DOI 10.1140/epje/i2006-10063-7

Eur. Phys. J. E 21, 231–242 (2006) THE EUROPEANPHYSICAL JOURNAL E

Lateral vibration of a water drop and its motion on a vibratingsurface

L. Dong1, A. Chaudhury2, and M.K. Chaudhury1,a

1 Department of Chemical Engineering, Lehigh University, Bethlehem, PA 18015, USA2 Cornell University, Ithaca, NY, USA

Received 5 September 2006 and Received in final form 8 November 2006Published online: 5 January 2007 – c© EDP Sciences / Società Italiana di Fisica / Springer-Verlag 2007

Abstract. The resonant modes of sessile water drops on a hydrophobic substrate subjected to a small-amplitude lateral vibration are investigated using computational fluid dynamic (CFD) modeling. As thesubstrate is vibrated laterally, its momentum diffuses within the Stokes layer of the drop. Above the Stokeslayer, the competition between the inertial and Laplace forces causes the formation of capillary waves onthe surface of the drop. In the first part of this paper, the resonant states of water drops are illustratedby investigating the velocity profile and the hydrostatic force using a 3d simulation of the Navier-Stokesequation. The simulation also allows an estimation of the contact angle variation on both sides of the drop.In the second part of the paper, we investigate the effect of vibration on a water drop in contact with avertical plate. Here, as the plate vibrates parallel to gravity, the contact line oscillates. Each oscillation is,however, rectified by hysteresis, thus inducing a ratcheting motion to the water droplet vertically downward.Maximum rectification occurs at the resonant states of the drop. A comparison between the frequency-dependent motion of these drops and the variation of contact angles on their both sides is made. The paperends with a discussion on the movements of the drops on a horizontal hydrophobic surface subjected to anasymmetric vibration.

PACS. 47.55.D- Drops and bubbles – 47.61.-k Micro- and nano- scale flow phenomena – 68.08.Bc Wetting

1 Introduction

Recently, Daniel et al. [1,2] studied the movements of liq-uid droplets due to surface energy gradient in the presenceof wetting hysteresis. Basic observation was that smallliquid droplets do not move towards the more wettablepart of the gradient, because they cannot overcome theforce due to contact angle hysteresis. The hysteresis forceon a gradient surface is, however, spatially asymmetric,its magnitude against the gradient being larger than thatalong the gradient. If a periodic force is applied to thedrop, the force against the gradient is reduced whereasit is enhanced along the gradient; consequently the dropsmove with enhanced speeds towards the region of higherwettability. The situation is somewhat similar to thecommon observation that small water drops remain stuckon window panes or on plastic drinking cups even thoughthey are attracted downward by gravity. The explanationof this effect in view of the early works by Frenkel [3] andothers [4,5] lies, again, in wetting hysteresis. It is also acommon observation that slight tapping of these surfacesoften dislodges the water drops temporarily causing them

a e-mail: mkc4@lehigh.edu

to move downward, as the drops overcome the effect ofhysteresis. The relationship between vibration and wet-ting hysteresis has been studied systematically by severalinvestigators [6–9]. The general conclusion of these studiesis that the wetting hysteresis can be partially or fully miti-gated by vibration. This effect was used by Daniel et al. [1,2] to induce motion of small drops on flat surfaces. Severalother authors [10–12] have also demonstrated similareffects with an asymmetrically rough surface. In all thesestudies it has been found that matching of the forcingfrequency with the natural frequency of drop vibration iscritical to obtaining maximum rectification of the periodicdisturbances by any asymmetric property of the surface,be it asymmetric roughness or wettability gradient.

The resonant frequencies of vibration of sessile or pen-dant liquid drops have in the past been studied exten-sively [13–23] due to their relevance to various techno-logical processes, such as liquid–liquid extraction, syn-thesis of ceramic powders, crystal growth in micrograv-ity, and the measurement of dynamic surface tension toname a few. The first basic result of oscillation of freeliquid drops dates back to Kelvin [13] and Rayleigh [14].Later Lamb [15], ignoring the viscous damping in the drop,

232 The European Physical Journal E

developed a general expression for the different vibrationmodes of a free liquid drop as follows:

ω =

√

γ

3πml(l − 1)(l + 2) , (1)

where ω is the resonant frequency, l is an integer value of2 or higher, γ and m are the surface tension and mass ofthe drop, respectively. Later research by Strani and Sa-betta [24] showed that a drop in partial contact with asurface has an extra low-frequency mode related to thecenter-of-mass oscillation. These authors [24] also showedthat the solid support could also raise slightly the resonantfrequencies of Rayleigh modes. Even though extensive ef-forts have been made to model the oscillation of dropsin partial contact with a substrate undergoing vertical vi-bration, similar studies for the case of a drop undergo-ing lateral vibration are rather scanty. Nevertheless, a fewrecent studies [25–27] have made significant advances tothis important problem. Lyubimov et al. [25] provided anelegant solution to the lateral oscillation of an invisciddrop by taking into account the oscillation of contact line.These authors predicted a low-frequency rocking vibration(rocking mode) of the drop in contact with a solid surfaceas well. Celestini and Kofman [26] analyzed this rockingmode quantitatively and compared their predictions withexperimental observations. Moon et al. [27], on the otherhand, identified a new mode associated with the rotationalmotion of the drop subjected to lateral oscillation.Although not studied for lateral vibration, Noblin et

al. [28] carried out a systematic study of the effect of ver-tical vibration of large water drops on a hydrophobic sur-face. These authors observed two types of behavior. At lowamplitude, the contact line remains pinned and the droppresents various shapes, whereas at higher amplitude, thecontact line exhibits an oscillatory motion by overcominghysteresis. This type of oscillation, in the case of lateralvibration, leads to a net translation of the drop when thesymmetry is broken either by hysteresis [1,2,10–12] or in-ertial forces [29].In this paper, we investigate the resonant modes of

a sessile water drop undergoing a lateral vibration us-ing a 3d numerical simulation of Navier-Stokes equation.The velocity profile, hydrostatic force, and contact anglevariation are investigated to identify the resonant states.We are particularly interested in finding out how muchthe contact angles of both sides of the drop differ andhow they depend on the frequency of vibration, as it isthis difference of the contact angles that determines theforce and velocity of drops on surfaces. Finally, we attemptto make a qualitative comparison between the frequency-dependent contact angles of the vibrating water drop andits frequency-dependent motion induced by gravity. Fi-nally we discuss a special case of drop motion induced byasymmetric vibration.

