Contact-line instabilities of driven liquid films · 2014-05-10 · solvent film, which is used, for instance, in the manufacture of circuit boards [32]. Rivulet formation limits

Contact-line instabilities of driven liquid films

J.S. Marshall & R. EttemaIIHR – Hydroscience and Engineering, The University of Iowa, IowaCity, Iowa 52242USA

Abstract

Liquid-film contact lines driven over a surface (substrate) by gravity, centrifugalforce, Marangoni force, or wind, exhibit a wealth of instabilities that controlmany coating and cleaning processes. Starting from a sheet with anapproximately straight contact line advancing over the substrate, theseinstabilities lead to fingering of the sheet to form rivulets, meandering andoscillation of the rivulets, and break-up of the rivulets to form droplets ormultiple rivulets. Droplet flows on a flat substrate can themselves exhibitdeformation, elongation and eventual bifurcation of the droplets to form smallerdroplets. The chapter presents a review whose objective is to give a broadexplanation of the underlying physics of these various phenomena and to presentan overview of targeted literature on these subjects. The chapter also examinessome new results regarding the effect of surface inhomogeneity on rivuletformation, the physics of wind-blown rivulets (including their break-up asdroplet flows), and rivulets subject to microgravity.

1 Introduction

Contact-line instabilities, leading to formation of rivulet and droplet flows,are an important feature of fluid flow in a wide variety of applications centralto the aerospace and chemical industries. For instance, during flight ofaircraft in heavy rain, formation of rivulets in the water sheet covering thewings increases the effective surface roughness of air flow over the wings,leading to significant degradation of aerodynamic efficiency [53,54].Rivulets control ice formation within the mid-section of the wings underfreezing-rain conditions, where the ice patterns commonly form a series of

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2 Instability of Flows

"ice rivulets" extending in the chordwise direction. The ability to accuratelymodel, and if possible suppress, these ice rivulets is important for efficientoperation of anti-icing measures, such as heating of the solid surface [1,2].Rivulets are commonly observed in coating flows in the chemical industry,such as the slide-coating process used in the photographic industry forspreading of silver halide on film [57]. In centrifugal spin-coating processes,liquid drops impact a solid substrate attached to a spinning disk. Thecentrifugal force imposed by the spinning disk spreads the drop into a thinliquid sheet that flows radially across the substrate. Rivulet formation limitsthe minimum thickness of this sheet and results in inefficient and non-uniform coating of the substrate [21]. Coating processes of this type are alsoimportant for cleaning of organic films from a substrate with a thin liquidsolvent film, which is used, for instance, in the manufacture of circuit boards[32]. Rivulet formation limits cleaning efficiency by decreasing the liquid-solid contact area. The presence of rivulets also has a significant effect onheat and mass transfer rates across the substrate surface, which has beennoted in studies concerning applications as diverse as solar-collector design[52] and chemical reaction rates in packed-column reactors [7,22,46].Furthermore, the formation of rivulets is of great interest in pesticideapplication on leaf surfaces, for which rivulet formation is to be avoided anddroplet retention encouraged to enhance the health of the plant [23].

The current chapter examines different phases in the evolution of driven contact-line problems, and in particular aspects concerned with the formation, stability andbreak-up of rivulet flows. The second section reviews the lubrication model for thin-film flows, as well as extensions to lubrication theory to account for effects such assurface inhomogeneity. This model forms the basis for many theoretical andcomputational analyses of contact-line instability problems. The implementation of thelubrication theory in the present chapter employs a precursor-film approach to resolvethe moving contact-line singularity [16]. Several authors have compared the precursor-film method with the local-slip method for removal of the contact-line singularity andfound that both methods yield similar results [15,51], although the precursor-filmmethod offers several computational advantages. The third section reviews numericalmethods that are used for solution of the thin-film equations. The fourth sectionexamines theoretical and computational results on the fingering instability and its rolein formation of rivulets. After reviewing the classic fingering instability analysis for ahomogeneous surface, we examine nonlinear effects caused by pre-existing dropletsand surface inhomogeneity. The fifth section examines the structure of stable rivuletsand instabilities that cause them to first meander in a steady path and then to oscillate intime over the substrate surface and periodically shed sub-rivulets. The sixth sectionexamines break-up of rivulets and droplet dynamics for flow driven by wind shear.Conclusions are presented in the seventh section.

