of hydrophilic and hydrophobic surface

119866119904

119886 119887 ℎ Apparent contact angleHydrophobic surface minus40 10 10 20 425∘

Hydrophobic surface A minus28 10 10 20 1193∘

Hydrophobic surface B minus24 10 10 20 1382∘

Hydrophobic surface C minus24 5 10 20 1518∘ (hierarchical micromorphology)

hierarchical micromorphology are more hydrophobic andexplain why lotus leaf is superhydrophobicmdashlotus leaf hasmicron-sized mastoids that are decorated with nanoscalewaxy crystals

As the apparent contact angles are 1233∘ and 1408∘respectively for gas-liquid interfacial fractions 119891V = 0333and 119891V = 0667 according to Cassie-Baxterrsquos wettabilitystate equation cos 120579CB = (1 minus 119891V) cos 120579119884 minus 119891V we canget the value of intrinsic contact angle 120579

119884= 1089∘

easily which is in good agreement with the analytic resultIt is proved that Shan-Chen model of lattice Boltzmannmethod is capable of considering both material propertiesand microtopography simultaneously so it is very suitablefor studying problems of microflows on complex wettabilitysurfaces

4 The Numerical Simulation of theDynamic Behaviors of Droplets Impactingon Hydrophobic Surface

The two dimensionless values used to describe the dynamicbehaviors of droplets impacting are Weber number We andReynolds Re The former one can be thought of a measureof the relative importance of the fluidrsquos inertia compared toits surface tension while the latter one quantifies the relative

importance of the fluidrsquos inertia to its viscous forces The twonumbers are defined blow

We =120588119863011988120

120590

Re =11986301198810

]119897

(10)

where 120588 is the density of droplet 1198630is the initial droplet

diameter1198810is the droplet impacting velocity 120590 is the surface

tension and ]119897is the kinematic viscosity of the fluid

The computational domain is still a rectangular space of500 times 500 lattice units Periodic scheme is used in the leftand right boundaries while standard bounce back schemeis used in the top and bottom fixed boundaries A circularliquid droplet with a diameter of 150 and an initial downwardvelocity is placed in the middle of the bottom wall Thedroplet is subjected to a uniform vertical downward forcefield Because this study focused on the impact of the wet-tability of solid surface on the dynamic behaviors of dropletsWe and Re are set to constants We = 50 and Re = 36 Thewettability is adjusted by changing the absorption parameterand geometric parameters of microtopography The studyobjects are typical hydrophilic surfaces and hydrophobicsurfaces whose properties are shown in Table 1


Figure 4 The dynamic behaviors of droplet impacting on hydrophilic surface (CA = 425∘)

Figure 5 The dynamic behaviors of droplet impacting on hydrophobic surface (CA = 1193∘)

Figures 4ndash7 show the dynamic process of droplets impact-ing on different solid surfaces respectively It can be foundfrom the five figures that droplets on different solid surfacesall experienced the two processes spreading and shrinking

After the droplet is in contact with the solid surface fluidspreads along the radius under the action of inertial force thatis spreading when the droplet spreads to its maximum fluidshrinks along the radius under the action of surface tension






u uV0

Dmax

(a)

u u

(b)

uu

(c)

Figure 8 The sketch map of dynamic behaviors of droplet impacting on different solid surfaces

that is shrinking However as a result of different surfacewettability droplet motions on different solid surfaces showsignificant difference

On the hydrophilic solid surface (CA = 425∘ shown inFigure 4) the droplet spreads on the surface and dips intothe microtopography gap quickly forming the shape of flyingsaucer Over time the edge of the droplet gradually tilts as thecenter portion becomes slightly thinner The droplet changesinto a disk shape When the droplet spreads to its maximumit begins to shrink under the action of surface tension andmove toward the centerline forming an upward flow whichwill not depart from the surface owing to adhesion