2 Mathematical formulation

We consider the oscillation of a sessile water drop vibratedparallel to the substrate as shown in Figure 1. When con-

R

Vibration

x

z

miny

x

z

ymax

R

x

z

miny

x

z

ymax

θ

θθ

R

Vibration

x

z

miny

x

z

ymax

R

x

z

miny

x

z

ymax

θ

θθ

Fig. 1. Schematic of an oscillating drop on a vibrating sub-strate. The undisturbed profile of the drop is shown by thesolid line, whereas the disturbed profile is shown by the dashedline. The contact line is assumed to remain pinned. The drophas an initial contact angle θ. θmax and θmin are, respectively,the maximum and minimum contact angles exhibited by thevibrating drop.

tact angle hysteresis is significant and the external forceis sufficiently weak, the contact line remains pinned, butthe water drop deforms. In order to elucidate the dynam-ics of the drop deformation, we used a FLUENT pack-age to implement the numerical analysis, in which the 3dNavier-Stokes and continuity equations are solved usinga well-established volume of fluid (VOF) model. Here weprovide a brief description of the VOF model based on thedetailed discussions of references [30] and [31]. In the VOFmodel, a single Navier-Stokes equation is solved along withthe continuity equation for a domain comprising two fluidphases, which are water and air in our case. These equa-tions are solved in conjunction with an extra VOF advec-tion equation, in which a variable α, denoting the volumefraction of one of the phases is introduced. α = 1 signifiesthat a given computational cell is filled with one of thephases (i.e. water), whereas α = 0 signifies that the cellis filled with the other (air). The interface is tracked bythe condition 0 < α < 1. The variables and propertiesof a computational cell are characterized by the volume-average values of the phases. The 3d Navier-Stokes, con-tinuity, and VOF advection equations used to solve theproblem of the oscillating drop are as follows:

∂

∂t(ρv) +∇ · (ρvv)=−∇p+∇·[η(∇v+∇vT )]−ρgk, (2)

∂ρ

∂t+∇ · (ρv)=0, (3)

∂α

∂t+v · ∇α=0, (4)

where v = (u, v, w) and p are velocity vector and pressure.ρ = ρwα + ρa(1 − α) is the density in the computationalcell, ρw and ρa being the density of water and air, re-spectively. η = ηwα + ηa(1 − α) is the viscosity in thecomputational cell, ηw and ηa being the viscosity of wa-ter and air, respectively. k = (0, 0, 1) is a unit vector inthe z-direction for horizontal vibration and k = (1, 0, 0) isa unit vector in the x-direction for vertical vibration. Inorder to account for the effect of the interface, a sourceterm f is added to the right side of equation (2), which

L. Dong et al.: Lateral vibration of a water drop and its motion on a vibrating surface 233

n = 1 n = 2 n = 3 n = 4

(a)

(b)

(c)

Fig. 2. First four resonance modes of a vibrating water drop: (a) simulated images, (b) pressure contours, and (c) experimentalimages. The first mode (n = 1) corresponds to the rocking mode. The second (n = 2), third (n = 3), and fourth modes(n = 4) here correspond to the Rayleigh first, second, and third modes, respectively. In (b), the pressure decreases from redcolor representing higher pressure to sky-blue representing lower pressure. The plate amplitude for these simulations is 0.5mm,the diameter of the drop is 2mm and the equilibrium contact angle is 110◦. The images of the vibrating water drops as shownin panel (c) were captured using a high-speed camera (Midas) at a frame rate of 2000. The drops were placed on a horizontalsilanized glass slide, which was vibrated along its long axis at an amplitude of 0.3mm [32]. The details of the apparatus usedfor these studies have been described in references [2,29] and [33].

is a volume force originating from the surface tension andcurvature of the interface. This is the continuum surfaceforce (CSF) model as developed by Brackbill et al. [31]. fis defined in the CSF model as

f = γ κ∇α (5)

where γ is surface tension, and κ is the curvature of in-terface. According to equation (5), f is finite within acomputational cell that contains the interface. Away fromthe interface it vanishes as ∇α = 0. In the CSF model, theinterface curvature is calculated according to the outwardunit normal vector n by

κ = −∇ · n, (6)

where n = −∇α/|∇α|. The interface interpolation algo-rithm used in this study is the geometric reconstructionscheme, which tracks the interface using a piecewise linearapproach as described in reference [30]. Using this scheme,the water-air interface is assumed to have a linear slope ineach interface-containing computational cell. In the VOFmodel, the cells that contains the interface are first locatedby the condition 0 < α < 1. The position of the linear in-terface relative to the center of the cell is then calculated

according to the volume fraction α and the unit normalvector n.The boundary conditions on the solid surface at z = 0

are the no-slip and the no-penetration conditions,

u = Up , v = 0, w = 0 (7)

where, Up is the velocity of the vibrating substrate. Fora sinusoidal periodic vibration, Up = 2πωA cos(2πωt), Aand ω being the amplitude of displacement and frequencyof vibration, respectively. In the VOF model, the contactangle between the water and wall is defined by adjustingthe curvature of the surface and the unit normal vector inthe computational cells near a wall [30,31].In our study, the numerical analysis was carried out

on a uniform rectangular mesh with the size of 40µm ×40µm×40µm, which was found to be optimum in terms ofaccuracy and computation time. We arrived at this meshsize after experimenting with various mesh sizes of de-creasing order until the simulation result no longer de-pended on the mesh size. In the calculation, the pressureand velocity are coupled with pressure-implicit with split-ting of operator (PISO) algorithm, which uses a guess-and-correct procedure for the calculation of pressure anda high degree of approximation relation between pressure

234 The European Physical Journal E

and velocity correction. A 2nd-order upwind scheme wasused for the discretization of the model equations, whichcalculates the momentum quantity at the cell face fromthe value of the upstream cell using a second-order linearinterpolation scheme [30].

