modeling the effects of contact angle hysteresis on the sliding of droplets down inclined surfaces

European Journal of Mechanics B/Fluids 48 (2014) 218–230

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European Journal of Mechanics B/Fluids

journal homepage: www.elsevier.com/locate/ejmflu

Modeling the effects of contact angle hysteresis on the sliding ofdroplets down inclined surfacesGulraiz Ahmed a,∗, Mathieu Sellier a,∗, Mark Jermy a, Michael Taylor ba Mechanical Engineering Department, University of Canterbury, Christchurch, New Zealandb Environmental Science and Research (ESR), Christchurch, New Zealand

a r t i c l e i n f o

Article history:Received 10 February 2014Accepted 17 June 2014Available online 7 July 2014

Keywords:DropletsContact angle hysteresisLubrication approximationSpreading/sliding

a b s t r a c t

Contact angle hysteresis is an important phenomenon that occurs both in natural and industrial dropletspreading/sliding applications. As they slide, droplets adopt a different contact angle at the front andrear, the advancing and a receding contact angles, respectively. This work investigates the different stagesinvolved in the motion of droplets down inclined surfaces in the lubrication approximation framework.A simplified hysteresis model is proposed, implemented, and tested. This model automatically locatesthe section of the contact line which is advancing and the section which is receding. This enables theapplication of different contact angles at the advancing and receding fronts and therefore takes intoaccount contact angle hysteresis. For validation purposes, experiments of fluid droplet spreading/slidingon inclined surfaces have also been performed tomeasure the terminal sliding velocity.With the inclusionof contact angle hysteresis, simulation results are shown to be in much better agreement with theexperimental ones. This paper also presents a simplemodel based onNewton’s second lawwhich is shownto have reproduced remarkably well the steady and dynamic results if the shape of the droplets does notdepart too much from a spherical cap configuration.

© 2014 Elsevier Masson SAS. All rights reserved.

1. Introduction

Spreading/sliding droplets feature a very important phe-nomenon which is crucial to their dynamics, i.e. contact angle hys-teresis. It is always present in natural as well as most industrialprocesses. This phenomenon is best understood by considering adroplet resting on an inclined substrate, like raindrops on a carwindscreen. Gravitational pull favors the sliding of the dropletdownslope while hysteresis resists it in static conditions. There-fore the droplet shape is asymmetric in comparison to the symmet-ric shape on a horizontal substrate. The droplet becomes thin witha lower contact angle at the back, known as receding contact an-gle, θr , and becomes thick with a higher contact angle at the front,known as advancing contact angle, θa. When the droplet reachesa definite size, it starts to slide down the incline maintaining itsasymmetric shape with θa in the front and θr at the back, see Fig. 1.

The difference between the advancing and receding contactangles is known as contact angle hysteresis. The presence of

∗ Corresponding authors.E-mail addresses: [email protected], [email protected]

(G. Ahmed), [email protected] (M. Sellier).

http://dx.doi.org/10.1016/j.euromechflu.2014.06.0030997-7546/© 2014 Elsevier Masson SAS. All rights reserved.

this hysteresis means that the system is in a metastable state.There are three major sources of hysteresis: chemical compositionof the fluid, surface roughness, and chemical heterogeneities[1]. Fluids such as polymers or surfactants may leave behind afilm on the substrate; the presence or absence of this film willdirectly influence hysteresis. Hysteresis is greatly influenced bysubstrate roughness. Dettre and Johnson investigated the changein hysteresis as roughness is varied [2]. They detected an unusualchange in θr as roughness is decreased. Chemical heterogeneitiesin the substrate may also play a vital role. Differences in substratewettability may cause unexpected sliding dynamics [3].

Contact angle hysteresis is also present in coating procedures,digital micro-fluidics, droplet evaporation, ink-jet printing, andpesticides spraying applications. Hysteresis is problematic in afew industrial applications (immersion lithography), but is crucialin others (coating and spray painting) [4]. Controlling the extentof hysteresis has a significant importance in most industrialoperations.

The prominence of contact angle hysteresis, has ledmany scien-tists to study its origin and its effect on the motion of the droplets.Adam and Jessop first appreciated the existence of contact an-gle hysteresis due to the frictional resistance of the substrate [5].De Gennes reviewed the early experiments that explain the phe-nomenon [1]. Furmidge investigated spray (droplet) retention in

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G. Ahmed et al. / European Journal of Mechanics B/Fluids 48 (2014) 218–230 219

Fig. 1. Sliding droplet with advancing, θa , and receding, θr contact angles.

the agricultural sector. Droplet retention is one of the sources andeffects of contact angle hysteresis. Furmidge suggested a simpletechnique to describe the retention of the droplets sliding downthe inclined substrates [6]. Dussan and co-workers studied wa-ter droplets sliding at a constant velocity down the inclined sub-strates of different levels of roughness [7–9]. In these studies, theauthors demonstrated a theoretical explanation of the retentionforces responsible for the sticking of droplets on solid substrates.Various authors have analyzed contact angle hysteresis for vari-able substrate properties, such as surface roughness and surfaceheterogeneity, both computationally and experimentally [10–13].Hysteresis of one or two degrees may essentially be due to the un-certainty in measurement and can be neglected [10]. In most prac-tical droplet sliding applications, the hysteresis is around 10° [1].Topographical heterogeneities such as peaks or ridges may causehysteresis to increase up to 50° [14].

