Chemical Engineering Science 62 (2007) 7214–7224www.elsevier.com/locate/ces

Numerical simulations of drop impact and spreading on horizontal andinclined surfaces

Siddhartha F. Lunkad, Vivek V. Buwa∗, K.D.P. NigamDepartment of Chemical Engineering, Indian Institute of Technology, Delhi, New Delhi 110 016, India

Received 25 April 2007; received in revised form 3 July 2007; accepted 9 July 2007Available online 25 July 2007

Abstract

The phenomenon of drop spreading is important to several process engineering applications. In the present work, numerical simulations ofthe dynamics of drop impact and spreading on horizontal and inclined surfaces were carried out using the volume of fluid (VOF) method.For the horizontal surfaces, the dynamics of impact and spreading of glycerin drops on wax and glass surfaces was investigated for whichthe experimental measurements were available [Šikalo, Š., Tropea, C., Ganic, E.N., 2005a. Dynamic wetting angle of a spreading droplet.Experimental Thermal and Fluid Science 29, 795–802; Šikalo, Š., Tropea, C., Ganic, E.N., 2005b. Impact of droplets onto inclined surfaces.Journal of Colloid and Interface Science 286, 661–669]. The influence of surface wetting characteristics was investigated by using staticcontact angle (SCA) and dynamic contact angle (DCA) models. The dynamics of drop impact and spreading on inclined surfaces and thedifferent regimes of drop impact and spreading process were also investigated. In particular, the effects of surface inclination, surface wettingcharacteristics, liquid properties and impact velocity on the dynamics of drop impact and spreading were investigated numerically and theresults were verified experimentally. It was found that the SCA model can predict the drop impact and spreading behavior in quantitativeagreement with the experiments for less wettable surfaces (SCA > 90◦). However, for more wettable surfaces (SCA < 90◦), the DCA observedat initial contact times were order of magnitude higher than SCA values and therefore the DCA model is needed for the accurate predictionof the spreading behavior.� 2007 Elsevier Ltd. All rights reserved.

Keywords: Drop; Dynamic simulation; Drop impact and spreading; Volume of fluid method; Computational fluid dynamics

1. Introduction

Many engineering and technology applications involve flowof liquids over solid surfaces, for example in trickle bed reac-tors, structured reactors and monoliths, packed beds, surfacecoatings, printing and many more. In these reactors/processes,the impact and spreading of liquid droplets on solid surfacesplay a crucial role. For example, a fraction of the total exter-nal catalyst area wetted by flowing droplets governs the cata-lyst efficiency and therefore the process performance. Owing toits importance in various applications, the phenomena of dropimpact and spreading over horizontal and inclined surfaces re-ceived a substantial attention in the literature (Pasandideh-Fard

∗ Corresponding author. Tel.: +91 11 26591027; fax: +91 11 26581120.E-mail address: vvbuwa@chemical.iitd.ac.in (V.V. Buwa).

0009-2509/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.doi:10.1016/j.ces.2007.07.036

et al., 1996; Bussmann et al., 1999, 2000; Kang and Lee, 2000;Fukai et al., 2000; Gunjal et al., 2005, Šikalo et al., 2005a–d).

For many years, the dynamics of drop impact and spreadinghas been a challenging problem for physicists and engineers.The important factors which govern the drop dynamics on asolid surface are the liquid properties (density, surface tensionand viscosity), the surface characteristics (contact angle androughness), the drop impact velocity and the surface inclina-tion. The experimental investigations of Šikalo et al. (2005a–c)with liquids of varying surface tension and viscosity (e.g., iso-propanol, water and glycerin) showed that the drop volume,the surface inclination and impact velocity have a significanteffect on the drop dynamics and the regimes of drop impact.Fig. 1 shows the regimes of drop the impact observed experi-mentally by Šikalo et al. (2005b) viz. (a) splash, (b) spreading,(c) spreading and sliding, (d) partial rebound, (e) rebound and(f) deformation. Šikalo et al. (2005a–c) attempted to quantify