2 Lubrication theory for thin films with contact lines

We consider a thin fluid film having thickness ),,( tyxh on a flat substrate


Instability of Flows 3

coinciding with the x-y plane. The volumetric flow rate ),,( tyxQ is defined by

dzh

vQ ∫=0

, (1)

where yx vu eev += is the projection of the velocity vector in the x-y plane.Integration of the continuity equation 0=⋅∇ u in z over the interval ),0( h ,where zyx wvu eeeu ++= is the velocity vector, and use of the kinematicboundary condition hthw ∇⋅+∂∂= v/ at hz = give

Q⋅−∇=∂∂

th . (2)

The pressure is decomposed as

)(),,,(ˆ),,,( zhgtzyxptzyxp N −+= ρ , (3)

where Ng is the component of body force per unit mass in the z− direction,normal to the substrate surface. Substituting eqn (3) into the x and y componentsof the Navier−Stokes equation gives

⎟⎟⎠

⎞⎜⎜⎝

⎛

∂

∂+∇++∇−−∇=

2

22ˆ

zghgp

DtD

xTNvvev µρρρ , (4)

where Tg is the body force per unit mass in the x-direction (tangential to thesubstrate) and ∇ is the projection of the del operator in the x-y plane. Thelubrication approximation assumes that the inertial forces are small compared toviscous forces and that the interface slope is small compared to unity. Under thisapproximation, the inertia term, )/( DtDvρ , and the horizontal derivatives inthe viscous diffusion term, v2∇µ , can be neglected, such that eqn (4) can berearranged as

xHV ghgpz

ev ρρµ −∇+∇=∂

∂ ˆ2

2. (5)

Since under the lubrication approximation the z-component of the momentumequation indicates that p̂ is approximately independent of z, integration of eqn(5) twice in the z-direction gives



DCev ++−∇+∇= zghgpzxTN ]ˆ[

2

2ρρ

µ, (6)

where C and D are coefficients of integration. The boundary conditions are no-slip on 0=z and a specified constant shear stress xeτ on hz = , or

0),0,,( =tyxv , xthyxz

ev τµ =∂∂ ),,,( , (7)

which yields ]ˆ[ hggph NxTx ∇+−∇−= ρρτ eeC and 0=D . Substituting eqn(6) into eqn (1) gives the volumetric flow rate as

xNxTh

hggph eeQµ

τρρ

µ 2]ˆ[

3

23+∇+−∇−= . (8)

The liquid upper-surface pressure can be written as a sum of a capillary pressureand a disjoining pressure −Π as

Π−∇−= hp 2ˆ σ , (9)

where σ is the surface tension. Physically, the disjoining pressure approximatesthe effect of van der Waals forces and other near-surface effects [24]. Thedisjoining-pressure model used in this paper is similar to that used by Schwartz[48] to examine the effect of surface contamination on hysteresis during thespreading of a droplet and by Schwartz and Eley [49] to examine bifurcation ofa droplet placed on a substrate with strong static contact-angle variation. Thelatter paper shows reasonable comparison between simulations using thedisjoining-pressure model and experimental results.An expression for disjoining pressure is given in terms of the ratio of liquid-layer thickness to the constant thickness *h of the precursor-film by

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟

⎟⎠

⎞⎜⎜⎝

⎛=Π

mn

hh

hhB

**, (10)

where B, m, and n are constants such that 1>> mn . The disjoining pressure,Π− , is negative for 1/ * <hh and positive for 1/ * >hh , so that layer

thickness is always forced toward the specified precursor film thickness. Thiseffect becomes small for 1/ * >>hh . The constant B may be expressed in termsof the equilibrium contact angle Eθ as [48]



)cos1()(

)1)(1(* E

mnhmn

B θσ −−

−−= . (11)

A typical value for (n,m) is (3,2), although as shown by Schwartz and Eley[49], the dynamics of the flow is not particularly sensitive to the valueschosen for these coefficients.Substitution of eqn (8) into eqn (2) gives a differential equation for theliquid sheet thickness ),,( tyxh as

[ ] ⎟⎟⎠

⎞⎜⎜⎝

⎛⋅∇−

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

+−Π+∇∇⋅−∇=∂∂

xxTNhghghh

th ee

µτρρσ

µ 2)(

3

22

3(12)