Contact angle hysteresis affects the sliding velocity of thedroplet sliding down an inclined substrate. Various authors in thepast have developed scaling laws to estimate the sliding velocityof droplets using the small contact angle assumption to apply thelubrication approximation [7,15]. Kim and co-workers performedexperiments to measure the sliding velocity of partially wettingviscous droplets on a smooth surface. They constructed a scalinglaw which does not assume a small contact angle and demon-strated the dependence of sliding velocity on various parameters[16]. As the droplet slides down with a particular hysteresis, thereexists a no-slip boundary condition at the liquid–solid interface.A moving contact line is a clear violation of the no-slip boundarycondition so it is necessary to use a model which can alleviate thesingularity which arises from the contact line motion.

The presence of contact angle hysteresis in the advancing andthe receding regions and the formation of a sharp corner at thetail of the spreading droplet above a critical speed is observedexperimentally [17,18]. Podgorski and co-workers recorded theshape of a droplet from above and analyzed the velocity withwhich it runs down an inclined solid substrate [17]. Le Grand andco-workers suggested a few improvements in the experimentalsetup of Podgorski and co-workers [18]. Droplets start to developa corner at the rear which change to a cusp at a greater velocity.At an even higher velocity a satellite droplet at the tail separatesand pearling phenomenon occurs [18]. Similar shape patterns areobserved when solving the same problem numerically [19,20].Schwartz and Eley investigated the hysteretic behavior of dropletmotion using the lubrication approximation on a chemicallyheterogeneous surface [3]. The introduction of disjoining pressurein the lubrication formulation can alleviate the contact linesingularity. Hysteretic effect was demonstrated by applyingdifferent equilibrium contact angles at different positions on thesubstrate to measure the contact line velocity. We expand here onthework of Gaskell and co-workers who implemented a hysteresismodel in the context of the lubrication approximation [21].

Nature demonstrates contact angle hysteresis phenomena inour daily surroundings. The approach to include this physicalaspect in models is far from settled. The present work exploresa new way of modeling contact angle hysteresis based on thegeometry of the sliding droplet to predict the advancing andreceding fronts of a sliding droplet and its subsequent effect on thedroplet terminal velocity.

In the past, contact angle hysteresis has received considerableattention but to the best of our knowledge, three-dimensionalmodeling of a fluid droplet spreading/sliding with contact an-gle hysteresis on an inclined substrate using the lubrication ap-proximation has not previously been reported. The objective ofthis paper is to present a way to introduce contact angle hys-teresis and to study its effect on the motion of droplets as theyspread/slide down inclined substrates. Section 2 provides detailsof the experimental setup and physical properties of the fluidsand substrates. Section 3 describes the mathematical model, thegoverning equations, and the numerical technique employed tosolve the governing equations. Lastly, Section 4 presents the resultsfrom the numerical simulations for fluid droplets spreading/slidingon inclined substrates under the influence of contact anglehysteresis.

2. Experimental methodology

2.1. Experimental setup

Experimental studies such as [17,18] are very accurate and de-scribe the dynamics of droplets sliding down the incline. Podgorskiand co-workers have calculated the velocity of the droplet as themean velocity of the fluid and LeGrand and fellow researchers havemeasured the velocity of the contact line. Both approaches are le-gitimate, but as we shall show later, using the center of gravity asa reference point for the droplet location appears to be more intu-itively correct. In this proposed model, the velocity of the slidingdroplet is determined by the position of its center of gravity as itslides.

Different experimental setups are described in the literatureand the choice largely depends on the aspects of the spreadingthat is of interest. In the present study, the interest lies in theterminal speed of the droplet on the inclined surface and the shapeit acquires as it spreads to form the footprint. The experimentalsetup proposed by Le Grand and co-workers makes the basis ofexperimentation for the present work, see Fig. 2.

A solid substrate (glass) is placed on a custom-built anglevariation device made of stainless steel. The dimensions of thebase of this device are 30 × 19.3 cm. The base can rotate abouta horizontal axis to vary the inclination angle from 0° up to 90°.Precise measurement of the angle of the inclination is made usingthe magnetic protractor/angle locator. The dimension of the glassplate used in the experiments is 30×20 cm. A 10×10 cmmirror isattached to the substrate using a hinge at the back and a 45° hollowwedge attached to the top corner of the mirror. The purpose ofplacing this mirror at 45° is to capture simultaneously the dropletfrom the side and from above. The experimental setup is side andback lit by a Kaiser Halogen video light 2000 W. The base andthe substrate are placed at right angles to each other so that thesubstrate (glass) is not covered by the base so that the substrate(glass) can be illuminated from beneath in order to have a strongcontrast for the top image. To avoid direct exposure to back-lighta diffusing screen made of sand-blasted glass is placed in betweenthe base and the substrate (glass). At least nine identical trials areperformed for each fluid droplet at a particular inclination angle toensure data reproducibility.

220 G. Ahmed et al. / European Journal of Mechanics B/Fluids 48 (2014) 218–230

Fig. 2. Experimental setup [18].

Fig. 3. Actual experimental setup.