S.F. Lunkad et al. / Chemical Engineering Science 62 (2007) 7214–7224 7215

Fig. 1. Experimentally observed regimes of the drop impact and spreading on inclined surfaces (Šikalo et al., 2005b, reproduced with permission). (a) Splash(the 3.3 mm isopropanol drop impacting on the glass surface with the inclination of 45◦ and with the impact velocity of 2.098 m/s, We = 544); (b) spreading(the 2.72 mm water drop impacting on the glass surface with the inclination of 10◦ and with the impact velocity of 3.25 m/s, We = 391); (c) spreading andsliding (the 2.72 mm water drop impacting on the wax surface with the inclination of 10◦ and with the impact velocity of 3.25 m/s, We = 391); (d) partialrebound (the 2.45 mm glycerin drop impacting on the glass surface with the inclination of 9◦ and with the impact velocity of 1.0361 m/s, We=51); (e) rebound(the 2.45 mm glycerin drop impacting on the glass surface with the inclination of 8◦ and with the impact velocity of 1.0361 m/s, We = 51) and (f) deformation(the 1.8 mm isopropanol drop impacting on the glass surface with the inclination of 8◦ and with the impact velocity of 1.63 m/s, We = 179).

the drop dynamics based on the Weber number (We) (ratio ofinertial force to surface tension force), the contact angle and thesurface inclination. Fukai et al. (2000) and Kang and Lee (2000)investigated the dependence of advancing and receding contactangles on the wall temperature and the contact line velocity ex-perimentally. Sedev et al. (1993) carried out an experimentalstudy with the droplets of octane, dodecane, and hexadecaneand reported the dependence of the DCA on the contact line ve-locity, the surface roughness and the time of solid–liquid con-tact. In an experimental and numerical investigation by Fukaiet al. (1995) for surfaces of different wettabilities, it was seenthat the effect of impact velocity on the droplet spreading wasmore pronounced when the wetting was limited. The theoret-ical model satisfactorily captured the deformation of impact-ing droplet in spreading as well as recoiling and oscillationphases. Reznik and Yarin (2002) reported a theoretical studybased on a analytical method for a viscous liquid drop spreadingover a horizontal surface with an impact velocity in the rangeof 0.1–0.9 m/s. Mundo et al. (1995) reported an experimentalstudy of the effect of the wall roughness on the drop dynamicsin the splashing regime. Pasandideh-Fard et al. (1996) investi-gated the effects of capillary forces on the drop spreading oversolid surfaces.

Besides the experimental investigations discussed above, sev-eral numerical studies on the dynamics of liquid droplet spread-ing over solid surfaces have been reported in the literature(Fukai et al., 1995; Bussmann et al., 1999; Pasandideh-Fardet al., 2002; Gunjal et al., 2005, Šikalo et al., 2005d). Differentnumerical methods are available for computations of flows withmoving interfaces, for example, the level set method (Osherand Sethian, 1988; Sussman and Osher, 1994), the front track-ing method (Unverdi and Tryggvason, 1992; Tryggvason et al.,2001) and the lattice-Boltzmann method (Gunstensen et al.,1991; Grunau et al., 1993, Shan and Chen, 1993; Shan and

Doolen, 1995; Nourgaliev et al., 2003) and the volume of fluid(VOF) method (Hirt and Nichols, 1981). However, the VOFmethod is more suitable for the simulation of drop spreadingbecause of its inherent mass conservation property, its suitabil-ity for problems where large surface topology changes occurand reduced computational costs. However, it is less accuratein interface calculations than the other methods like the levelset or the front tracking. In spite of this limitation, it is stillthe most preferred method for the computations of drop impactand spreading where strong topological changes of interfacesoccur (Fukai et al., 1995; Bussmann et al., 1999; Gunjal et al.,2005; Šikalo et al., 2005d).

Fukai et al. (1995) investigated the effect of the surface wet-tability on the spreading behavior of a drop. They observed thatthe impact velocity greatly influences the droplet spreading be-havior. The incorporation of advancing and receding angles inthe numerical model with adaptive mesh refinement improvedtheir predictions. Pasandideh-Fard et al. (2002) studied thethree-dimensional solidification of a molten drop on horizontaland inclined surfaces with an interface tracking algorithm anda continuum surface force (CSF) model. Gunjal et al. (2005)carried out an experimental and VOF based numerical studyof the drop impact over horizontal surfaces. Their predictionssuccessfully captured the spreading, splashing, rebounding andbouncing regimes of the drop dynamics over horizontal surfacesof different wettabilities. Most of the numerical simulationsof drop spreading discussed above were carried for horizontalsurfaces and using the static contact angle (SCA) model. How-ever, the predictive capability of the numerical methods to sim-ulate the dynamics of drop impact and spreading on inclinedsurfaces and the different regimes observed during the impactand spreading process is not well investigated.