It is assumed that the layer thickness approaches a constant value ∞h farupstream, which provides a characteristic length scale in the vertical (z)direction. The characteristic length scale in the horizontal (x-y) plane is L andthe characteristic time scale is ULT /= , where U is the forward advectionvelocity of the liquid front in an equilibrium state. We denote the dimensionlessprecursor-film thickness by ∞= hh /*δ and the aspect ratio by Lh /∞=ε .Dimensionless variables are defined as

∞=′ hhh / , Lxx /=′ , Lyy /=′ ,Ttt /=′ , 1/ −

∞Π=Π′ mh σδ . (13)

Substituting into eqn (12) and dropping the primes on the dimensionlessvariables, the dimensionless equation for liquid-layer thickness becomes

( ) x

xm

hP

hPhhPhhhPth

e

e

⋅∇−⎭⎬⎫

⎩⎨⎧ +∇−Π∇+∇∇⋅−∇=

∂∂ −

24

33

32

32

123

1

])([εδ

(14)

The parameters 41 ,..., PP are defined in terms of the tangential and normalBond numbers, σρ /2

TT ghBo ∞= and σρ /2NN ghBo ∞= , and the shear

parameter στ 2/3 ∞= hS . Setting the time scale T to balance the left-handside of eqn (14) with the horizontal gravity or stress terms that drive theliquid sheet forward, these dimensionless parameters become

SBoP

T +=

31

ε ,SBo

BoPT

N

+=

ε2 ,

SBoBoPT

T

+=3 ,

SBoSP

T +=4 . (15)



For a gravity-driven flow 13 =P and 04 =P , whereas for a shear-drivenflow 03 =P and 14 =P . The length scale L in the horizontal direction is setsuch that 11 =P , so the only remaining dimensionless parameters in eqn (14)are the normal body-force parameter 2P and the parameter 21 / εδ −m

multiplying the disjoining-pressure term.An equilibrium solution )(0 xh for liquid-layer thickness of a sheet that

is driven forward without any variation in the lateral (y) direction isobtained by adding a term xUh to the right-hand side of eqn (14), whichrepresents a translation of the coordinate system with the advectionvelocity of the liquid front. Setting 0/ =∂∂ th in this advected coordinatesystem and integrating once in the x-direction, the equation for theequilibrium solution becomes

ChPhPhhPhhhUh xxm

xxx =++−Π++−−

204

303,0

302,

302

1,0

300 ε

δ , (16)

subject to the boundary conditions 1→h and 0→xh as −∞→x andδ→h and 0→xh as ∞→x . Here, C is a constant of integration and a

subscript denotes differentiation. The equilibrium disjoining pressure, Π , isgiven by

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛=Π

mn

hhB

00

δδ . (17)

Applying the boundary conditions on 0h at ±∞=x yields a set of twoequations for U and C, which may be solved to yield

δδδ

−−+−

=1

)1()1( 24

33 PP

U ,⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−

−−−=

+

δ

δδδ

1

11 )()( 42

3 PPC .

(18)

The numerical solution for equilibrium liquid layer thickness )(0 xh isobtained by prescribing some initial condition and solving the unsteadyequation (14), with 0/ =∂∂ yh and the term )/( xhU ∂∂ added to the right-hand side, until a steadily advecting interface shape is obtained. Thenumerical solution is obtained using a Crank−Nicholson method, whichresults in solution of a pentadiagonal matrix at each time step. The solution



achieves a steadily propagating form within a dimensionless time of aboutunity. Tests with different time step and spatial step size are found to havenegligible effect on the asymptotic result for interface shape.

3 Numerical solution methods

A fairly simple method for solving the lubrication theory equation (14) fornonlinear evolution of a thin liquid layer is obtained by using an ADI method forpartial differential equations with fourth-order spatial derivatives, similar to thatdeveloped by Conte and Dames [11] for the biharmonic equation. Derivatives areapproximated using second-order central differences. Nonlinear terms andderivatives of third and lower order are treated explicitly. In some cases in which itis desirable to move the coordinates with the driven-layer front, a convection term

xUh is added to the right-hand side of eqn (14), where U is the front advectionspeed from the equilibrium theory. The discretized equations have the form

( )[ ] )(,

)(,

4)2/1(,

4)(,

)(,

)2/1(,

nji

njiy

njix

nji

nji

nji BhhAthh +∇+∇∆+= ++ (19a)