2.2. Visualization

Droplets are released from a syringe, which is connected toa syringe pump. The rate at which the fluid is dispensed isfixed to 0.235 ml/min for all the experimental runs. At this ratethe approximate volume of the water droplet released from theneedle is 27.17 ± 0.2 µl and for the glycerine solution droplet is24.16±0.2µl. After dispensing a single droplet, the syringe pumpis stopped and the droplet spreading dynamics on the inclinedsubstrate is recorded. The recorded data which is of interest iswhen the droplet has achieved a stationary profile after impactingthe substrate and there are no perturbations due to the impact.The experiments are performed at room temperature andpressure.Fig. 3 shows the actual experimental setup.

Before the start of the experiment, the substrate (glass) isfirst cleaned with Virkon 1% solution and ethanol to kill micro-organisms on the substrate. After treating the surface with Virkonand ethanol, the substrate is cleaned with isopropanol. Thesubstrate is then rinsedwith distilled water so that the substrate isfree from the disinfectants mentioned above and then dried in hotair to remove any water residue.

Water droplets are transparent, which makes them hard toobserve without adequate lighting. Food color is added in waterto increase the contrast. The motion of the droplet is recorded bya Photron FASTCAM SA 1.1 camera at a frame rate of 1000 fps. Thecaptured images are processed using the Photron FASTCAMViewerVersion 3.0 and Image J software. Fig. 4 shows the top and side viewof the water droplet sliding down a 60° inclined glass substrate.

Fig. 4. Water droplet spreading/sliding on a 60° inclined glass substrate and post-processing in Image J.

Table 1Measured physical properties of water.

Physical property Value Published values range

Density at 22 °C, kg/m3 998.6 ± 2 1000Viscosity at 22 °C, Pa s 0.001Surface tension at 22 °C, N/m 0.072 ± 0.003 0.072Contact angle on clean glass, ° 11 ± 1 8.1 [22]

2.3. Fluid property measurement

Surface tension and contact angle of the fluids are measuredusing the contact angle and surface tension meter with FireWireCCD camera CAM 200 and provided software CAM 2008. Thesemeasurements are made using a 1.25 mm diameter hypodermicneedle and a plastic syringe. The needle is cleaned with 1% Virkonsolution to avoid any organic residue, rinsed thoroughly withethanol and distilled water. Amean value for the contact angle andsurface tension is calculated and reported in Tables 1 and 2.


Table 2Measured physical properties of glycerine 50% by weight solution.

Physical property Value Published values range

Density at 25 °C, kg/m3 1122.6 ± 1 1123.8Viscosity at 25 °C, Pa s 0.0051 ± 0.0002 0.0050–0.0053Surface tension at 25 °C, N/m 0.06663 ± 0.00111 0.066–0.068Contact angle on clean glass, ° 14.3 ± 2.00 13–17 [22]

Fig. 5. Sketch of the droplet geometry.

3. Problem Specification and mathematical formulation

In the present work, the spreading and sliding of a droplet areinvestigated. The problem is schematically illustrated in Fig. 5.A three-dimensional problem is considered, where the mass andmomentum balance equations are given as,

∇.U ′ = 0, (1)

ρ

∂U ′

∂t ′+ U ′.∇U ′

= −∇p + µ∇

2U ′ + ρg, (2)

where U ′= u′ i+v′ j+w′ k and g = gsinθi i+gcosθi k. The primes

denote dimensional variables. These dimensional variables are firstconverted into non-dimensional ones (without primes). Let H0 bethe characteristic droplet thickness at rest on ahorizontal substrateand L0 = R0 be the radius of the droplet at equilibrium. The lubrica-tion approximation is based on the assumption that: ϵ =

H0L0

≪ 1.The scalings used here are usual and have been used extensively inthe past for Newtonian rheology, i.e. U0 =

L0T0

, P0 =σϵL0

[3,23]. The

time scale first proposed by Orchard is used here, i.e. T0 =

L0µσϵ3

[24]. The velocity and the space coordinates are made dimension-less using u′

= U0u, v′= U0v, w′

= ϵU0w, x′= L0x, y′

=

L0y, z ′= H0z where U0 is the velocity scale in the x direction.

The equilibrium configuration of the droplet is defined, where V0is the volume of the droplet lying on the substrate, R0 is the char-acteristic radius of the droplet, H0 is the maximum height, and θethe equilibrium contact angle. The equilibrium droplet shape is as-sumed to be a paraboloid of revolution, for which simple formulacan be found for V0 =

π2 H0R2

0, ϵ =H0R0

, and the contact angle,

θe = arctan

2H0R0

≈

2H0R0

. The reference scales, R0 =

4V0πθe

1/3and H0 =

θeR02 are calculated from expressions for V0 and θe, be-

cause these are easily observable quantities. Neglecting terms of

order ϵ2=

H20

L20or higher, Eq. (2) can be written in dimensionless

form as,

∂

∂z

∂u∂z

=

∂p∂x

−Bo sin θi

ϵ, (3)

∂

∂z

∂v

∂z

=

∂p∂y

, (4)

∂p∂z

= −Bo cos θi, (5)

with boundary conditions, u = v = w = 0 on z = 0, and∂u∂z =

∂v∂z = 0 and p = −∇

2h − Π(h) on z = h. Bo =ρgL20σ

isthe Bond number representative of the ratio between the gravi-tational and surface tension forces. A thin precursor film ahead ofthe contact line is assumed to be present in order to alleviate thesingularity at the dynamic contact line. In order to prescribe a re-alistic equilibrium shape, a disjoining pressure term of the form