Šikalo et al. (2005d) reported a numerical study on the DCAmodel for drops impacting over flat surfaces. Using the VOF

7216 S.F. Lunkad et al. / Chemical Engineering Science 62 (2007) 7214–7224

method, Bussmann et al. (1999) successfully simulated the dropspreading on an inclined surface by accounting the time varia-tion of the DCA and the contact line velocities. They investi-gated the impact and spreading of a 2 mm water droplet on a45◦ inclined steel surface. The experimentally measured DCAswere implemented as a wall boundary condition. The predictedevolution of drop shapes was in a remarkably good agree-ment with the experimental observations. Recently, Afkhamiand Bussmann (2006) extended the work of Bussmann et al.(1999) by incorporating the variation of drop height into theVOF algorithm and discussed the effect of different implemen-tations of the contact angle and the contact line velocity on thepredictions of drop spreading. However, it should be noted thatthe predictions of both Bussmann et al. (1999) and Afkhamiand Bussmann (2006) were validated experimentally only for asingle experimental case of the 2 mm water droplet impactingon the 45◦ inclined steel surface.

Though it is understood that the experimental data on thetime variation of the contact angle is always not easy to obtainand appropriate DCA models are lacking, the first step towardsimproving the predictive capabilities of the numerical methodswould be to implement the experimentally measured time vari-ation of the contact angles (advancing, �A and receding, �R)

in numerical simulations (Bussmann et al., 1999). In order toprogress further, it is imperative to understand the predictivecapabilities to simulate the drop impact and spreading on hori-zontal surfaces using both SCA and DCA models. In particular,it will be worthwhile to simulate the drop impact and spreadingbehavior for the different regimes that are observed for differ-ent surface inclinations, surface wetting characteristics, liquidproperties and drop impact velocities (see Fig. 1) and validatethe predictions.

In the present work, we have carried out the numericalinvestigations of drop impact and spreading on horizontaland inclined surfaces. The experimental data reported byŠikalo et al. (2005a,b) were used to validate the predictions.The numerical simulations were carried out by using theVOF method and using the commercial flow solver (Flu-ent 6.3). The simulations of glycerin drop impacting on thehorizontal surfaces of wax and glass surfaces for the im-pact velocity of 1.0361 m/s (corresponding to We = 51) wereperformed using SCA and DCA models and the predictionswere validated using the measured spread factors (of Šikaloet al., 2005a,b). The simulations of the liquid drop (water, glyc-erin and iso-propanol) impact and spreading on inclined sur-faces (wax and glass surfaces with 10–45◦ inclinations) werecarried out using the SCA model for different impact veloci-ties (corresponding Weber numbers in the range of 51–544).The experimental images and the measured spread factors (ofŠikalo et al., 2005a,b) were used to validate the predictions.

2. Computational model

2.1. The VOF method

In a computational domain under consideration, the partic-ular fluid phase is defined by the volume fraction (�q) in a

control volume as the fraction of the qth phase inside a cell as

�q ={

0 if the cell is empty (of qth phase),1 if the cell is full (of qth phase).

(1)

If 0 < �q, < 1, the cell contains the interface between the qthphase and the other phase(s). Depending upon local values ofvolume fraction, a single set of Navier–Stokes equations forNewtonian fluid under laminar flow conditions were solved inthe entire computational domain as

�

�t(�) + ∇ · (��v) = 0, (2)

�

�t(��v) + ∇ · (��v�v)

= −∇P + ∇ · [�(∇�v + (∇�v)T)] + ��g + �F , (3)

where �v is velocity vector, P is pressure, �F is surface tensionforce per unit volume, �g is the gravitational acceleration, � is thedensity (as defined by Eq. (4)) and � is the viscosity (as definedby Eq. (5)). Depending upon volume fraction values, the flowvariables and the fluid properties in any given cell are eitherpurely representative of one of the phases, or representative ofa mixture of the phases. Based on the local value of �q , theappropriate fluid properties and flow variables were assigned toeach control volume within the domain. In a two-phase system,for example, if the phases are represented by the subscripts 1and 2 and the volume fraction of the phase 2 is known, thedensity and the viscosity in each cell are given as follows:

� = �2�2 + (1 − �2)�1, (4)

� = �2�2 + (1 − �2)�1. (5)

In the VOF model, the motion of a moving interface is com-puted by solving an advection equation for the volume fractionof the qth phase (secondary-phase):

��q

�t+ �v · ∇�q = 0. (6)

The volume fraction for the primary-phase is obtained from thefollowing equation:

n∑q=1

�q = 1. (7)

The surface tension model incorporated in FLUENT is the CSFmodel proposed by Brackbill et al. (1992). With this model,the contribution of the surface tension force to the VOF calcu-lation results in a source term in the momentum equation andis modeled as

�F = ���∇�2

12 (�1 + �2)

, (8)

where � is the surface tension coefficient and � is the surfacecurvature calculated as (Brackbill et al., 1992).