( )[ ])(,

4)1(,

4)(,

)2/1(,

)1(,

njiy

njiy

nji

nji

nji hhAthh ∇−∇∆+= +++ , (19b)

where )(,njiA are coefficients of the fourth-order derivatives of h and )(

,njiB denote

the remaining terms. Solving for )2/1(,+n

jih in eqn (19b) and substituting into eqn

(19a) yields

( )[ ] )( 3)(,

)1(,

4)1(,

4)(,

)(,

)1(, tOBhhAthh n

jinjiy

njix

nji

nji

nji ∆++∇+∇∆+= +++ , (20)

which indicates that the local error in the computational method is )( 3tO ∆ . Forcomputations performed on a rectangular grid spanning the interval

),( maxmin xx and ),( maxmin yy with uniform grid spacing, typical boundaryconditions have the form

1),( min =yxh , 0),( min =yxhx ,δ=),( max yxh , 0),( max =yxhx ,0),( min =yxhy , 0),( min =yxhyy ,

0),( max =yxhy , 0),( max =yxhyy . (21)

The value of the Bond number only enters into the problem in the disjoining-pressure term, and the results are not sensitive to small change in Bond number.Several investigators have shown that contact-line instabilities, such as the fingering



instability, are sensitive to variation in the precursor-film thickness [6,34]. Theappropriate selection of δ depends on the specific experimental conditions underconsideration. Kataoka and Troian [34] report good comparison betweenexperimental studies and computations for fingering stability in shear-driven flowwith 01.0=δ . Values of δ in the range 0.01−0.03 have been used by numerousinvestigators [6,20,34,48,49,56] and have consistently yielded results in at leastqualitative (and often quantitative) agreement with experiments.One limiting consideration is that complete resolution of the flow field requiresthat the grid spacing x∆ be on the order of, or smaller than, the precursor-filmthickness. This restriction is necessary due to a small undershoot in the liquid-layer thickness close to the contact point. This undershoot is shown in Fig. 1b,which gives a close-up view of the region indicated by a dashed circle in Fig. 1a.The length of this undershoot region is proportional to the precursor-filmthickness (with a typical length of about δ6 ), such that the smaller theprecursor-film thickness the smaller the grid spacing necessary to resolve thisregion. Failure to maintain a sufficiently small grid increment using the standardADI method results in numerical instability.A variety of improvements to the basic ADI method described above havebeen developed in order to improve the stability of the method, specificallyfor small values of the precursor-film thickness δ. A survey of various ADIschemes applied to the lubrication equations is presented by Witelski andBowen [60]. A variety of “positivity-preserving” schemes for thelubrication theory have been formulated, which ensure that the computedlayer thickness will remain positive throughout the computation [13,25,62].Such methods are able to remain stable for arbitrarily small values

Figure 1: Plots of (a) equilibrium layer thickness profile versus position alongthe substrate with 03.0=δ and (b) a close-up showing the undershootin layer thickness in the region indicated by a dashed half-circle in (a).The symbols in (b) are the numerical computation points.

(a) (b)



of the precursor-film thickness, even if the “undershoot region” is notresolved. One caution is that for problems employing a disjoining-pressureto simulate variation in contact angle, it may still be necessary to fullyresolve this undershoot region near the contact line since the disjoiningpressure term is greatest in this region. It is also noted that fourth-orderparabolic equations, such as the lubrication equations for a liquid film, areknown to exhibit physical singularities in finite time [5], which furthercomplicates numerical solution of these equations.

4 Fingering instability of driven contact lines

4.1 Homogeneous surfaces

The front of a driven liquid layer develops a ridge just behind the contact linewith larger layer thickness than the liquid layer as a whole. This ridge exhibits afingering instability in which perturbations on the liquid layer front grow suchthat thicker regions of the ridge move forward faster than thinner regions of theridge. The fingering instability leads to formation of narrow rivulets of liquidthat penetrate into the unwetted part of the substrate. Following the fingeringinstability, the liquid flows preferentially within a discrete number of rivulets,resulting in a reduction in efficiency of the coating process. Experimentalstudies of fingering of a driven contact line are reported by de Bruyn [8],Huppert [27], and Cazabat et al [9]. These studies provide data on the onset offingering instability and quantities such as dominant instability wavelength andfinger growth rate. An example showing growth of fingers on a sheet driven byMarangoni force is given in Fig. 2 (from Cazabat et al [9]). A fluorescentimaging method was developed by Johnson et al [30] that allows globalmeasurement of film thickness and contact lines. This method was used to studyfingering and rivulet formation by Johnson et al [31], providing quantitative dataon contact angle and film thickness across the rivulet.