The droplet on the hydrophobic solid surface (CA =1193∘ as shown in Figure 5) also experiences spreading andshrinking but compared with motions on the hydrophilicsurface the droplet immerses into the gap only at the initialstage of contacting with the surface In the shrinking processfluid moves toward the center forming an upward flow Asthe droplet is under stretching the contact area of dropletand solid surface gradually decreases Although the droplethas a tendency to depart it cannot get away from the solidsurface for having no sufficient kinetic energy to overcomethe adhesion force

For more hydrophobic solid surfaces (CA = 1382∘ andCA = 1518∘ shown in Figures 6 and 7 resp) dropletsalso experience spreading and shrinking However dropletshave not dipped into the gap and they spread and shrinkmore quickly In the shrinking process fluid moves towardthe center forming an upward flow As droplets are understretching the contact area of droplet and solid surfacedecreases gradually until droplets can bounce up away fromthe solid surface Then droplets move along the horizontaland the vertical directions repeatedly They rise in a mannerof squirming fall under the action of vertical force afterreaching the highest point and then experience anotherperiod of spreading and shrinking It can be found bycomparing Figures 6 and 7 that if the surface hydrophobicityis stronger the maximum height of the droplet is greater andthe dropletwill undergomore spreading and shrinking beforereaching a steady state eventually

Next characteristics of dynamic behaviors of dropletsimpacting on solid surfaces will be analyzed from the view of

energy According to the principle of minimum energy thestationary droplet on solid surfaces has a minimum surfaceenergy Droplets will go through two phases after hitting asolid surface spreading and shrinkingThe spreading behav-iors of droplets on solid surfaces with different wettabilityare similar Droplets with initial kinetic energy 119864V = 1198981198812

02

deviate from the center and move outward under inertiaforce against the adhesion force from solid surface alongwith increase of the surface area and surface energy In thisphase part of the kinetic energy of the droplet is convertedinto viscous dissipation energy119882 and part of it is convertedinto surface energy 119864

119904 When 119864V is running out the droplet

velocity drops to zero while 119864119904reaches maximum Then

the droplet shrinks inward under surface tension with thesurface area decreasing and the velocity increasing Energyvariation during the shrinking phase is to convert a portionof 119864119904into kinetic energy 119864V119891 in order to overcome the

acting of adhesion forces If the rebound phenomenon occursafter the shrinking phase it should happen along with theenergy transition between kinetic energy and gravitationalpotential energy So when the droplet is completely reboundtheminimum gravitational potential energy transferred fromkinetic energy is 119864

ℎ= 119898119892119877 Since the adhesive force to

droplets on the hydrophobic surface is far less than thaton the hydrophilic surface the viscous dissipation is quietsmall for the former droplets which makes droplets stillhave a relatively high kinetic energy even after spreading andshrinking phases This leads to the unique dynamic behaviorof droplets on a hydrophobic surface

(1) If the kinetic energy satisfies 119864119904119891

le 119864V119891 lt 119864119904+ 119864ℎ

after the shrinking phase the rebound phenomenonwill not occur even for droplets with a tendency topop up as shown in Figure 8(b)

(2) With the hydrophobicity of solid surface enhancedviscous dissipation during spreading and shrinkingbecomes smaller If the kinetic energy satisfies 119864V119891 ge119864119904+ 119864ℎafter shrinking droplets can move upward

from the solid surface and the rebound phenomenonoccurs as shown in Figure 8(c)


5 Conclusion

This paper simulates the wettability of hydrophobic sur-faces and the dynamic behaviors of droplets impacting onhydrophobic surfaces with lattice Boltzmann method andgets the following conclusions

(1) Lattice Boltzmann method is capable of consideringbothmaterial properties andmicrotopography simul-taneously gives a relatively high precision and showsgreat perspective in the study of the dynamic behav-iors of droplets impacting on hydrophobic surfaces

(2) The dynamic behaviors of droplets impacting on solidsurfaces are affected by the wettability of the surfaceThe more hydrophobic the solid surface the morelikely the droplets rebound