� = −(∇ · n̂) = 1

|n|[(

n

|n| · ∇)

|n| − (∇ · n)

], (9)

S.F. Lunkad et al. / Chemical Engineering Science 62 (2007) 7214–7224 7217

Side View

θR

θAφ

φPressure Inlet

L

Wall

W

Pressure Inlet

H

PressureInlet

RefinementWall

Fig. 2. Solution domain, boundary conditions and grid used in the simulationsof drop impact on an inclined surface.

where

n = ∇�q . (10)

The control-volume formulation requires convection and diffu-sion fluxes through the control-volume faces to be computedand balanced with source terms within the control volume it-self. For the calculation of face fluxes in the VOF algorithm,the geometric reconstruction scheme (Youngs, 1982) based onthe piecewise linear approach was used (see Fluent 6.3 UsersGuide, 1996 for further details; Rider and Kothe, 1998).

The numerical treatment of a moving three-phase contactline needs a further discussion. For fluids exhibiting a non-zero contact angle, the presence of wall also affects the surfacenormal and therefore wall adhesion forces were included inthe present work. The wall adhesion was modeled in a mannersimilar to that of the surface tension in the case of a gas–liquidinterface (Brackbill et al., 1992), except that the unit normal n̂

in this case was evaluated using the contact angle �w as follows:

n̂ = n̂w cos(�w) + t̂w sin(�w), (11)

where n̂w and t̂w are the unit vectors normal and tangential tothe wall, respectively (which are directed into fluid and wall,respectively). The contact angle is the angle between the walland tangent to the interface at the wall as shown in Fig. 2. Thecontact angle (�w), static or dynamic was specified from theexperimentally measured values.

Table 1Liquid properties (Šikalo et al., 2005b)

Liquid Density (�)

(kg m−3)

Viscosity (�)

(mPa s)Surface tension(�) (N m−1)

Water 996 1.0 0.073Glycerin 1220 116.0 0.063Isopropanol 786 2.4 0.021

While a no-slip boundary condition is imposed for the fluidvelocities at the solid surface (other boundary conditions dis-cussed separately in Section 2.2), the moving contact line feelsa slip. In order to avoid stress singularities at the contact point,often a Navier slip condition (uslip = �(�u/�z)), where � isthe slip length, is used (Renardy et al., 2001; Spelt, 2005). Inmost of the applications, this slip length is expected to be muchsmaller than the mesh size used in numerical computations(Renardy et al., 2001). The available computational resourcesdo not allow the mesh size to be refined in such a way that thetrue slip length is captured correctly. In the present work, there-fore, no explicit slip length was imposed. However, it shouldbe noted that the algorithm itself introduces the implicit sliplength of the order of the grid spacing (see Renardy et al., 2001for further details).

The model equations given above were solved using thecommercial flow solver Fluent 6.3. The spatial derivatives inEqs. (2), (3) and (6) were discretized using the QUICK scheme(Leonard, 1979) while the temporal derivatives were discretizedusing a first-order implicit method. The pressure implicit withsplitting of operator (PISO) scheme was used for the pressurevelocity coupling in the momentum equation (Issa, 1985).

In the beginning of a simulation, a spherical drop was patchedat the corresponding position in the computational domain(liquid-phase volume fraction, �q , was set to unity). This patchwas initialized with an impact velocity in vertically downwarddirection (at time t = 0). A time step varying from 0.5 × 10−7

to 1 × 10−6 s was used and 20–30 internal iterations per timestep were carried out, with which the normalized residuals wereless than 10−4.

2.2. Solution domain, boundary conditions and grid

Numerical simulations were carried out using three-dimensional rectangular boxes of different sizes (horizontal orwith inclinations of 10–45◦) with a Cartesian grid (see Fig. 2).The extents of the solution domain (L, H, W) in the x-, y- andz-direction were decided based on drop diameter, impact ve-locity, liquid properties and surface wettability data. Typically,the solution domain for simulation of the 3.3 mm diameterisopropanol drop impacting on the 45◦ inclined glass surface(SCA = 0◦) was 15 mm × 15 mm × 6 mm and for simulat-ing the 2.45 mm glycerin drop impact on the horizontal waxsurface (SCA = 93.5◦), the domain size was 10 mm × 10 mm× 5 mm. Numerical simulations of drop with diameters ofthe range of 1.8–3.3 mm and impact velocities in the range of1–3.25 m/s (surface inclinations of 10–45◦) were investigatedin the present work.