(a) (c)

(d)

(b)

Figure 2: Growth of fingers in a liquid film driven up a vertical wall byMarangoni force (Cazabat et al [9]).



A linear stability analysis for fingering of a driven contact line wasdeveloped by Troian et al [56] for flow down an inclined plane. In thisanalysis, it is shown that a driven contact line, modeled using a lubrication-theory approach as developed in Section 2 with no body force normal to thesubstrate surface, is always unstable to sufficiently long waves along theliquid ridge just behind the contact line, where the instability growth ratedecreases with increase in the precursor-film thickness. This analysis wassubsequently extended to films driven by Marangoni convection [34], to flowwith weak inertia [36], and to flow with non-zero normal body force [6].Bertozzi and Brenner [6] find that a normal body force (such as gravity orientednormal to an inclined plane) suppresses the fingering instability in terms of boththe range and the growth rate of unstable wave numbers. A sufficiently largenormal body force can make the contact line stable for all wave numbers. Theeffect of normal body force is shown in Fig. 3, where we plot the change inequilibrium layer thickness profile and in the instability growth rate for differentvalues of the normal force parameter P2.

Nonlinear aspects of the fingering instability are examined theoretically byKalliadasis [33] and computationally by a number of investigators, including Eres etal [20], Moyle et al [42], Diez and Kondic [14], and Kondic and Diez [35]. Themajority of these studies focus on the shape of the rivulets once they form. Inparticular, it is found that the rivulets may have either a finger-like form, with sidesthat are approximately parallel, or a saw-tooth form, in which the rivulet moreclosely resembles a triangle. This issue was also taken up experimentally

(a) (b)

Figure 3: Effect of the normal body-force parameter P2 on (a) the equilibriumlayer thickness profile )(0 xh and (b) the growth rate β as a function ofwave number k. Data are given for 02 =P (upper solid curve), 0.5(dashed curve), 1.0 (dashed-dotted curve) and 2.0 (lower heavy solidcurve). For all cases the precursor-film thickness is 03.0=δ(Marshall and Wang [37]).



Figure 4: Computational results showing rivulet shape of a gravity-drivencontact line for three different inclination angles (a) 13.9°, (b)27.9°, and (c) 90° (Kondic and Diez [35]).

by Hocking et al [26] and Johnson [29]. Kondic and Diez [14] examine therivulet form for flow down an inclined plane with several different inclinationangles. They show that the shape of the rivulet depends significantly on theinclination of the substrate plane, with more sawtooth-like rivulets forming forsmall inclination angles (nearly horizontal planes) and more finger-like rivuletsforming for high inclination angles (nearly vertical planes). Computationalresults from Kondic and Diez [35] illustrating this variation for three inclinationangles are shown in Fig. 4.

The structure of a rivulet formed from fingering instability of a drivencontact line is examined in Figs. 5 and 6. A droplet is observed to form on theend of the rivulet and the ridge forms a V-shape that leads to the rivulet spine.Streamlines of the flow rate Q relative to the equilibrium flow rate for thedriven film are plotted in Fig. 6. As the rivulet develops, fluid travels



(a) (b)

Figure 5. Structure of a rivulet formed from fingering instability, showing (a)contours of layer thickness over a time series as the rivuletdevelops (drawn in a frame moving with the equilibrium contact-line velocity) and (b) three-dimensional shape of rivulet (with thevertical coordinate scaled differently from the horizontalcoordinates) (Wang [58]).

Figure 6. Streamlines of flow-rate Q relative to the equilibrium flow rate ofthe moving contact line during rivulet development (Wang [58]).



along the ridge and is fed into the rivulet. It is interesting that a weak series oftraveling waves forms behind the rivulet, which are usually too weak to beapparent in the contour plots but are clearly seen in the relative flow-ratestreamlines in Fig. 6.