(3) If the dropletrsquos kinetic energy is greater than thesum of surface energy and the minimum conversiongravitational potential energy when the spreadingand shrinking finished the rebound phenomenonwill occur As the hydrophobic surfacersquos viscous dis-sipation is much smaller than hydrophilic surfacersquosthe droplet still has high kinetic energy even afterspreading and shrinking which is advantageous tothe rebound of droplets

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

The work is supported by the National Natural ScienceFoundation of China (11502210 51279165 and 51479170) andthe Specialized Research Fund for the Doctoral Program ofHigher Education of China (20126102110009)

References

[1] M Nosonovsky and B Bhushan ldquoSuperhydrophobic surfacesand emerging applications non-adhesion energy green engi-neeringrdquo Current Opinion in Colloid amp Interface Science vol 14no 4 pp 270ndash280 2009

[2] J-P Zhao X-H Shi X-G Geng and Z-M Hou ldquoLiquidslip over super-hydrophobic surface and its application in dragreductionrdquo Journal of ShipMechanics vol 13 no 3 pp 325ndash3302009

[3] Z L Dou J D Wang F Yu and D R Chen ldquoFabrication ofa micro-structured surface based on interfacial convection fordrag reductionrdquo Fluid Mechanics vol 56 no 7 pp 626ndash6322011

[4] M R Davidson ldquoBoundary integral prediction of the spreadingof an inviscid drop impacting on a solid surfacerdquo ChemicalEngineering Science vol 55 no 6 pp 1159ndash1170 2000

[5] M Pasandideh-Fard S Chandra and J Mostaghimi ldquoA three-dimensional model of droplet impact and solidificationrdquo Inter-national Journal of Heat and Mass Transfer vol 45 no 11 pp2229ndash2242 2002

[6] H Fujimoto Y Shiotani A Y Tong T Hama and H TakudaldquoThree-dimensional numerical analysis of the deformation

behavior of droplets impinging onto a solid substraterdquo Inter-national Journal of Multiphase Flow vol 33 no 3 pp 317ndash3322007

[7] Z L Guo and C G Zheng Theory and Applications of LatticeBoltzmann Method Science Press Beijing China 2010

[8] Y L He Y Wang and Q Li Lattice Boltzmann Method Theoryand Applications Science Press Beijing China 2011

[9] J J Huang C Shu and Y T Chew ldquoLattice Boltzmann study ofdroplet motion inside a grooved channelrdquo Physics of Fluid vol21 no 2 Article ID 022103 pp 1ndash11 2009

[10] Z Y Shi GHHu andW Zhou ldquoLattice Boltzmann simulationof droplet motion driven by gradient of wettabilityrdquo ActaPhysica Sinica vol 59 no 4 pp 2595ndash2600 2010

[11] P L Bhatnagar E P Gross andMKrook ldquoAmodel for collisionprocesses in gases I Small amplitude processes in charged andneutral one-component systemsrdquo Physical Review vol 94 no3 pp 511ndash525 1954

[12] X W Shan and H D Chen ldquoLattice Boltzmann model for sim-ulating flows with multiple phases and componentsrdquo PhysicalReview E vol 47 no 3 pp 1815ndash1819 1993

[13] X W Shan and H D Chen ldquoSimulation of nonideal gases andliquid-gas phase transitions by the lattice Boltzmann equationrdquoPhysical Review E vol 49 no 4 pp 2941ndash2948 1994

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Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


2 Numerical Method

21 Lattice Boltzmann Method Lattice Boltzmann methodbased on mesoscopic kinetic models has developed into analternative and promising numerical scheme for simulatingfluid flows in recent years Lattice Boltzmann method pre-sented in this paper is based on the streaming and collisionof discrete particle density distribution function in the fixedlattice point and the corresponding evolution function isexpressed as [11]

119891120572(x + e

120572120575119905 119905 + 120575

119905) minus 119891120572(x 119905)

= minus1

120591[119891120572(x 119905) minus 119891(eq)