3 Results and discussion

3.1 Drops on a vibrating horizontal surface

We first discuss the case where the water drop vibrateson a horizontal surface, i.e. the direction of vibration isperpendicular to that of gravity. After discussing variousways to identify the natural resonant frequencies of a liq-uid drop, we discuss in the next section the case of thedrop on a vibrating vertical surface in which the dropmoves down by gravity. In Section 3.3, we briefly discussthe drop motion on an asymmetric vibrating surface.By varying the vibration frequency, the resonant

modes of a water drop can be easily identified from thephase contour of the drop at different times. The shapes ofa water drop at various resonant modes as obtained fromthe CFD simulation are compared with those obtainedexperimentally in Figure 2.Here the first mode (n = 1) corresponds to the rocking

vibration. The second (n = 2), third (n = 3), and fourth(n = 4) modes correspond to the Rayleigh first (l = 2),second (l = 3), and third (l = 4) modes, respectively.At the Rayleigh modes, regular capillary waves appear onthe drop surface with its crests varying with the resonantmodes. The pressure contours (Fig. 2b) also clearly high-light the different higher-pressure spots corresponding tovarious modes.The understanding of the dynamics of a laterally os-

cillating drop can begin with the close observation of theprofile of the horizontal velocity along the vertical direc-tion (Fig. 3). The velocity profile for a 2mm size drop cor-responding to the vibration frequency of 130Hz (Fig. 3)shows three characterized regions: the Stokes layer region(∼ (2µ/ρω)1/2), the free-surface region, and a sublinearvelocity gradient region. Within the Stokes layer, the fluidhas a very large velocity gradient due to the diffusion ofthe substrate momentum. Above this layer, the fluid ex-periences an inertial force, which causes the drop to de-form. The deformation is characterized by the oscillationof the center of mass, which leads to a contact angle dif-ference between two sides of the drop (Fig. 1). The defor-mation is also associated with an increase of the surfacearea of the drop and the variation of the local curvaturealong the drop surface. Thus a restoring force arises dueto the Laplace pressure acting inside the deformed drop,which attempts to decrease the surface area and restorethe drop to its original shape. Competition of these twoforces causes capillary waves to develop on the free surface.As Laplace pressure varies according to the local curva-ture, the velocity profile in the free-surface region showsan oscillating behavior. The drop performs something likea swinging oscillation of a one-end pivoted spring in thesublinear velocity gradient region. The drop deformation

t = 0.008 st = 0.010 s

t = 0.012 st = 0.014 s

t = 0.010 s

t = 0.012 st = 0.014 s

Surface Layer Region

Sublinear Velocity

0.00

0.25

0.50

0.75

1.25

1.50

-1 0 1U

Z

Stokes Layer Region

Gradient Region

Stokes Layery

z

x

1.00

Fig. 3. Horizontal-velocity profile along the vertical directionin the central (x = 0 and y = 0) part of the drop at differ-ent times. These simulations were carried out with a drop of2mm diameter at a vibration frequency of 130Hz. The plateamplitude and the equilibrium contact angle are 0.5mm and110◦, respectively. The liquid-solid interface is at z = 0. Here,the dimensionless velocity, U = u/U∗p , U

∗

p being the maximumvelocity of the plate (2πAω). The dimensionless vertical coor-dinate Z = z/H, where z is the dimensional vertical coordinateand H is the equilibrium height of the drop. The difference ofthe interface position is due to the evolving shape of the dropat different times. This plot shows that the momentum dissi-pation is confined within the Stokes layer (∼ (2µ/ρω)1/2). Thevelocity profile of an oscillating drop can be roughly dividedinto three regions: Stokes layer, surface layer, and a sublinearvelocity gradient region.

dU/dZ

Z = 0.298

Z = 0.373

Z = 0.484

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0 0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4 2.7

ω ω/ *

Fig. 4. The horizontal-velocity gradient along the vertical di-rection at different planes inside a vibrating drop versus thevibration frequency. U and Z are the dimensionless velocityand vertical distance, respectively. ω∗ = (γ/m)1/2, where m isthe mass of the drop. These are averages of maximum valuesof dU/dZ obtained from several cycles, which are calculatedat various values of Z as indicated in the figure. All thesepositions are outside the Stokes layer, and in the sublinearvelocity gradient region. At resonant frequencies, the drop ex-periences larger deformation, thus larger velocity gradient [34].The first three resonant frequencies are around 60Hz, 130Hz,and 290Hz, respectively.

L. Dong et al.: Lateral vibration of a water drop and its motion on a vibrating surface 235

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0 0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4 2.7

ω / ω*

F

A ω2 = Constant

Aω = Constant

A = Constant

Fig. 5. The hydrostatic force (integration of pressure gradi-ent over the entire drop) for a water drop of 2mm diameteras a function of the vibration frequency. The dimensionlesshydrostatic force F = F ∗/mg, where F ∗ is the dimensionalhydrostatic force, m is the mass of the drop as defined in Fig-ure 4 and g is the gravitational acceleration. ω∗ is defined inthe caption of Figure 4. The figure shows that the hydrostaticforce has minima at the resonant frequencies.

can be characterized by the amplitude of this swinging os-cillation, which can be quantitatively estimated from thevelocity gradient. Moreover, one can compare the veloc-ity gradients at different vibration frequencies to identifythe resonant states. Figure 4 shows the vertical gradientof the horizontal velocity (dU/dZ) for a 2mm diameterwater drop as a function of the vibration frequency. As ex-pected, the velocity gradient dU/dZ exhibits a sinusoidaloscillation at the steady state. The dU/dZ data reportedin Figure 4 are the average maximum over several cyclesat a specific frequency. All the three positions reported inFigure 4 are outside the Stokes layer, but are in the sub-linear velocity gradient region. It is evident from the datasummarized in Figure 4 that the drop undergoes larger de-formations at resonant frequencies compared to the non-resonant frequencies. Figure 4 identifies three resonant fre-quencies at 60Hz, 130Hz, and 290Hz, respectively.At any given instance, the inertial force of the entire

drop is balanced by the pressure (hydrostatic) and viscousforces as shown in equation (8), which is the volume inte-gral form of equation (2) (ignoring the gravitational termfor horizontal vibration):

ρ

(∫

V

vtdV +

∫

V

(v · ∇)vdV

)

=

−

∫

V

∇pdV +

∫

V

∇ ·[

η(

∇v +∇vT)]

dV. (8)

Numerical simulation shows that all the three terms ofequation (8), i.e. the inertial force (the term on the left),the hydrostatic force (first term on right), and the viscousforce (second term on the right), change at resonance, withthe hydrostatic force varying with a different phase thanthe other two forces. While any of the above forces canbe used to identify the resonance modes of a drop, we usethe hydrostatic force to achieve this goal. Numerical sim-

0

100

200

300

400

500

600

700

0 100 200 300 400 500 600

Rocking Mode

n = 2

n = 3

ω* (Hz)

ω (

Hz)