Π(h) =(1−cos θe)(n−1)(m−1)

h∗ε2(n−m)

h∗

hij

n−

h∗

hij

mis used [3], where n

and m are constants and h∗ is the precursor film thickness equalto 0.01 (i.e. 1% of the characteristic droplet thickness). Eqs. (3) and(4) are integrated twice with respect to z over the film or dropletthickness (0 ≤ z ≤ h) to give,

u =

Bo sin θi

ϵ−

∂p∂x

12

h2

− (h − z)2, (6)

v =

−

∂p∂y

12

h2

− (h − z)2. (7)

Using this velocity profile and the continuity equation in the in-tegral form lead to the time dependent lubrication approximationfor Newtonian fluid given by,

∂h∂t

= −13

∂

∂x

h3Bo sin θi

ϵ−

∂p∂x

+

∂

∂y

h3

−∂p∂y

, (8)

and the pressure field in the droplet,

p = −∇2h − Π(h) + Bo cos θi(h − z). (9)

Eqs. (8) and (9) form the second order coupled non-linear differ-ential equations for h and p. Previous investigations reviewed inOron et al. [25] have substituted Eq. (9) into (8) to form a fourth or-der time-dependent differential equation purely in h. Later studies[23,26–29] have shown that the two coupled second order non-linear form lubrication equations are easier to solve.

The spatial discretization of the lubrication approximation forEqs. (8) and (9) is performed by central finite differencing withuniform mesh spacing in the x and y directions, leading to a setof ODEs for h and p at mesh points (i, j). Time discretizationis performed using the Crank–Nicolson method. Temporal errorcontrol as stated in [30]was used to achieve the automatic adaptivetime stepping [27]. The multigrid solver approach is a fast wayof solving the discretized equations using a hierarchy of grids.Its main advantage is its fast convergence rate by efficientlysmoothing and correcting the errors on multiple grid levels(mesh sizes) and exact solution is only required on the coarsestgrid level. The discretized equations are solved using the fullapproximation storage (FAS) and full multigrid (FMG) technique asmentioned in [27]. The Red–Black Gauss–Seidel Newton iterationscheme is used with linearized Newton iterative step to performrelaxation [27].


Fig. 6. Contact angle hysteresis model.

3.1. Implementation of contact angle hysteresis model

Contact angle hysteresis is introduced into the numerical code.The first step towards approximating the pointwhich separates theadvancing contact line from the receding contact line in a dropletsliding scenario is to assume that the droplet is symmetric aboutthe x − z plane. Let l(x) be a length defined as a function of x, fromthe centerline of the spreading droplet to the contact line. Then thepoint xhys, where l(x) is maximum, delineates the advancing andreceding regions. x > xhys is the advancing part and x < xhys is thereceding part in the droplet profile. At every time step the locationof point of hysteresis is determined and the hysteresis is appliedby introducing static values of θa and θr to the disjoining pressureterm. The model is schematically described in Fig. 6.

3.2. Analytical model

A number of analytical models are present in the literatureto predict the velocity of the sliding/moving droplets [16,31,32].In this section, a simple model is used to describe the constantterminal velocity achieved by the droplet as it slides down theinclined substrate. This motion is the result of a force balancebetween the gravitational force, capillary force present in thevicinity of the contact line, Fcl, and the viscous resistive force, Fr .According to Newton’s second law,

mdU ′

dt= mg sin θi − Fcl − Fr , (10)

where U ′ is the velocity with which the droplet slides down theincline. The left-hand side is equal to zerowhen the droplet reachesconstant terminal velocity, i.e.

mg sin θi − Fcl − Fr = 0. (11)

Fig. 7 shows the sketch of the various forces on the droplet as itslides down the incline. If the droplet has a circular shape duringthe sliding motion [16,31], then

Fcl = 2σR0

π

0cos θc cosαdα, (12)

where σ denotes the surface tension of the fluid, α denotes theangular position, θc is the contact angle the droplet makes aroundthe contact line. θc = θa at the front of the droplet and θc = θr atthe rear of the droplet. Fig. 8 shows the schematic of the contactangle, θc , as a function of the angular position.

If the footprint of the droplet is assumed to be circular, then Fclis given by,

Fcl = 2σR0 (cos θr − cos θa) . (13)

Fig. 7. Schematic of the forces on the droplet as it slides down the inclinedsubstrate.

Fig. 8. Illustration of the contact angle θc as a function of the angular position α.

Wall shear stress is found to be [32],

τw =3µU ′

h. (14)

The resistive force is found by integrating thewall shear stress overthe entire footprint of the droplet,

Fr =

R0−ϵc

0τw (2πr) dr, (15)

where ϵc is the cut-off length to prevent the singularity at thecontact line, h is the thickness of the droplet and is dependenton the radius, r . The profile of the droplet is approximated by aparaboloid,

h (r) = H0

1 −

r2

R20

, (16)

where H0 is the characteristic central height and R0 is thecharacteristic base radius. Eq. (11) can be written as,

mg sin θi − 2σR0 (cos θr − cos θa)

−

R0−ϵc

0

3µU ′

h(2πr) dr = 0. (17)

Eq. (17) can be solved for the constant terminal sliding velocity, U ′.Eq. (10) can be written as,

mg sin θi − 2σR0 (cos θr − cos θa)

−

R0−ϵc

0

3µU ′

h(2πr) dr = m

dU ′

dt. (18)

Eq. (18) can be solved to get the rate of change of velocity of thesliding droplet in case of variation of substrate inclination withtime. Further extending this, it is possible to find the thresholdinclination angle, θthreshold after which the droplet starts to move.If the droplet is immobile, Eq. (18) can be simplified to,

mg sin θthreshold = 2σR0 (cos θr − cos θa) . (19)

This model applies best when the footprint of the sliding dropletis close to a circle. If the equilibrium contact angle is lower, as


Fig. 9. Satellite droplets [17].