4.2 Effect of pre-exising droplets

Bertozzi and Brenner [6] point out that experimental results for liquid-film flowdown an inclined plane [8] indicate that the driven front becomes unstable andrivulets grow even under conditions for which linear analysis indicates that thefingering stability is suppressed by the normal component of the gravitationalbody force. These authors argue that this discrepancy may be due to transientgrowth of perturbations in precursor-film thickness that are amplified by passageof the contact line, where the amplification factor varies inversely with theambient precursor-film thickness. This issue is further examined by Ye andChang [61], who report a spectral analysis of the response of the layer front torandom variations in thickness of the thin precursor film.

The impact of a driven contact line on droplets of various sizes wasexamined in a series of numerical simulations by Wang [58], and is summarizedbelow. The droplet is modeled by a Gaussian variation in the initial layerthickness centered at a point ),( DD yx ahead of the front. The droplet has aGaussian decay radius R and a peak thickness maxh∆ above the precursor-filmthickness. The initial layer thickness is given by

}/])()[(exp{)()0,,( 222max0 Ryyxxhxhyxh DD −+−−∆+= . (22)

The computations are performed over a region 1515 <<− y and1515 <<− x , with initial front location at 0=x and initial droplet center at

5.5=Dx . The grid size is 03.0=∆=∆ yx , the time-step size is 01.0=∆t , andthe precursor-film thickness is 1.0=δ . All length scales are non-dimensionalized by the upstream liquid-layer thickness.

A series of plots showing the impact of a driven liquid layer front on a dropletwith radius 2=R and maximum thickness 2.0max =∆h is shown in Fig. 7 attimes just prior to impact, near the middle of impact, and following passage of thefront through the droplet. Regions shaded gray indicate the thickest parts of theliquid layer, and a dashed circle in Fig. 7c indicates the initial radius and locationof the droplet. A plot showing the maximum value of layer thickness along lines

const=y at the same three times is given in Fig. 8. During the initial part ofimpact with the droplet, the front is observed to accelerate into the droplet, suchthat mass conservation leads to a subsequent decrease in the thickness of theliquid ridge near the impact point. Acceleration of the front with increase inlayer depth is evident from the expression (18) for U with small values of δ.After the contact line passes over the droplet and propagates out the far side,liquid mass from the droplet is added to the liquid ridge and the front propagation



Figure 7: Contours of layer thickness at three times as a driven liquid layerfront passes over a pre-existing droplet. Ten evenly spaced contoursof layer thickness are plotted over the interval )3.1,2.0( , and grayshading indicates regions where 3.1>h . A dashed circle in (c)indicates the initial position and radius of the droplet (Wang [58]).

Figure 8: Plots of the maximum value of h over lines const.=y as a functionof y at times 3=t (dashed curve), 5 (dashed-dotted curve), and 10(solid curve) (Wang [58]).



speed returns to its ambient value. These effects ultimately lead to an increase inridge thickness and an outward deviation of the front near the part of the ridgethat impacts with the droplet, as well as a decrease in the ridge thickness oneither side of this region.

A parametric study was performed of the effect of droplet radius, maximumthickness and normal force on the droplet impact with the front. Results fordroplet radii 1=R , 2, and 5 with droplet thickness 2.0max =∆h are shown inFigs 9 and 10 for a time after the front passes through the droplet. All three casesexhibit the same series of processes explained in the previous paragraph, leadingto an eventual increase in liquid-ridge thickness in the impact region and adecrease in thickness on either side of the impact region. The increase in ridgethickness in the impact region is smaller for the 1=R case, but about the samefor the 2=R and 5=R cases; however, the outward deviation of the frontfollowing impact with the droplet is considerably larger in the 5=R case thanfor smaller droplet radii.

Results for values of the droplet maximum thickness of 1.0max =∆h ,0.2 and 0.5 are shown in Figs 11 and 12 for cases with 2=R after passageof the front through the droplet. As might be expected, cases with largerdroplet thickness result in larger increases in thickness of the liquid ridgein the impact region, and correspondingly larger decreases in ridgethickness within neighboring regions of the ridge.

Figure 9: Plot showing effect of droplet radius on contours of h withdroplet radius (a) 1, (b) 2, and (c) 5 (indicated by a dashedcircle). Contour values and notation are the same as in Fig.8 (Wang [58]).