7218 S.F. Lunkad et al. / Chemical Engineering Science 62 (2007) 7214–7224

Fig. 3. Comparison of the evolution of drop shapes obtained using the (a) SCA and (b) DCA based simulations for the 2.45 mm glycerin drop impacting at1.0361 m/s (We = 51) on the horizontal wax surface (SCA = 93.5◦, DCA as shown in Fig. 4).

The no slip boundary condition was specified at the wall(bottom face) and all the remaining faces were defined as thepressure inlets (as shown in Fig. 2). For the horizontal surface,the DCA approach uses the experimental time variation of thecontact angle data reported by Šikalo et al. (2005a) (shown inFigs. 4 and 6). The user defined functions were used to im-plement the DCA as the wall boundary condition. Bussmannet al. (1999) studied the effect of the grid resolution using 10,16 and 25 cells per radius (cpr) of the drop. They reportedthat the grid resolution of 10 cpr was sufficient to capture ac-curately the dynamics of drop spreading on inclined surfacesand further refinement in the grid resolution did not alter thepredictions significantly. In the present work, the effect of gridresolution was investigated using the grid resolutions of 6, 10and 14 cpr. For the grid resolutions of 10 cpr and above, thepredicted spread factors and the apex height variation were al-most independent of the grid resolution (Results are discussedin Section 3.1). Therefore, the grid resolution of 10 cpr wasused for all simulations unless mentioned otherwise. The gridnear the wall (in a direction normal to the wall) was refinedusing a non-uniform grid spacing (see Fig. 2).

3. Results and discussion

3.1. Drop impact on horizontal surfaces

The dynamics of a glycerin drop impacting on different sur-faces (horizontal), for which experimental measurements wereavailable (Šikalo et al., 2005a), was investigated. The simula-tions were carried out for the 2.45 mm diameter glycerin droplet(physical properties given in Table 1) impacting at 1.0361 m/s(We = 51) on the horizontal glass (SCA = 15◦) and wax sur-faces (SCA=93.5◦) using both the SCA and DCA approaches.The time variation of the DCA reported by Šikalo et al. (2005a)was used and implemented using the user defined subroutine.Figs. 3(a), (b) and 5(a),(b) show the images of the evolution ofdrop shapes at different times for the SCA and DCA models, re-spectively. The quantitative comparison of the measured (Šikaloet al., 2005a) and the simulated spread factor (d/D) is shownin Figs. 4 and 6.

For the case of the glycerin drop impacting on the wax sur-face (Figs. 3(a) and (b)), it can seen that the spreading of the

0

0.8

1.6

2.4

3.2

10 10.10.01

Dimensionless Time (t* = tu/D)

DC

A (

rad

ian

), d

/D,

h/D

DCA-Experimental (Sikalo et al., 2005a)

d/D-Experimental (Sikalo et al., 2005a)

d/D-Simulated (SCA) d/D-Simulated (DCA)h/D-Simulated (SCA) h/D-Simulated (DCA)

Fig. 4. Comparison of the experimental and the numerical spread factorsfor the 2.45 mm glycerin drop impacting at 1.0361 m/s (We = 51) on thehorizontal wax surface (SCA = 93.5◦, DCA as shown in the figure).

drop in the horizontal direction is almost the same for the SCAand DCA, but there is appreciable change in the shape and thedrop height from 1 ms onwards. This is more apparent fromthe comparison of the spread factors (d/D and h/D) shown inFig. 4. The predicted d/D using both the SCA and DCA wasin a very good agreement with the experiments. However, theh/D predicted by the SCA and DCA were different at manytime instants. The measured data of h/D, however, was notavailable in Šikalo et al. (2005a). Interestingly, for the case ofthe glycerin drop impacting on the glass surface (see Figs. 5(a)and (b)), the evolution of the drop shapes predicted using theSCA and DCA was significantly different. The comparison ofd/D (see Fig. 6) indicates that unless the DCA is implemented,the drop spreading (d/D and h/D) cannot be accurately pre-dicted. The SCA of 15◦ led to significant lateral spreading ofthe drop (see Fig. 5(a)) and thus the d/D was over-predicted(see Fig. 6). However, when the DCA was implemented, thecontact angles were in the range of 160–75◦ (for t∗=0.01–7.0),which were order of magnitude higher than the SCA of 15◦.This led to the recoiling of the drop (e.g., see the drop shapeat 15 ms) and the predicted d/D were in excellent agreementwith the measurements (see Fig. 6).