Fig. 6. The experimental and computed resonant frequenciesof water drops as a function of ω∗ or the mass of the drop. Theopen symbols correspond to the experimentally obtained data,which were reported previously by Daniel et al. [2,29,33]. Theclosed symbols correspond to the frequencies obtained fromthe CFD simulations for water drops of different masses (thusfor different values of ω∗). The closed-diamond, closed-square,and closed-triangle symbols correspond to the frequencies ob-tained from the calculation of the hydrostatic force (as shownin Fig. 5). The closed-circle symbols correspond to the fre-quencies from the calculation of dU/dZ (as shown in Fig. 4).Some of the simulated data calculated from two ways collapsetogether. The solid lines correspond to the first and secondRayleigh modes (Eq. (1)). The dashed line corresponds to thelinear regression through the points corresponding to the ex-perimental rocking-mode data.

ulations show that the hydrostatic force exhibits minimaat resonance frequencies as shown in Figure 5. These min-ima are due to the fact that the pressure gradient in theStokes layer is at 180◦ out of phase with that above theStokes layer at resonance. Away from the resonance, thisphase difference is less than 180◦. This results in the max-imum cancellation of the hydrostatic forces in the two re-gions at resonance, but not at non-resonance frequencies.In Figure 5, the hydrostatic force is non-dimensionalizedby dividing it with the weight of the drop instead of Aω2,although non-dimensionalization with the latter would bea more logical thing to do. The objective here is to pro-vide an idea of the magnitude of the hydrostatic force incomparison to the weight of the drop and to show thatthe position of the resonance frequency does not dependon the choice of either the constant amplitude A, constantAω, or constant Aω2 criterion. We, however, find that theplot with the constant Aω allows us to identify the peakposition most clearly, which is therefore used in the sub-sequent analysis.

The resonance frequencies obtained from the max-ima of the dU/dZ values (Fig. 4), however, are in goodagreement with those obtained from the minima of thehydrostatic force (Fig. 5). These data are summarizedin Figure 6. As expected from equation (1), the reso-nant frequencies decrease with the drop mass followinga square-root relationship, which conforms well to Lamb’s

236 The European Physical Journal E

0

5

10

15

20

25

30

0 0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4 2.7.6 2.

70°110°

90

95

100

105

110

115

120

125

130

0 0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4 2.7

/ *

/ *

Co

nta

ct A

ngle

(°)

θ max- min

(°)

θmax

min

(a)

(b)

ω ω

ω ω

θ

θ

Fig. 7. Contact angle as a function of the vibration frequencyfor a water drop. (a) θmax and θmin correspond to the maxi-mum and minimum angles at both sides of the drop when theequilibrium contact angle is 110◦. (b) The contact angle dif-ference θmax − θmin for the equilibrium contact angle is 110

◦

(solid squares) and 70◦ (solid diamonds). These calculationswere carried out with drops of the same mass (3.2× 10−6 kg).The solid lines are used only as a guide to the eye.

analysis [15]. The CFD results also agree well with the ex-perimental resonant frequencies of the drops as reportedearlier by Daniel et al. [2,29,33]. Moreover, both the ex-perimental and theoretical data show good agreementwith the predictions based on Rayleigh equation (Eq. (1))for the second and third modes (Rayleigh first and secondmodes).

Another method to identify the resonance frequencieswas suggested by Celestini and Kofman [26], which isbased on the examination of the contact angles on bothsides of the vibrating drop. The difference of these anglesexhibits maxima at the resonance frequencies, which wereused by Celestini and Kofman [26] to identify the rockingmode of vibration of liquid mercury on glass. We used thisapproach to identify not only the rocking mode, but alsothe higher modes by following the frequency-dependentcontact angles as obtained from the CFD simulated im-ages of the oscillating drops. After capturing these images,the contact angles were measured using an image analysisprogram written in Matlab. As is the case with the velocity

z

y

x

gravity

0.00

0.25

0.50

0.75

1.00

1.25

-1 0 1

t = 0.0225 st = 0.0205 st = 0.0185 st = 0.0165 s

Surface Layer Region

Sublinear velocity

U

Stokes Layer Region

gradient region

Z

Fig. 8. Horizontal-velocity profile in the central (x = 0 andy = 0) part of a drop on a vertically vibrating surface. Uand Z are defined in Figure 3. These calculations were carriedout with a 2µl size drop at a vibration frequency of 90Hz. Theplate amplitude and the equilibrium contact angle are 0.12mmand 97◦, respectively. This figure shows that the drop deformsmore along the direction of gravity than against it. Duringthe vibration, the plate experiences a maximum accelerationof 3.85m/s2, which is considerably lower than the magnitudeof the gravitational acceleration (9.83m/s2).

gradient, the contact angles on both the left and the rightsides of the drop exhibit sinusoidal oscillation at steadystate. θmax (maximum contact angle) and θmin (minimumcontact angle) reported in Figure 7a are the average maxi-mum and minimum values of both left- and right-side con-tact angles over several cycles. As anticipated (based onthe analysis of Celestini and Kofman [26]), both θmax andθmin vary as a function of the vibration frequency. θmaxpasses through maximum and θmin passes through mini-mum values leading to large values for θmax− θmin at theresonant frequencies. Thus, at the resonant frequencies,the two sides of an oscillating drop experience maximumcontact angle differences. It is also important to noticethat the general pattern of the behavior of the oscillat-ing drop is not too sensitive to the equilibrium contactangle. For example, the resonant frequencies of a drop(3.2 × 10−6 kg) on a surface of equilibrium contact angleof 110◦ are about 4 to 8% lower than those obtained witha drop of the same mass, but with a lower equilibriumcontact angle (70◦) (Fig. 7b).

In the simulation, it was assumed that the drop re-mains pinned on the surface. In reality, when the contactangle difference between two sides of the drop is largerthan the contact angle hysteresis, the drop would moverelative to the vibrating substrate. Since the drop experi-ences larger contact angle difference at resonant frequen-cies, one expects that the drop would overcome the hys-teretic threshold force most readily at the resonant fre-quencies and exhibit forward and backward motions rela-tive to the vibrating substrate. With an asymmetric hys-teresis or when an asymmetric inertial force acts on thedrop, the drop should translate on the substrate with max-imum velocities at the resonant frequencies.

L. Dong et al.: Lateral vibration of a water drop and its motion on a vibrating surface 237

ω* (Hz)

ω(H

z)

0

100

200

300

400

500

600

700

0 50 100 150 200 250 300 350

Rocking Mode

n = 2

n = 3

Fig. 9. The resonant frequencies of water drops on both thehorizontal vibrating surface and the vertical vibrating surfaceas a function of ω∗ or the mass of the drop. The closed symbolscorrespond to the horizontally vibrating surface, whereas theopen symbols correspond to the vertically vibrating surface.The solid lines correspond to the first and second Rayleighmodes (Eq. (1)). The dashed line corresponds to the linearregression through the points corresponding to the data of therocking mode for the drop on a horizontal vibrating surface.ω∗ is defined in Figure 4.