Table 3Properties of the silicon oil droplet spreading on inclined glass substrate.

Physical property Silicon oil [17]

Density, kg/m3 924Viscosity, µ, Pa s 0.00915Surface tension, N/m 0.0205Equilibrium contact angle, θe° 45Advancing contact angle, θa° 50Receding contact angle, θr ° 40Substrate inclination, θi° 10–75Volume of droplet, µl 8.3

in the case of water and glycerine solution droplets sliding onthe glass substrate, the footprints formed by the droplets are notcircular. These droplets form a large tail and a footprint whichis complex and difficult to define mathematically. Podgorski andco-workers used silicon oil droplets on fluoro-polymer coatedsubstrate having a θe of 45°. In the next section, the analyticalmodel is used alongside numerical results and compared withexperimental results.

4. Results and discussion

The numerical results are presented in terms of the positionof the center of gravity of the droplet as a function of time. Theposition of the center of gravity is computed according to xcg =

Ω (h−h∗)xdωΩ (h−h∗)dω and ycg =

Ω (h−h∗)ydωΩ (h−h∗)dω , where Ω = [0, 15] × [0, 4] is

the computational domain. The purpose of measuring the positionof center of gravity of the droplet throughout the simulation is tohave an interpretation about the rate (speed) at which the dropletslides down the inclined plane.

4.1. Comparison with the experimental literature

Podgorski and co-workers reported experiments about thechange in shape of the spreading silicon oil droplet along aninclined fluoro-polymer (FC725 from 3M) coated substrate [17].At higher inclination angle, the speed attained by the spreadingdroplet increases. At a critical speed, the droplet shape is no longerrounded and develops a corner at the trailing edge. The shape ofthe corner becomes sharper up to 60° of inclination, where it startsto shed smaller droplets behind, see Fig. 9 [17]. In the study ofPodgorski et al., the capillary number, Ca =

µU ′

σ, is plotted against

Bo sin θi =ρgV2/3

σsin θi. In the present study, the velocity of the

droplet, U ′ is computed from the gradient of the plot between theposition of the center of gravity of the droplet and time.

The physical properties of silicon oil on glass substrate arereported in Table 3, [17].

Mesh convergence study for precursor film thickness, h∗ equalto 0.01 is performed, see Fig. 10. Initially, the mesh is 960 ×

128 which is then doubled to 1920 × 256 to investigate theeffect of mesh refinement. Fig. 10 shows that increasing the meshdensity increases the accuracy at the expense of the computationalresources. Above 1920 × 256 mesh resolution the results are very

Fig. 10. Mesh convergence without including contact angle hysteresis model: acomparison with experimental results from Podgorski et al. [17].

Fig. 11. Comparison of experimental results [17] andnumerical oneswith a contactangle hysteresis of 10°.

weakly affected. A noteworthy difference between the results ofPodgorski et al. and the numerical results is the change of slopeobserved experimentally beyond Bo sin θi = 1.3 but not apparentin the simulation. It is thought to be an artifact of the way theposition of the droplet is recorded: the present study uses thecenter of gravity as the reference point while the reference pointused by Podgorski et al. is not as clearly defined. Note that beyondBo sin θi = 1.3, a cusp starts to develop at the rear of the dropletswhich is the likely cause for this change of slope.

Fig. 12(a) demonstrates the shape of the droplet at t = 10for different substrate inclination angles. As the inclination of thesubstrate is increased, the speed with which it slides increases.For lower inclination angles, the distance traveled by the dropletis small and the droplet approximately maintains its spherical capshape. As the substrate inclination increases, the droplet forms ateardrop shape. At higher inclinations, the tail of the droplet thinsfurther.

In the results above, contact angle hysteresis is not included.The motion of the sliding droplet is affected by the inclusion ofhysteresis. In the next instance, contact angle hysteresis, θhys, of10° is imposed, i.e. θa = θe +

θhys/2

and θr = θe −

θhys/2

,

where θe = 45°.Fig. 11 demonstrates that the application of contact angle

hysteresis reduces the speed of the sliding droplet and it comes


(a) θhys = 0°. (b) θhys = 10°.

Fig. 12. Thickness contours obtained from numerical simulations of the silicon oil droplets sliding down the incline without hysteresis for t = 10 for different substrateinclinations.

Fig. 13. Comparison between experimental, numerical and analytical results.

closer to experimental observations. It is generally observed thatthe advancing contact angle increases and receding contact anglereduces with increase in speed of the sliding droplet [33]. Theassumption of a uniform contact angle hysteresis of 10° for theentire range of substrate inclinations is only an approximation. Thisis well demonstrated from the plot in Fig. 11. At lower inclinationangles the droplet has a low velocity and a hysteresis of 10°whichcauses the droplet to slow down more than the experimentalobservations.