It is also important to note that the predictions of the dropspreading d/D using the SCA and DCA were in a very good

S.F. Lunkad et al. / Chemical Engineering Science 62 (2007) 7214–7224 7219

Fig. 5. Comparison of the evolution of drop shapes obtained using (a) SCA and (b) DCA simulation for the 2.45 mm glycerin drop impacting at 1.0361 m/s(We = 51) on the horizontal glass surface (SCA = 15◦, DCA as shown in Fig. 6).

0.0

0.8

1.6

2.4

3.2

10.001.000.100.01

Dimensionless Time (t* = tu/D)

DC

A (

rad

ian

), d

/D, h

/D

DCA-Experimental (Sikalo et al., 2005a)

d/D-Experimental (Sikalo et al., 2005a)

d/D-Simulated (SCA) d/D-Simulated (DCA)

h/D-Simulated (SCA) h/D-Simulated (DCA)

Fig. 6. Comparison of the spread factors for the 2.45 mm glycerin dropimpacting at 1.0361 m/s (We=51) on the horizontal glass surface (SCA=15◦,DCA as shown in the figure).

agreement with the experiments for less wettable surface (wax)for which the measured variation of the DCA (from 165◦ to90◦ for t∗ = 0.01–5.0) was not significantly different than theSCA (=100◦). However, for more wettable surface (glass), themeasured variation of the DCA (from 160◦ to 75◦ for t∗ =0.01–7.0) was significantly different than the SCA (=15◦) andtherefore the predictions of the d/D using the DCA were dif-ferent than that using the SCA and were in excellent agreementwith the measurements.

The effect of the grid resolution on the predicted d/D andh/D ratios for the 2.45 mm glycerin drop impacting on thehorizontal glass surface (using the DCA as shown in Fig. 6) wasstudied using the grid resolutions of 6, 10 and 14 cpr and theresults are shown in Fig. 7. It can be seen that the effect of gridresolution on the predicted d/D and h/D ratios was negligiblefor a grid resolution greater than 10 cpr and therefore, in allfurther simulations a grid resolution of 10 cpr was used unlessmentioned otherwise.

3.2. Drop impact on inclined surfaces

The numerical investigations for inclined surfaces were pri-marily focused to simulate the regimes of drop impact observed

0

0.5

1

1.5

2

2.5

1010.10.01

Dimensionless Time (t* = tu/D)

d/D

, h

/D

d/D-Experimental (Sikalo et al., 2005a) d/D-Simulated (6 cpr) d/D-Simulated (10 cpr) d/D-Simulated (14 cpr)

h/D-Simulated (6 cpr) h/D-Simulated (10 cpr)h/D-Simulated (14 cpr)

Fig. 7. Effect of the grid resolution on the predicted spread factors for the2.45 mm glycerin drop impacting at 1.0361 m/s (We = 51) on the horizontalglass surface (using DCA as shown in Fig. 6).

by Šikalo et al. (2005b) (shown in Fig. 1) viz. (a) splash (b)spreading (c) spreading and sliding (d) partial rebound (e) re-bound and (f) deformation. The simulation results based on theSCA model for corresponding regimes of Fig. 1 are shown inFigs. 8(a)–(f). It can be clearly seen that the regimes of dropsplash, rebound and deformation were not captured using theSCA approach. This can be attributed to the fact that for thetypical cases of drop splash, rebound and deformation (shownin Fig. 1) qualitatively appear to have higher values of the DCA(> 90◦ i.e., less wettability characteristics) during the initial pe-riod of contact because of which the liquid drop tends to recoilat the solid surface. The time variation of the contact angle forsplash, rebound and deformation regimes was not reported byŠikalo et al. (2005b). Based on the SCA given in Table 2 andDCA (Figs. 4 and 6), it can be understood that the SCA data (forglass surface-isopropanol SCA = 0◦ and glass surface-glycerinSCA = 15◦) appear to have order of magnitudes differencesfrom DCA values at the initial contact times.

The simulation results for the 2.72 mm water droplet impact-ing at 3.25 m/s (We = 391) on the 10◦ inclined wax surface(static—�R =105◦, �A =95◦) were in good agreement with the

7220 S.F. Lunkad et al. / Chemical Engineering Science 62 (2007) 7214–7224

Fig. 8. Simulated regimes of the drop impact and spreading over inclined surfaces. (a) Splash; (b) Spreading; (c) Spreading and sliding; (d) Partial rebound;(e) Rebound; (f) Deformation. (Other details same as given in the caption of Fig. 1.)