3.2 Drops on a vibrating vertical surface

The CFD simulation of a drop on a vibrating vertical plate(Fig. 8 inset) was carried out as before, except that thegravity now acts parallel to vibration. During the firsthalf of the stroke, the inertial and the gravitational forcesjointly deform the drop, whereas during the next half ofthe stroke, the inertial force is reduced by gravity. Overall,the drop vibrates asymmetrically as is shown by the non-symmetrical velocity distribution inside the drop duringthe forward and reverse cycles (Fig. 8).The resonance frequencies of the drop oscillating ver-

tically were determined as before from the integration ofthe pressure gradient. The results summarized in Figure 9show that these resonance frequencies are not noticeablydifferent from those of the horizontal vibrations. A sim-ilar picture emerges from the contact angles of the dropas well. Figure 10a plots the largest difference of the con-tact angles for a 2µl drop on both sides of the drop asa function of frequency. As expected, a larger differenceof the contact angles occurs when the drop trajectory isdownward as opposed to the upward movement. Althoughthe frequencies corresponding to the first, second and thirdmodes are not all that different from that of the horizontalvibration, the amplitude of the drop vibration correspond-ing to each resonance is higher for the case of verticalvibration. Furthermore, the first mode (i.e. the rockingmode) for the case of asymmetric vibration is consider-ably stronger than the second mode, which is opposite tothe trend observed with the vibration on the horizontalsurface.Apparently, some of the gravitational energy is chan-

neled to the vibration modes of the drop during this asym-metric vibration. Furthermore, the strengthening of thefirst peak relative to the second is indicative of the en-

0

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0 0.5 1 1.5 2 2.50

5

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00

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0 0.5 1 1.5 2 2.50.00000

0.00005

0.00010

0.00015

0.00020

0.00025

0.00030

0.00035

0.00040

Ca

(a)

(b)

/ *

θm

ax–

θm

in (°)

θ d,r

d,a

u,a

u,r

ω ω

/ * ω ω

θ θ

θ

Fig. 10. (a) Simulated contact angle difference as a functionof the vibration frequency for a 2µl water drop subjected toa vibration on a horizontal (solid diamond) and vertical sur-face. Here, the solid triangle and solid square correspond tothe downward and upward movements of the drop, respec-tively. (b) The experimental velocities of a 2µl drop movingvertically downward on a vibrated surface. The surface wasa silanized silicon wafer. In all cases, the maximum vibrationvelocity (2πAω) is 69mm/s, while the maximum plate accel-eration changes from 1.3m/s2 at 30Hz to 13.0m/s2 at 300Hz.The average of the advancing and receding contact angles atrest is 97◦. All the simulations of panel (a) were performedwith the equilibrium contact angle of 97◦. The solid lines areused only as a guide to the eye.

ergy exchange between the two modes. These simulationresults, however, could not be confirmed in the same wayas that of the horizontal vibration since the large dropsmove down by gravitational force and the small dropsmove downward in the presence of vibration. The simu-lation results, however, have important implications withrespect to the vibration-induced motion of drops on verti-cal surfaces [35], which provide an indirect test of the sim-ulation results. In order to understand the implications ofthese results, we have conducted a simple experiment witha small liquid droplet on a homogeneous surface inclinedvertically. The surface is a silanized silicon wafer, whichexhibits advancing and receding contact angles with water

238 The European Physical Journal E

of 104◦ and 89◦ (15◦ hysteresis), respectively. Small dropsof water (∼ 2µl) remain stuck to this surface becausethe hysteresis force wins over the gravity force. However,as the plate is vibrated, the contact line oscillates asym-metrically. Each oscillation is rectified [36] by hysteresisand the drop moves downward with a speed dependingon the frequency of vibration. When the drop speed isplotted as a function of frequency, clear velocity maximacould be identified at the resonance modes of vibration.Figure 10b summarizes these results, where the velocity(V ) of the drop is non-dimensionalized by dividing it withthe capillary velocity (γ/η) to yield the capillary numberCa(V η/γ). It should be noted that the actual velocity ofa drop varies with time. V represents the average driftvelocity.

In agreement with simulation results, the most in-tense velocity peaks correspond to the first and third res-onances, respectively. As a first-order approximation, thevelocity of the drop (or Ca) is proportional to the dif-ference of the cosines of dynamic receding and advanc-ing angles [37,38]. However, as discussed in reference [29],some amount of wetting hysteresis is necessary to breakthe symmetry of any periodic pulse. Thus, for the dis-cussion to follow, we assume that a small hysteresis isintrinsically present at the contact line. The discussion tofollow is not correct in its detailed features, but it pro-vides an approximate framework with which to examinethe droplet motion. We consider that the ultimate effect ofall the hydrostatic, inertial and viscous forces is to modifythe local contact angles on the two sides of the drop. Letθ be the intrinsic Young’s contact angle, and θd,r and θd,abe the receding and advancing contact angles of the dropduring the first half of the downward stroke of the drop.If θd,a > θ and θd,r < θ, uncompensated forces would acton the advancing and receding edges, the magnitudes ofwhich are γ(cos θ−cos θd,a) and γ(cos θd,r−cos θ), respec-tively, per unit length. The net uncompensated force ex-perienced by the drop is ∼ γR(cos θd,r−cos θd,a), R beingthe base radius of the drop. If most of the resistance to themotion of the drop arises from the contact line region, theviscous drag force experienced by the drop is proportionalto ηRV . Balance of the above two forces leads to Cad ∼cos θd,r − cos θd,a, where Cad is the capillary number cor-responding to the downward motion of the drop. Duringthe other half of the stroke, Cau ∼ cos θu,r−cos θu,a (herethe subscript u denotes upward). Note that in the aboveexpressions, both θd,r and θu,r have to be smaller thanthe intrinsic receding angle (θr), and both θd,a and θu,ahave to be larger than the intrinsic advancing angle (θa)before the drop moves on the surface. The net velocity isobtained from the difference of the above two terms, i.e.Ca ∼ (cos θd,r − cos θd,a)− (cos θu,r − cos θu,a).