Fig. 12(b) shows the droplet thickness contours of the dropletssliding with θhys = 10° at t = 10 for different substrate inclinationangles. Fig. 12(a) and (b) clearly show that the hysteresis reducesthe speed with which the droplet slides down. This additionaldissipative force explains why the numerical and experimentalresults are in better agreement, see Fig. 11. For large inclinationangles, the droplet forms a teardrop shape, but the tail formed isnot as narrow as in the without hysteresis case.

Using the physical properties mentioned in Table 3, Eq. (17) isused to calculate the velocity using the analyticalmodel. Analyticalresults are dependent on the value of the cut-off length, ϵc .

Fig. 13 shows a comparison of the experimental, analytical andnumerical results. It is clearly depicted that ϵc has an impact on

the analytical results. For ϵc equal to 1 × 10−5 m, the analyticalsolution has an excellent agreement with the experimental resultswhen the droplet has a near constant circular footprint. At higherinclination angles, the footprint of the droplet develops a cusp.As soon as this cusp is developed, the analytical model is aweaker approximation. When ϵc is increased to 1 × 10−4 m, theanalytical solution has come closer to the numerical results wherethe inclination angles are lower and the droplet footprint is closeto a circle. Analytical and numerical results deviate due to cuspformation at higher inclination angles. It is clear from these results,that the cut-off length ϵc plays a critical role in the prediction ofthe terminal velocity for the system considered, it appears that theoptimal value of ϵc for this particular system is 1 × 10−5 m.

As a further test of the correct modeling of contact anglehysteresis, it can be checked that the motion of the droplet stopswhen a varying substrate inclination reaches a threshold value.Increasing inclination angles beyond this threshold inclinationangle should start the sliding dropletmotion at a particular contactangle hysteresis. Therefore as a final illustrative example, silicon oildroplets for the physical properties stated in Table 3 are allowed toflow down an inclined glass substrate with a variable inclination.Starting from an initial inclination of 10° the inclination is rampedup in increments of 10° every 2 dimensionless time units up to aninclination angle of 70° in Case 1. Case 2 is the reverse cycle. Thiscan be described with the help of Fig. 14.

The silicon oil droplet is released from a stationary equilibriumposition. There are three different situations that are reportedhere. Firstly, the droplet slides down the inclinewithout hysteresis(θhys = 0). In the second and third cases θhys is introduced using themodel defined in Section 3.1 to investigate its effects on themotionand shape of the droplet. A hysteresis of 10° and 20° is introducedfor Cases 1 and2, such that θa = θe+

θhys/2

and θr = θe−

θhys/2

.

The optimum cut-off length, ϵc is 1.4×10−4 m for these particularcases.

Fig. 15 shows the three hysteresis situations for Case 1. Itdisplays the contours of the droplet thickness. With no hysteresis,see Fig. 15(a), the droplet starts to move on the substrate at verysmall inclination angles around 10°. As the inclination is increased,the speed gained by the droplet increases due to an increase ingravitational pull. It is visible from the contours that the shape ofthe droplet changes to an oval shape and then to a more conicalshape. Fig. 15(b) describes the effects of a 10° hysteresis on themotion and shape of the droplet. In the start, the droplet moves


Fig. 14. Protractor to show the variation of inclination angle, θi .

very slightly but the equilibrium shape of the droplet changes.Contact angle hysteresis restricts the motion of the droplet. Aninteresting point to note is that the conical shape of the contoursstarts to appear when the inclination angle is large. Fig. 15(c)demonstrates the effect of θhys = 20° on the shape of the slidingdroplet. An interesting point to note here is that the conical shapeis not developed and the sliding velocity is reduced further.

Fig. 16 shows the position of the center of gravity of thedroplet against time for the three situations. Results from theanalyticalmodel are also plotted. Both results show the same trend.There is an offset developing between both sets of results as thesubstrate inclination increases. This is primarily due to the changein footprint shape froma circle to amore oval shape, refer to Fig. 15.

Fig. 17 shows the three situations in Case 2. The Case 2 scenariois important in the sense that in Case 1, the dropletwas acceleratedgradually with time. In Case 2, the droplet is first accelerated dueto high inclination angles then its speed is reduced due to the re-duction in the inclination angle. Fig. 17(a) illustrates the motion ofthe droplet as it slides down the incline without hysteresis. In thestart, the droplet slides down quickly forming a conical shape, butas the inclination is reduced the sliding speed is reduced. This hap-pens due to reduction in gravitational forces with reduction in the

Fig. 16. Position of center of gravity for contact angle hysteresis, θhys = 0°, 10°, 20°for Case 1 scenario for 14 non-dimensional time units.

tilt angle. An important point to note here is that the droplet shapechanges fromconical to amore oval shape. Reduction in speed doesnot cause the droplet to stop entirely for the case without hystere-sis, it slides very slightly. Fig. 17(b) and (c) represents the dropletshapes as it slides down with θhys of 10° and 20°. It clearly showsthat as the hysteresis is increased the sliding velocity is reducedand the corresponding shape of the droplet ismore oval. An impor-tant thing to note here is that the droplet stops tomove at very lowinclinations and the part of the droplet that expanded previouslydue to high inclinations starts to collect and form an oval shape awell-known and observed consequence of contact angle hystere-sis. This is in line with the theory that the hysteresis tends to stopthe moving droplet until there is a change in physical condition.