Figure 10: Plot showing effect of droplet radius on maximum value of h overconst=y lines with droplet radius 1 (dashed curve), 2 (solid

curve) and 5 (dashed-dotted curve) at the same times as in Fig. 9(Wang [58]).

Figure 11: Plot showing effect of droplet thickness on contours of h withdroplet maximum thickness (above the ambient precursor-filmthickness) of (a) 0.1, (b) 0.2, and (c) 0.5. Contour values andnotation are the same as in Fig. 7 (Wang [58]).



Figure 12: Plot showing effect of droplet thickness on maximum value of hover const.=y lines with droplet maximum thickness (above theprecursor-film thickness) of 0.1 (dashed curve), 0.2 (solid curve)and 0.5 (dashed-dotted curve) at the same times as shown in Fig. 11(Wang [58]).

Figure 13: Plot showing effect of normal force parameter 2P on contours of hwith (a) 02 =P , (b) 0.5, and (c) 1.0. Contour values in (a) are thesame as in Figure 7. Ten evenly spaced contours are drawn over theinterval (0.2, 1.23) in (b) and over the interval (0.2, 1.15) in (c).Gray shading denotes regions where h is larger than the maximumcontour line in each plot, and the droplet radius and initial positionare indicated by a dashed circle (Wang [58]).



Figure 14: Plot showing effect of normal force parameter on maximumvalue of h over const.=y lines with 02 =P (solid curve), 0.5(dashed curve) and 1.0 (dashed-dotted curve) at time 10=t(Wang [58]).

Results for values of the normal force parameter 02 =P , 0.5 and 1.0 areshown in Figures 13 and 14 for cases with 2=R and droplet thickness

2.0max =∆h after passage of the front through the droplet. The contour linesand shaded regions are adjusted for each of these three cases to beproportional to the maximum layer thickness in the equilibrium solution. Theplots indicate that the increase in ridge thickness in the region of impact withthe droplet is much reduced as the normal force parameter 2P increases. Forinstance, for 02 =P the maximum layer thickness at the ridge increases bynearly 10% of the ambient value after passage through the droplet, whereasfor 02 =P the same droplet yields only a 1.5% increase in ridge thickness.The normal body force suppresses the front response to impact with dropletsby inducing a tangential pressure force that acts to force fluid from thickerregions of the liquid layer to thinner regions.

4.3 Effect of substrate surface inhomogeneity

An alternative explanation for the observed sub-critical instability of thedriven liquid-layer front is that it occurs due to response of the front tovariation in contact angle on the surface caused by variation in surfaceroughness or chemical composition, as might occur due to an oil film orother contamination. In the precursor-film method, the effect of contact-



angle variation is introduced through the disjoining-pressure term. A studyusing the precursor-film method to investigate the effect of contact-anglevariation on droplet hysteresis is reported by Schwartz [48], and a similarnumerical model was used by Schwartz and Eley [49] to examine bifurcationof a droplet placed on a surface with strong contact-angle variation. Thework of Schwartz [48] was concerned with contact-angle hysteresis, so thespots of contact-angle variation are consequently selected to have a smalllength scale, with contamination-spot radius measuring approximately 5%and spot-separation distance approximately 40% of the upstream liquid-layerthickness. Because of the small length scales used in the study, no rivuletformation due to the surface inhomogeneities was observed.

A recent study by Marshall and Wang [37] utilizes the precursor-filmlubrication theory with the disjoining-pressure term proposed by Schwartz[48] to investigate the effect of surface inhomogeneities on rivuletformation. Since the length scale for rivulet formation is governed by themost unstable wavelength of the fingering instability, the study examinedmuch larger contamination-spot sizes and separation distances than used bySchwartz [48]. Most of the computations were performed withcontamination spots of radius equal to twice the upstream liquid-layerthickness and with spot-separation distances ranging from 5 to 10 times theupstream layer thickness. Marshall and Wang [37] examine both the effectof isolated contamination spots and the effect of arrays of contaminationspots on the development of fingers from the driven contact line. Caseswith isolated contamination spots are shown in Figs 15 and 16 for spotswith Gaussian variation in contact angle and maximum change in contactangle of 01.0min, −=∆ Eθ and 0.01, respectively. The growth rate andwavelength of the most unstable perturbation from linear fingering theory,indicated by a dashed line in Figs 15d and 16d and an arrow in Figs 15cand 16c, agree closely with the fingering that develops downstream of thecontamination spots.