In Figure 11, the experimentally obtained capillarynumbers are compared with the driving force [(cos θd,r −cos θd,a)−(cos θu,r−cos θu,a)]. Again, the general trends ofboth sets of data are very similar and the positions of theexperimentally obtained velocity peaks corresponding tothe first and third resonance show a very good correlationwith those of the driving forces. There is, however, a dis-

0.00

0.05

0.10

0.15

0.20

0.25

0 0.5 1 1.5 2 2.5

0.00000

0.00005

0.00010

0.00015

0.00020

0.00025

0.00030

0.00035

0.00040

Ca

∆(cos r–cos a)

/ * ω ω

θθ

Fig. 11. The experimental velocities (solid diamond) of a 2µlwater drop moving on a vibrating vertical surface are comparedwith the characteristic driving force (solid square) as a functionof vibration frequency. Here, ∆(cos θr − cos θa) = (cos θd,r −cos θd,a)− (cos θu,r − cos θu,a) (see Fig. 10a for the definitionsof subscripts.) The velocity as well as the forcing frequency arenon-dimensionalized as Ca and ω/ω∗, Ca is defined in the textand ω∗ is defined in Figure 4. The solid lines are used only asa guide to the eye.

crepancy. The computational analysis predicts a small ve-locity peak corresponding to the second resonance, whichis absent in the experimental velocity spectrum. Therecould be various sources for this discrepancy. First of all,our experiments may not have the required sensitivity todetect small differences of velocity to properly identify thesecond peak. Secondly, the computational analysis hasbeen carried out with pinned contact line and by ignoringwetting hysteresis. Hysteresis, which is the key to obtain-ing rectification [29] of periodic pulses, also suppresses orobliterates any excitation less than the threshold force.

For reasons related to hysteresis, there are some dif-ferences between the numerically obtained contact anglesand those found experimentally. Figure 12 summarizes thevariations of the experimentally obtained contact anglesof the two sides of the drop for oscillation frequencies of110Hz, 230Hz, and 310Hz, respectively. The velocity ofthe drop is maximum at 110Hz (ω/ω∗ = 0.56) which cor-responds to the first resonance mode. According to thenumerical simulation, the maximum difference of the con-tact angles, at this frequency, should fluctuate between+25◦ and −15◦, respectively. The experimental values, onthe other hand, fluctuate between +44◦ and −26◦, respec-tively. In the presence of hysteresis, the drop undergoesa larger deformation as one of the edges approaches theadvancing angle, which is larger than the equilibrium an-gle, whereas the other edge approaches the receding angle,which is smaller than the equilibrium angle. The drivingforces along both the downward and upward directionsare, however, reduced and the corresponding capillarynumbers are described by: Cad ∼ (cos θd,r− cos θd,a)−H,and Cau ∼ (cos θu,r − cos θu,a) − H, respectively. HereH signifies a hysteresis threshold that needs to be over-come for the drop to move. The net capillary number is,

L. Dong et al.: Lateral vibration of a water drop and its motion on a vibrating surface 239

-2.5

-2.0

-1.5

-1.0

0.5

0

0 0.01 0.02 0.03 0.04 0.05

Dis

tan

ce

(mm

)

Plate

Front Edge

Back Edge

Time (s)

-30-101030

0 0.01 0.02 0.03 0.04 0.05

Time (s)

310 Hz

230 Hz

110 Hz

-30-101030

-30-10103050

110 Hz

Con

tact

An

gle

Dif

feren

ce (

°)

≈

(a)

(b)

Fig. 12. (a) Periodic rectification of the advancing and reced-ing edges of a 2µl water drop on a vertical surface subjected toa vibration frequency of 110Hz. The tracking of both edges ofthe drop was performed with a high-speed camera at a framerate of 2000. The positions of both the advancing and recedingedges were performed relative to a fixed reference point. In eachcycle of vibration, both the advancing and receding contact lineoscillates on the surface, but with a net drift downwards. Thedifferences of the contact angles between the bottom and up-per edges of the drop were estimated from the video framescaptured with a high-speed camera (2000 fps). Typical resultsobtained for the vibration frequencies of 110Hz, 230Hz, and310Hz are shown in panel (b).

however, given by the following equation:

Ca ∼1

τ

∮

[(cos θr − cos θa)−H]dt, (9)

where τ is the period of oscillation. In equation (9), onlypositive values of (cos θr − cos θa) − H are considered,as the drop cannot move during the duration for which(cos θr − cos θa) < H. We thus have stop-go events: thedrop moves when (cos θr − cos θa) > H, but it stops when(cos θr−cos θa) < H. This jump discontinuity is important

in order to achieve a net drift motion of the drop as hasbeen discussed in reference [29]. It is possible that duringhalf of a cycle (cos θr−cos θa) > H, in which case the dropwould move downward. However, during the other half ofthe cycle cos θr−cos θa < H, in which case there is no driv-ing force on the drop, thus it remains stuck on the surface.The evaluation of the cyclic integration needs to reflect theabove facts. However, if the drop moves relative to the sub-strate in both the downward and upward directions, as isthe case at the first and third resonance frequencies, its netdrift velocity (expressed in terms of Ca) is approximatelygiven by the difference of Cad and Cau. In this case, thehysteresis term H nearly cancels out and one obtains thesame expression as that on the non-hysteretic surface asdiscussed in references [2] and [3]. Based on the advanc-ing and receding contact angles (θd,r = 68

◦, θd,a = 113◦;

θu,r = 73◦, θu,a = 100

◦) on the right and left sides of thedrop, the term [(cos θd,r − cos θd,a)− (cos θu,r − cos θu,a)]is estimated to be about 0.3 at the first resonance peak.This leads to an estimate of the constant of proportion-ality between the capillary number (Ca ∼ 0.00035) and[(cos θd,r − cos θd,a) − (cos θu,r − cos θu,a)] as 0.0012. Us-ing a similar approach, the above proportionality factorat the third resonance is about 0.002. These proportion-ality constants are indicative of the frictional resistanceexperienced by the drop, a part of which originates atthe contact line. Recently, Daniel et al. [2] studied themotion of liquid drops of various viscosities driven by agradient of the surface energy. From these studies, theconstant of proportionality between the capillary numberand the driving force (cos θa − cos θr) can be estimatedto be about 0.005. Using a correction factor that comesfrom the fact that an oscillating drop on the average ex-periences a driving force that is π times [39] lower than[(cos θd,r−cos θd,a)−(cos θu,r−cos θu,a)], we expect a pro-portionality factor between capillary number and drivingforce to be 0.0016, which roughly corresponds to the num-bers reported above. This should, nevertheless, be consid-ered as a crude analysis at present as we have not con-sidered the energy dissipation occurring within the thinStokes layer adjacent to the substrate.