Table 2Static contact angle data for inclined surfaces (Šikalo et al., 2005b)

Liquid Glass Wax

�A(◦) �R(◦) �A(◦) �R(◦)

Water 10 6 105 95Glycerin 17 13 97 90Isopropanol ∼ 0 ∼ 0 ∼ 0 ∼ 0

experimentally observed spreading and sliding regime (shownin Fig. 9). The predicted spread factor (xfront/D) was in ex-cellent agreement with the measured xfront/D factor (Šikaloet al., 2005a) (see Fig. 10(a)). The predicted xback/D factor was

also in excellent agreement with the measured xback/D factor(Šikalo et al., 2005a) up to t∗ = 1. However, for t∗ > 1, thexback/D factor was under-predicted. The simulation results forthe 2.72 mm water droplet impacting at 3.25 m/s (We=391) onthe 10◦ inclined glass surface (static—�R = 6◦, �A = 10◦) areshown in Fig. 8(b). The corresponding spread factors (xfront/D

and xback/D) for this case were slightly over predicted as shownin Fig. 10(b). As shown before, this over prediction is becauseof the lower values of static �R and �A in comparison with thehigher values of the DCA present during initial contact period.

The effect of the surface inclination was further studiedfor two cases of 2.72 mm diameter water droplet impacting at1.55 m/s (We = 90) on the 45◦ inclined wax and glass surfacesfor which a quantitative data on the time variation of spread

S.F. Lunkad et al. / Chemical Engineering Science 62 (2007) 7214–7224 7221

Fig. 9. Simulation results for a 2.72 mm water droplet impacting at 3.25 m/s(We = 391) on 10◦ inclined wax surface. (a) Experimental image (Šikaloet al., 2005b, reproduced with permission); (b) side view; (c) top view (timeinterval between the images is 1 ms).

factor (xfront/D and xback/D) was reported by Šikalo et al.(2005a). Both of these water–wax and water–glass systemswere simulated using the SCA approach. The simulated evo-lutions of drop spreading for the water–wax and water–glasssystem are shown in Figs. 11 and 13, respectively. The plotsin Figs. 12 and 14 show the comparison of the measured andthe predicted spread factors (xfront/D and xback/D) as a func-tion of the dimensionless time (t∗ = tu/D). Similar to the ob-servation made for 10◦ inclined surfaces, an excellent agree-ment was found for the water–wax system (Fig. 12), but thespread factors were over predicted for the water–glass sys-tem (Fig. 14). It should be noted that the SCA approach hasbetter predictive accuracy for the less wettable wax surface(static—�R = 105◦, �A = 95◦) than the more wettable glasssurface (static—�R = 10◦, �A = 6◦). As shown in Fig. 14, thepredicted h/D for the water–glass system was also in excellentagreement with the measurements of Šikalo et al. (2005b). Oneexperimental image for the 2.72 mm diameter water droplet im-pacting at 1.55 m/s (We=90) on the 45◦ inclined wax surface attime t = 8.08 ms was reported by Šikalo et al. (2005a) (shownin Fig. 15(a)). The corresponding simulated result is shown inFig. 15(b) and it can be inferred that the experimentally andnumerically observed drop shapes are in acceptable agreementas there is a collection of some liquid at the advancing edge, ashallow central region, and a receding edge as seen in Fig. 15(a).

In order to predict the dynamics of drop spreading observedin the regimes of splash, rebound and deformation (shown in

-2

0

2

4

6

0.010.11.00.0

t* (tu/D)

x/D

xfront/D-Experimental (Sikalo etal., 2005b) xfront/D-Simulated (SCA)

xback/D-Experimental (Sikalo etal., 2005b) xback/D-Simulated (SCA)

t* (tu/D)

xfront/D-Experimental (Sikalo etal., 2005b) xfront/D-Simulated (SCA)

xback/D-Experimental (Sikalo etal., 2005b) xback/D-Simulated (SCA)

-2

0

2

4

6

0.010.11.00.0

x/D

Point of

contactx front

xback

Fig. 10. Comparison of the predicted spread factors with the experimentsof Šikalo et al. (2005b) for the (a) wax–water and (b) glass–water system(We = 391, 10◦ inclination).

Fig. 11. Predicted spreading of the 2.72 mm water droplet impacting at1.55 m/s (We = 90) on the 45◦ inclined wax surface.

Figs. 1(a), (d), (e), (f)) quantitatively, the measured time varia-tion of �R and �A (DCA approach) can be implemented throughuser defined sub-routines in the commercial solver. Bussmannet al. (1999) have demonstrated the such an implementationwhich could successfully predict the drop spreading behav-ior. However, such predictions could not be included in thismanuscript due to lack of data on measured time variation of�R and �A. The work on the implementation of DCA model

7222 S.F. Lunkad et al. / Chemical Engineering Science 62 (2007) 7214–7224

-0.8

0.0

0.8

1.6

2.4

3.2

00.0100.101.010.0

t* = (tu/D)

x/D

xfront/D-Experimental (Sikalo et al., 2005a)

xback/D-Experimental(Sikalo et al., 2005a)

xfront/D-Simulated (SCA) xback/D-Simulated (SCA)

Fig. 12. Comparison of the predicted spread factors for the wax–water system(We = 90◦, 45◦ surface inclination) with the measurements of Šikalo et al.(2005a).