Fig. 18 represents the position of the center of gravity of thedroplet with time for the three situations for Case 2 scenario.Initially the droplet sliding velocity increases for all the threesituations, but contact angle hysteresis plays a significant role indecelerating the droplet. Results from the analytical model areclose to numerical results, but they are different in the start. In the

(a) θhys = 0°. (b) θhys = 10°. (c) θhys = 20°.

Fig. 15. Thickness contours obtained from numerical simulations showing the effect of contact angle hysteresis on the shape and displacement of silicon oil droplets forCase 1 scenario and ∆ti = 2.





start, the inclination angles are high, causing the droplet footprintto change instantaneously to an oval shape, whereas the analyticalmodel assumes the droplet footprint to be circular, see Fig. 17.

Up to this point, we have presented the effects of hysteresishas on two different scenarios in which the inclination angleis changing after 2 dimensionless time units for three differenthysteresis situations. ∆ti is doubled next to investigate theinfluence on the shape and speed of the sliding droplet.

Fig. 19 illustrates the shape of the droplet as it slides down theincline for Case 1 scenario. Fig. 19(a), shows that the droplet slidesfaster as the inclination angle is increased. At higher inclinationangles, the droplet takes the form of a teardrop with a very narrowtail which elongates with time. Fig. 19(b) and (c) describe theeffects of increasing the contact angle hysteresis on the shape ofthe sliding droplet. As the hysteresis is increased, one of the directeffects is that the sliding velocity decreases; the other effect isthat the teardrop shape formed at higher inclinations becomes lessnarrow.

Fig. 20 represents the position of the center of gravity of thedroplet. It shows that the droplet moves slowly at the start,but as the gravitational force increases due to the increase in

inclination, it moves down more rapidly. The increase in contactangle hysteresis reduces the pace of the sliding droplet. Resultsfrom the analytical model are close to the numerical results. Smalldiscrepancies in these results arise when the shape of the footprintis far from a circular shape i.e. oval or a tear drop shape, refer toFig. 19.

Fig. 21 illustrates the shape of the droplet as it slides down theincline for Case 2. Fig. 21(a) presents the situation in which thedroplet slides down the incline without hysteresis. At the start thedroplet gains velocity due to high inclination angles which causesit to expand initially to form a teardrop shape. As the inclinationangle is reduced, the tail of the teardrop shape becomes narrowto the extent that the sliding droplet breaks up leaving behindsatellite droplets. An important thing to observe here is that themajor drop volume slides quickly in comparison to the satellitedroplets. Fig. 21(b) shows the results for θhys = 10°. The speedwith which the droplet slides down is decreased in comparison tothe one achieved without hysteresis. Droplet breakup is delayeddue to reduction in the overall speed of the droplet, but the dropleteventually breaks up. In the last 4 s of the simulation, the dropletis almost stationary on a 10° inclination. Fig. 21(c) demonstratesthe effects of θhys = 20°. As it slides down the incline, the dropletfirst changes its shape to a teardrop then develops a narrow tail.This tail does not gets separated due to low sliding speeds at lowerinclination angles. In the last few seconds, the droplet is seen toretract, which results in the center of gravity shifting up slope ascan be seen in Fig. 22.

Fig. 22 represents the position of the center of gravity ofthe droplet with which it slides down for Case 2. It shows thatthe droplet moves rapidly in the start, but as the gravitationalforces decrease due to decrease in inclinations, it moves downslowly. Increase in hysteresis progressively reduces the pace of thesliding droplet. The apparent up slope motion of the droplet forθhys of 20° is due to stationary contraction of the droplet at lowinclination.

Making use of Eq. (19), the threshold inclination angle, θthresholdcan be calculated to further analyze the results from the numericalsimulations. Evaluated θthreshold for different contact angle hystere-sis is given in Table 4.

It is clear from Table 4 that for greater θhys the thresholdinclination angle becomes bigger. This is also supported from theplots for center of gravity with time, see Figs. 16, 18, 20 and 22.





Table 4Threshold inclination angle calculations using Eq. (19).

Contact angle hysteresis, θhys , ° Threshold inclination angle, θthreshold , °

0 0.0010 9.2120 18.56

It is clearly shown that for a hysteresis of 10°, the position of thecenter of gravity of the droplet does not change with time for aninclination of 10°. As the hysteresis is increased to 20°, the dropletis stationary for an inclination of 20° and if the inclination is lessthan 20° the droplet starts to retract.

4.2. Comparison with experiments

With the help of the Image J software, the side view is usedto calculate the position of the centroid of the spreading dropletalong the inclined axis. The error calculated with this method inthe prediction of the location of centroid is below 0.1 mm. This

error is calculated by free hand drawing the shape of the slidingdroplet while post-processing the experimental results in ImageJ software. Slight variations in the physical properties of the fluidand substrate surface properties have a critical effect on the spreadrate. The speed of the droplet is estimated by drawing a trendlinethrough different plots for centroid position variation with time.The slope of these trendlines gives an estimate of the terminalspeed with which the droplet is sliding on the inclined substrate.The variation in the results of the experiments, apparent in theerror bars, can be attributed to the uncertainty in the profile of thedroplet as it slides down the incline.