120572(x 119905)]

(1)

where 119891120572(x 119905) is the particle density distribution function

which depends on position x the particle discrete velocity e120572

and time 119905119891(eq)120572

(x 119905) is the equilibriumdistribution functionand 120591 is the dimensionless relaxation time related to thehydrodynamics viscosity

The left-side of lattice Boltzmann equation represents thestreaming of particles while the right-side represents thecollision process whichmakes119891 heading for the equilibriumdistribution 119891(eq) In D2Q9 model [11] the equilibriumdistribution 119891(eq) is given by

119891(eq)120572

(x 119905) = 120588120596120572[1 +

e120572sdot u1198882119904

+(e120572sdot u)2

21198884119904

minus1199062

21198882119904

] (2)

The discrete velocities are set as

[e0 e1 e2 e3 e4 e5 e6 e7 e8]

= [0 1 0 minus1 0 1 minus1 minus1 1

0 0 1 0 minus1 1 1 minus1 minus1] 119888

(3)

where 119888 = 120575119909120575119905is lattice velocity 120575

119909and 120575

119905are the lattice

length and the time step respectively 119888119904= 119888radic3 is the sound

speed in the discretemodel and theweighting factors dependon the link angle (120596

120572= 49 (120572 = 0) 120596

120572= 19 (120572 = 1 2 3 4)

120596120572= 136 (120572 = 5 6 7 8))The macroscopic density and velocity are defined as

follows

120588 = sum120572

119891120572

u =1

120588sum120572

119890120572119891120572

(4)

The macroscopic pressure is given by 119901 = 1205881198882119904directly

22 Shan-Chen Model To introduce intermolecular forcesbetween microscopic particles Shan and Chen proposed apseudopotential model [12 13] which easily coupled micro-forces the dominant role of microflow problems The modelreflects the physical nature of fluid dynamics and broadensthe application of lattice Boltzmann method in range ofsimulation of microflow on complex surface with wettability

In Shan-Chen model fluid particles at site of x receive forcesfrom nearest neighbor fluid particles and solid walls whichare written in the following form respectively

Fint (x 119905) = minus119866120595 (x 119905)8

sum120572=1

119908120572120595 (x + e

120572120575119905 119905) e120572 (5)

Fads (x 119905) = minus119866119904120595 (x 119905)

8

sum120572=1

119908120572swi (x + e

120572120575119905) e120572 (6)

119866 in (5) is the interaction strength and 119866119904in (6) is

called adsorption parameter representing the strength of theforce contributed by solid Varying the 119866

119904parameter allows

simulation of different solid surface wettability where swiis a ldquoswitchrdquo that takes on value one if the site at x + e

120572120575119905

is a solid and is zero otherwise 119908120572is the same direction-

dependent weighting factor used before 120595(x 119905) is the inter-action potential function which must be monotonicallyincreasing and boundedTherefore the equation used in thispaper is expressed as 120595(x 119905) = 120588

0[1 minus exp(minus120588120588

0)]

In Shan-Chen model the effect of interparticle forces isincorporated into the equilibrium distribution function feqby shifting the equilibrium velocity ulowast

120588 (x 119905) =8

sum120572=0

119891120572(x 119905)

120588 (x 119905) ulowast (x 119905) =8

sum120572=0

e120572119891120572(x 119905) + 120591 (Fint + Fads)

(7)

The equation of state is given by

119901 = 1205881198882119904+119866

61205952 (8)

3 Numerical Simulation of the Wettability ofHydrophobic Surfaces

A key parameter for characterizing the wettability ofhydrophobic surface is the apparent contact angle which ismainly determined by material properties and micromor-phology This provides a new research route first determinethe relationship between the adsorption parameter119866

119904and the

intrinsic contact angle at a specific interaction strength 119866 bysimulating the contact angle of a smooth solid surface thenconstruct microtopography on the surface and simulate theapparent contact angle of real solid surface