Overall, the intense velocity peak at the first reso-nance, as predicted by the numerical simulation, is consis-tent with the experimental observation. However, as men-tioned above, there are some disagreements between thesimulations and experiments, which are due to various rea-sons. Most significantly, the base of the drop is pinned inour simulation. With the simulated results obtained withpinned drops, we assumed that the drop would move aslong as the maximum difference of the contact angles onboth sides of the drop is larger than hysteresis. Secondly,the role of contact angle hysteresis has not been explic-itly taken into account in our simulation. As mentionedabove, hysteresis should allow a much larger deformationof the drop as one of its edge tries to attain the advancingangle while the other edge tries to achieve the recedingangle, all of which resulting in a larger asymmetric vi-bration. This larger asymmetry could excite the first res-onance (i.e. the rocking mode) even more strongly, and

240 The European Physical Journal E

0.0 sec

10 mm

0.0 sec

1.5 sec1.5 sec

3.2 sec3.2 sec

7.5 sec7.5 sec

0.0 sec0.0 sec

1.5 sec1.5 sec

3.2 sec3.2 sec3.2 sec3.2 sec

7.5 sec7.5 sec7.5 sec7.5 sec

(c)

0.0 sec0.0 sec

1.5 sec1.5 sec

3.2 sec

7.5 sec

0.0 sec

1.5 sec

4

6

8

10

12

14

16

18

0 0.5 1 1.5 2 2.5

Forward

Backward

0.15

0.00

0.15

0 0.004 0.008Time (s)

Posi

tion

(m

m)

0

θm

ax–

θm

in (°)

(a)

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

0 0.5 1 1.5 2 2.5

ω / ω*

V/ A

ω

(b)

ω / ω*

Fig. 13. (a) The difference between the contact angles of the right and the left sides of a drop subjected to an asymmetricvibration (see inset). These simulations were carried out with a small drop of 2mm diameter. The equilibrium contact angle is110◦. Here, three resonant frequencies are observed at 60Hz, 130Hz, and 290Hz, respectively. Based on these contact angles,it is expected that the drop would move forward when ω/ω∗ is about 0.40, but move backward when it is about 0.85. Anotherreversal of drop motion is expected to occur at ω/ω∗ and 1.90. Panel (b) (adapted from Ref. [29] with a slight modification)provides the experimental data of drop motion that is in qualitative agreement with the prediction of panel (a). Panel (c) showsthat two dissimilar-size drops indeed move in the opposite directions when excited by an asymmetric waveform (100Hz) of thetype shown in panel (a). ω/ω∗ of the smaller drop (∼ 0.68) corresponds to its second resonance mode, whereas that (∼ 1.35) ofthe larger drop corresponds to the pre-resonance of the third mode.

suppress the second mode, as is found experimentally. Es-timation of the intrinsic hysteresis is, however, a majorlimitation of the problem, as the release of the contact linefrom metastable energy barriers that gives rise to hystere-sis may depend on vibration frequency as well.

3.3 Drop motion using asymmetric vibration: polarizedratchet

The enhancement of the first resonance compared to thesecond in the presence of an external force, such as grav-ity, could lead to several interesting possibilities. If, forexample, a Maxwell stress or an inertial stress is set upin the drop and if this stress is switched on and off witha time-varying asymmetry, it may be possible to induce astrong first resonance in one instance, but a stronger sec-ond resonance in another instance. Thus, depending uponthe strengths of these external excitations, the drop maybe induced to go forward or backward. A situation some-what like that speculated above was recently observed byDaniel et al. [29,33]. These authors examined the fate of

a drop vibrated on a hydrophobic surface with an asym-metric waveform. The particular waveform used by theseauthors is either a sawtooth or an exponential fall/risewave that accelerates the plate fast in one stroke, but de-celerates it slowly in the other. The resulting asymmetricinertial force was rectified by hysteresis thus inducing amotion of the drop on the surface. Daniel et al. [29,33]observed that the direction of the motion of the drop de-pends on the frequency of vibration and the size of thedrop. One of the most interesting observations was thattwo drops of dissimilar sizes travel in the opposite direc-tions at a given frequency. This is what we denote here asa polarized ratchet. A plausible explanation of this effectcould be due to the asymmetric amplification of modes, asis the case with a drop moving down by gravity. In orderto test this hypothesis, we carried out a numerical solutionof a drop on a surface undergoing asymmetric vibration(fast linear rise and exponential fall of acceleration) usingthe methodologies described in the above sections. Themotion of the vibrating plate resulting from such a wave-form is shown in the inset of Figure 13a, which was inputin the simulation as a Fourier series. The contact angles of

L. Dong et al.: Lateral vibration of a water drop and its motion on a vibrating surface 241

a small drop (2mm diameter) exhibited three resonancemodes at 60, 130 and 290Hz, respectively, which are thesame as those of symmetric periodic vibrations (Fig. 4).Interestingly, however, the intensities of these modes dur-ing the forward and the backward strokes exhibit an oscil-latory behavior. For example, while the first rocking modeis stronger during the forward stroke, the second mode isstronger in the reverse stroke. Another reversal of orderoccurs near the third resonance. The consequence of all ofthese is that a drop of a given size should move in one di-rection at its first resonance frequency, but in the oppositedirection at the second resonance. If the frequency is keptconstant, two different-size drops, the resonance frequen-cies of which correspond to first and second modes, respec-tively, move in opposite directions as shown in Figure 13c.

4 Conclusions

Numerical analysis of the 3d Navier-Stokes equation hasbeen useful in identifying the resonance modes of vibrationof a sessile drop on a hydrophobic surface. The vibrationmode can be identified in various ways, i.e. calculating thevelocity gradient, the net hydrostatic force across the dropor just by finding the extreme values of the contact angleson both sides of the drop. The resonant frequencies iden-tified from these simulations are in good agreement withthe experimental results reported previously [2,29,33].A remarkable finding of the simulations with a drop

on a vibrating vertical surface is that gravity enhancesthe first resonant mode and weakens the second mode,even though the positions of the resonance peaks do notdiffer substantially from those observed with the horizon-tal vibrations. These results are in qualitative agreementwith the experimental observations of drops moving down-ward on a vibrating surface. When a drop is vibrated on ahorizontal surface with an asymmetric vibration, differentmodes are asymmetrically amplified during the forwardand reverse strokes of a vibration cycle. These asymme-tries lead to what we term as “polarized ratchet”, in whichdrops of different sizes move in opposite directions.The current analysis is however limited by the fact

that the contact line is pinned. In reality, the contact linewould slip on the surface whenever the dynamic angle islarger than the advancing, or lower than the receding an-gle. A detailed analysis of the problem by taking care ofthe contact line slippage and contact angle hysteresis is anatural extension of this study; however, all these param-eters are non-linearly coupled to each other. If successfullycarried out, such an analysis would shed considerable lighton the vibrated motion [29] of liquid drops on surfaces dueto asymmetric hysteresis or inertial force on quantitativeterms.