Fig. 23 shows the side-view of the glycerine solution dropletssliding down a 30° inclined glass substrate. The experimentaldroplet profiles are compared with the numerical results. Withand without hysteresis, it is clear that as time passes the dropletelongates because of smaller equilibrium contact angle. As seenfrom Fig. 23, hysteresis results are close to the experimentalresults, but are not precisely matched.

Fig. 24 represents Ca versus Bo sin θi for water, where θi is theangle of inclination of the substrate with the horizontal plane. Theequilibrium contact angle of the water varies between 10.5° and12° for the glass substrate, see Table 1. For different equilibriumcontact angles and zero hysteresis, the numerical results areplotted. The results from the numerical code tend to over-estimatethe sliding speed by nearly 50% relative to experimental results.The numerical results can be improved by introducing contactangle hysteresis. The advancing contact line is well defined whichis not the case for the receding contact line, because the dropletleaves a thin film as it slides down the inclined substrate causingerrors in the accurate estimation of the location of the centroidwith time.

In the numerical results without hysteresis, a constant equilib-rium contact angle is applied but actually the water droplet slidesdown the glass substrate with an advancing and receding contactangle which is different from equilibrium contact angle. The equi-librium contact angle for water is quite low so a contact anglehysteresis, θa−θr , of 2° and 4° is applied in the simulation for equi-librium contact angles of 10.5° and 11°, i.e. θa = θe + (1°, 2°) andθr = θe − (1°, 2°).

Fig. 24 demonstrates that with the increase in hysteresis theresults produced come into much closer agreement with the





experimental ones. Fig. 24 indicates that if a lower equilibriumcontact angle for water is used for the hysteresis analysis, it willproduce results with reasonable agreement to the experimentalinvestigation.

Fig. 25 shows the effect of hysteresis on the contours of heightforwater droplet at different inclinations and after 4 dimensionlesstime units. It can clearly be deduced from the figure that the shapeof the droplets under hysteresis is quite different. For a lowerhysteresis, the shape of the tail is thinner in comparison to thedroplets with larger θhys. This wider tail in case of large hysteresiscauses the motion of the center of gravity of the droplet to slowdown.

Fig. 26 represents Ca versus Bo sin θi for glycerine 50% byweightsolution. The equilibrium contact angle of this solution is 14.3° ±

2° for the glass substrate, see Table 2. Computationally, threeequilibrium contact angles are used. The reason for this choiceis due to the uncertainty on the equilibrium contact angle whenestimated with the help of the procedure stated earlier in theexperimentation section. The trend of experimental spreading of

Fig. 23. Comparison of the side-view of glycerine solution droplet profiles with thenumerics at θi = 30° sliding with and without hysteresis on glass substrate.

glycerine solution droplet is similar to the one from the numericalcode. The numerical results may be improved by introducingcontact angle hysteresis.

Fig. 26 also considers a hysteresis of up to 4° in the advancingand receding contact angles for the 12.3° and 14.3° equilibriumcontact angle. It is clear from this figure that the introduction ofhysteresis causes the results to converge towards the experimenta-tion. Analyzing the lower bound of the estimated equilibrium con-tact angle, i.e. 12.3°, with a hysteresis, θa − θr , of 4° demonstratesa further reduction in the speed with which the droplet spreads onan inclined substrate, see Fig. 26. Thus, increasing the hysteresis inthe simulations improves the agreement with experimentation.

5. Conclusion

In this work, a mathematical model based on the lubricationapproximation which describes the spreading/sliding motion ofgravity driven fluid droplets under the influence of contact angle


Fig. 24. Effect of contact angle hysteresis on spreading of water droplets for equilibrium contact angles of 10.5° and 11°.

Fig. 25. Thickness contours obtained from numerical simulations showing theeffect of hysteresis on the shapes of water droplets after 4 dimensionless time unitssliding on inclined substrates having, θe = 11°.

hysteresis is presented. Thismathematicalmodel consists of highlynon-linear PDEs which are solved numerically via the multigridtechnique. A model to incorporate the contact angle hysteresisis proposed and implemented to investigate the effect of contactangle hysteresis on the motion of the droplets as they slide downthe incline. A simple analytical model is also suggested to predictthe dynamics of the sliding motion of the droplet. The results fromthe analytical and numerical implementation are compared withthe experimental results available in the literature and a new setof data obtained in this study. The shape of the droplets observedin the numerical simulations tends to agree with the shapes of thedroplet described in the literature. Based on this, special cases arediscussed in which the droplets tend to break up leaving satellitedroplets behind. Droplets stick to the substrate in a few caseswhere the forces due to hysteresis overcome the forces present dueto gravitational pull. An important effect of equilibrium contactangle is observed, i.e. the droplets having small θe are likely to formlong tails as they slide down the incline substrates and the dropletswith large θe are more likely to have shapes closer to teardrops. Ithas been shown here that our numerical simulations are able tocapture the main effect of contact angle hysteresis.

Fig. 26. Effect of contact angle hysteresis on spreading of glycerine solution droplets for equilibrium contact angles of 12.3° and 14.3°.


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modeling the effects of contact angle hysteresis on the sliding of droplets down inclined surfaces

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