31 Determining the Intrinsic Contact Angle The intrinsiccontact angle of the smooth solid surface is adjusted bychanging the adsorption parameter 119866

119904for a specific inter-

action strength 119866 = minus58 The computational domain is arectangular space of 500 times 500 lattice units and a latticeunits is approximately 12120583minphysical scale Dimensionlesstime 120591 = 10 and gravity acceleration 119892 = 05 times 10minus6Periodic scheme is used in the left and right boundaries whileno-slip bounce back scheme is used in the top and bottomboundaries A circular liquid droplet with a diameter of 150is placed in the middle of the bottom wall


Gs = minus08

(a) CA = 1593∘

Gs = minus12

(b) CA = 1454∘

Gs = minus16

(c) CA = 1327∘

Gs = minus20

(d) CA = 1205∘

Gs = minus24

(e) CA = 1089∘

Gs = minus28

(f) CA = 968∘

Gs = minus32

(g) CA = 816∘

Gs = minus36

(h) CA = 681∘

Gs = minus40

(i) CA = 562∘




119904|


120579 = minus32091003816100381610038161003816119866119904

1003816100381610038161003816 + 18474 (9)






120579(∘)

|Gs|

180

150

120

90

60

30

000 10 20 30 40 50

SimulationFitting





(a) CA = 1089∘

Gs = minus24 a = 10 b = 5 h = 20

(b) CA = 1231∘

Gs = minus24 a = 5 b = 10 h = 20

(c) CA = 1397∘


(d) CA = 1518∘



119866119904







119884= 1089∘





We =120588119863011988120

120590

Re =11986301198810

]119897

(10)















u uV0

Dmax

(a)

u u

(b)

uu

(c)








02









le 119864V119891 lt 119864119904+ 119864ℎ





5 Conclusion





Competing Interests


Acknowledgments


References






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Gs = minus08

(a) CA = 1593∘

Gs = minus12

(b) CA = 1454∘

Gs = minus16

(c) CA = 1327∘

Gs = minus20

(d) CA = 1205∘

Gs = minus24

(e) CA = 1089∘

Gs = minus28

(f) CA = 968∘

Gs = minus32

(g) CA = 816∘

Gs = minus36

(h) CA = 681∘

Gs = minus40

(i) CA = 562∘




119904|


120579 = minus32091003816100381610038161003816119866119904

1003816100381610038161003816 + 18474 (9)






120579(∘)

|Gs|

180

150

120

90

60

30

000 10 20 30 40 50

SimulationFitting





(a) CA = 1089∘

Gs = minus24 a = 10 b = 5 h = 20

(b) CA = 1231∘

Gs = minus24 a = 5 b = 10 h = 20

(c) CA = 1397∘


(d) CA = 1518∘



119866119904







119884= 1089∘





We =120588119863011988120

120590

Re =11986301198810

]119897

(10)















u uV0

Dmax

(a)

u u

(b)

uu

(c)








02









le 119864V119891 lt 119864119904+ 119864ℎ





5 Conclusion





Competing Interests


Acknowledgments


References






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra











(a) CA = 1089∘

Gs = minus24 a = 10 b = 5 h = 20

(b) CA = 1231∘

Gs = minus24 a = 5 b = 10 h = 20

(c) CA = 1397∘


(d) CA = 1518∘



119866119904







119884= 1089∘





We =120588119863011988120

120590

Re =11986301198810

]119897

(10)















u uV0

Dmax

(a)

u u

(b)

uu

(c)








02









le 119864V119891 lt 119864119904+ 119864ℎ





5 Conclusion





Competing Interests


Acknowledgments


References






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra



















u uV0

Dmax

(a)

u u

(b)

uu

(c)








02









le 119864V119891 lt 119864119904+ 119864ℎ





5 Conclusion





Competing Interests


Acknowledgments


References






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










5 Conclusion





Competing Interests


Acknowledgments


References






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article dynamic behaviors and energy transition … · 2019. 7. 30. · using a lattice...

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