L.D. was supported by a grant from PITA (Pennsylvania In-frastructure Technology Alliance). M.K.C. acknowledges manydiscussions with F. Celestini and thanks him for sharing hiswork with us prior to publication. Some of the ideas regardingthe role of hysteresis in drop motion resulted from some earlier

suggestions by Prof. P.G. de Gennes (see Ref. [29]). All the nu-merical simulations of this work were carried out by L.D. Theexperiments with the motion of a drop on a vertical surfaceand the related analysis were carried out by A.C.

References

1. S. Daniel, M.K. Chaudhury, Langmuir 18, 3404 (2002).2. S. Daniel, S. Sircar, J. Gliem, M.K. Chaudhury, Langmuir

20, 4085 (2004).3. Y.I. Frenkel, J. Exp. Theor. Phys. (USSR) 18, 659

(1948), translated into English by V. Berejnov (http://arxiv.org/ftp/physics/papers/0503/0503051.pdf).

4. J.J. Bikerman, J. Colloid Sci. 5, 349 (1950).5. C.G.L. Furmidge, J. Colloid Sci. 17, 309 (1962).6. T. Smith, G. Lindberg, J. Colloid Interface Sci. 66, 363

(1978).7. C. Andrieu, C. Sykes, F. Brochard, Langmuir 10, 2077

(1994).8. E.L. Decker, S. Garoff, Langmuir 12, 2100 (1996).9. T.S. Meiron, A. Marmur, I.S. Saguy, J. Colloid Interface

Sci. 274, 637 (2004).10. O. Sandre, L. Gorre-Talini, A. Ajdari, J. Prost, P. Sil-

berzan, Phys. Rev. E 60, 2964 (1999).11. A. Buguin, L. Talini, P. Silberzan, Appl. Phys. A: Mater.

Sci. Processing 75, 207 (2002).12. A. Shastry, M.J. Case, K.F. Böhringer, Langmuir 22, 6161

(2006).13. Lord Kelvin, Math. Phys. Pap. 3, 384 (1890).14. Lord Rayleigh, The Theory of Sound (Macmillan, 1894).15. H. Lamb, Hydrodynamics (Cambridge University Press,

UK, 1932).16. S. Chandrasekhar, Hydrodynamic and Hydromagnetic Sta-

bility (Dover, Mineola, NY, 1961).17. V.I. Kalechits, I.E. Nakhutin, P.P. Poluektov, Sov. Phys.

Tech. Phys. 29, 934 (1985).18. R.W. Smithwick, J.A.M. Boulet, J. Colloid Interface Sci.

130, 588 (1989).19. V.I. Kalechits, I.E. Nakhutin, P.P. Poluektov, Yu.G.

Rubezhnyi, V.A. Chistyakov, Sov. Phys. Tech. Phys. 5,496 (1979).

20. N. Yoshiyasu, K. Matsuda, R. Takaki, J. Phys. Soc. Jpn.65, 7 (1996).

21. O.A. Basaran, D.W. DePaoli, Phys. Fluids 6, 2923 (1994).22. E.D. Wilkes, O.A. Basaran, Phys. Fluids 9, 1512 (1997).23. H. Rodot, C. Bisch, A. Lasek, Acta Astronautica 6, 1083

(1979).24. M. Strani, M.F. Sabetta, J. Fluid Mech. 141, 233 (1984).25. D.V. Lyubimov, T.P. Lyubimova, S.V. Shklyaev, Fluid

Dyn. 39, 851 (2004).26. F. Celestini, R. Kofman, Phys. Rev. E 73, 041602 (2006).27. J.H. Moon, B.H. Kang, H.Y. Kim, Phys. Fluids 18, 021702

(2006).28. X. Noblin, A. Buguin, F. Brochard-Wyart, Eur. Phys. J.

E 14, 395 (2004).29. S. Daniel, M.K. Chaudhury, P.G. De Gennes, Langmuir

21, 4240 (2005).30. Fluent 6.2 User’s Guide, Fluent Inc., 2003.31. J.U. Brackbill, D.B. Kothe, C. Zemach, J. Comput. Phys.

100, 335 (1992).

242 The European Physical Journal E

32. The images were taken at 0.3mm plate amplitude ratherthan at higher amplitudes because of two reasons. Firstly,we tried to avoid the distortion of the forcing signals thatsometimes develop at higher amplitudes in our experimen-tal apparatus. Secondly, wetting hysteresis allows a largerdeformation of the drop than the simulations. When we im-pose the condition of pinned contact line, we mean that thecontact angle at the three phase contact line is pinned to acertain value. All the macroscopically observable changesare due to the dynamic deformation of the drop. In real-ity, however, the contact angle on each edge of the drop isbound by two extreme values with multiple metastable an-gles in between. We expect a larger deformation of the dropin a real system because the metastable angles on each sidecan switch between two extremes values. Nevertheless, theresonant vibration frequencies and the overall pattern ex-hibited by the vibrating drop are not affected by the am-plitude of vibration in both simulations and experiments.

33. S. Daniel, The Effect of Vibration on Liquid Drop Motion,PhD Thesis, Lehigh University, 2005.

34. We checked the linear response of the drop vibration invarious ways. Particularly, we calculated the velocity gra-dient at a given frequency and the fixed position inside the

drop and observed a linear correlation between the velocitygradient and vibration amplitude.

35. Vibration-induced motion of objects in a gravitational fieldfor solid-solid case, where symmetry is broken by solid fric-tion has recently been studied by Buguin et al.: A. Buguin,F. Brochard, P.G. De Gennes, Eur. Phys. J. E 19, 31(2006).

36. Some elegant studies of the contact line oscillation in an-other system involving Faraday waves in a glass containerwere reported previously. See C.-L. Ting, M. Perlin, J.Fluid Mech. 295, 263 (1995); L. Jiang, M. Perlin, W.W.Schultz, Phys. Fluids 16, 748 (2004). In the paper by Jianget al., the authors report a non-linear relationship betweenwave frequency and amplitude near contact lines. We sus-pect that a similar non-linearity may also be present inour studies. However, we have not investigated this effectsystematically this time.

37. H.P. Greenspan, J. Fluid Mech. 84, 125 (1978).38. F. Brochard, Langmuir 5, 432 (1989).39. In either the downward or upward direction, the average

force is 2πtimes the maximum force. However, this force is

divided by 2 as the drop moves forward half of the cycleand backward for the other half.