Fig. 13. Predicted spreading of the 2.72 mm water droplet impacting at1.55 m/s (We = 90) on the 45◦inclined glass surface.

-0.8

0.0

0.8

1.6

2.4

3.2

10.001.000.100.01

t* = (tu/D)

x/D

an

d h

/D

xfront/D-Experimental(Sikalo et al., 2005a)

xback/D-Experimental(Sikalo et al., 2005a)

xfront/D-Simulated (SCA) xback/D-Simulated (SCA)

h/D-Experimental (Sikalo et al., 2005b) h/D-Simulated (SCA)

Fig. 14. Comparison of the predicted spread factors and apex height variationfor the glass-water system (We = 90◦, 45◦ surface inclination) with themeasurements of Šikalo et al. (2005a).

Fig. 15. Qualitative comparison of the drop shape for the 2.72 mm water dropimpacting on the 45◦ inclined wax surface: (a) experiments of Šikalo et al.(2005a), image at t = 8.08 ms (reproduced with permission), (b) predictionat time t = 8.0 ms.

proposed by Bussmann (2000) and generating experimentaldata on the time variation of �R and �A for the cases in theregimes of splash, rebound and deformation will be reportedseparately.

4. Conclusions

The dynamics of drop impact and spreading on the horizon-tal and the inclined surfaces was investigated using the VOFmethod and the predictions were validated using the experi-ments of Šikalo et al. (2005a,b). For the horizontal surfaces,the results indicated that when the surface is less wettable(SCA > 90◦), predictions of both the SCA and DCA modelsagree remarkably well with the measurements. For highly wet-table surfaces (SCA < 90◦), unless the measured time varia-tion of the contact angle (DCA) is provided, the drop spreadingcannot be accurately predicted.

The simulations of the drop spreading on the inclinedsurfaces using the SCA were performed to predict the dif-ferent regimes of splash, spreading, spreading and sliding,rebound and deformation observed experimentally by Šikaloet al. (2005b). The spreading and sliding regimes could bepredicted quantitatively well using the SCA approach. How-ever, the observed regimes of splash, rebound and deforma-tion regimes were not in good qualitative agreement. In such

S.F. Lunkad et al. / Chemical Engineering Science 62 (2007) 7214–7224 7223

regimes of splash, rebound and deformation, the experimentalimages reported by Šikalo et al. (2005b) indicate higher val-ues of DCA (> 90◦ i.e., less wettable characteristics) duringthe initial period of contact than the SCA because of whichthe liquid drop tends to recoil and exhibits partial or full re-bound regimes. Whereas the SCA reported for such regimeswere order of magnitude lower (e.g., for the glass surfaceSCA = 6–10◦) in comparison with the DCA in early times ofthe contact and therefore the simulations carried using suchlow values of SCA failed to predict the observed partial or fullrebound regimes. Therefore, it is imperative to develop andrigorously validate the DCA models which can account forthe time variation of contact angles for the above-mentionedregimes of drop impact on the inclined surfaces.

Notations

D drop diameter, md instantaneous drop spread m�F continuum surface force, kg m−2 s−2

g gravitational acceleration, m s−2

h instantaneous drop height, mn surface normal vectorn̂ unit normaln̂w unit normal at wallP pressure, Paq particular phase under considerationt time, st̂w unit tangent to the wallt∗ dimensionless time ( = tu/D)u drop impact velocity, m s−1

uslip slip velocity, m s−1

�v velocity vector, m s−1

We Weber number (=�Du2/�)

x drop instantaneous spread, mxfront displacement of the advancing tip of drop from

the point of contact over inclined surface, mxback displacement of the receding tip of drop from

the point of contact over inclined surface, m

Greek letters

� volume fraction of phase�A advancing contact on an inclined wall,◦�R receding contact angle on inclined wall, ◦�w contact angle on an horizontal wall, ◦� radius of curvature, m� slip length, m� viscosity, N s m−2

� density, kg m−3

� surface tension, N m−1

angle of surface inclination, ◦

Acronyms

SCA static contact angleDCA dynamic contact angleVOF volume of fluid method

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