Nonlinear effects can be observed for passage of the contact linethrough arrays of contamination spots. For instance, in Fig. 17a asubcritical instability is observed, in which fingering of the contact line anddevelopment of rivulets is observed at a wavelength equal to the spot-spacing distance, even though the spot-spacing distance is less than thecritical wavelength for fingering instability according to the linear theory.At very small values of the spot-separation length (Fig. 17b) the fingeringreverts to the most unstable wavelength from the linear theory. Subcriticalinstabilities are also observed for cases where the normal body force issufficiently large to suppress the fingering instability in the linear theory,but nonlinear effects allow the rivulets to form. This is illustrated,for instance, in Fig. 18a for a case with 12 =P . For sufficientlylarge normal body force (i.e., sufficiently large 2P ), thefingering instability will be eliminated even for passage of a drivencontact line through an array of contamination spots. This latter effect is



Figure 15: Time series showing impact of a driven liquid-layer front on anisolated surface contamination spot having 01.0min, −=∆ Eθ . In (d),the maximum value of layer thickness is plotted as a function oftime. The dashed line in (d) and the array in (c) indicate the growthrate and fastest-growing wavelength from linear stability theory(Marshall and Wang [37]).

apparent, for instance, in Fig. 18b for a case with 22 =P , whereperturbations that form on the driven contact line with spacing equal to the spotspacing appear to advect forward without growing in time.

5 Rivulet instabilities

Once formed, rivulets are subject to several instabilities that lead to a variety ofdifferent oscillatory behaviors and several different regimes of rivulet flow. Afairly comprehensive study of the regimes exhibited by rivulets on an inclined



Figure 16: Time series showing impact of a driven liquid-layer front on anisolated surface contamination spot having 01.0min, =∆ Eθ . In (d)the maximum value of layer thickness is plotted as a function oftime. The dashed line in (d) and the array in (c) indicate the growthrate and fastest-growing wavelength from linear stability theory(Marshall and Wang [37]).

plate is given by Schmuki and Laso [50]. This work identifies four regimes,illustrated schematically in Fig. 19, which are listed in order of increasing liquidflow rate. For very low flow rates, a droplet regime exists in which the liquidflows down the plate as a series of discrete droplets (Fig. 19a) that move at anapproximately constant speed. Droplet flows are discussed in detail in the nextsection. As the flow rate increases, the distance between droplets decreases untilthe droplets touch, at which point the liquid flow takes the shape of astraight rivulet. In this straight-rivulet regime, the liquid is transported down the



Figure 17: Nonlinear aspects of rivulet development in an ordered array ofcontamination spots: (a) fingering and rivulet development atwavelength below the critical wavelength for fingering instabilityand (b) reversion of the fingering instability to the most unstablewavelength from linear theory for very small spot spacing(Marshall and Wang [37]).

(a)

(b)

Figure 18: Effect of normal body force on fingering for a contact line driventhrough an array of negative relative contact-angle spots for normalbody-force parameter of (a) 12 =P and (b) 22 =P (Marshall andWang [37]).

(a) (b)



g

(a) (b) (c) (d)

Figure 19: Regimes of liquid flow down an inclined plate: (a) droplet flow, (b)straight rivulet flow, (c) meandering rivulet flow, and (d)oscillating rivulet flow (sketches from flow visualizations ofSchmuki and Laso [50]).

inclined plate in a narrow, straight stream (Fig. 19b). With further increase inflow rate, the rivulet enters a meandering-rivulet regime (Fig. 19c), in which therivulet traces a meandering path but remains stable, such that the path of therivulet does not change in time. With yet further increase in flow rate the rivuletenters an oscillating-rivulet regime (Fig. 19d) (sometimes also called “pendulumrivulet”), in which the rivulet path is no longer steady but oscillates back andforth in time. In this oscillating regime, the rivulet tends to break intermittentlyat particularly sharp bends, producing fluid streaks that break off from the mainrivulet and continue downstream as long liquid ribbons approximately parallel toeach other.

These various regimes of rivulet flow depend on a variety of parametersdescribing the liquid, gas and substrate surface, including liquid flow rate, liquiddensity and viscosity, plate inclination angle, surface tension of the liquid-gasinterface, and contact angle of the interface with the substrate. Figure 20, fromSchmuki and Laso [50], illustrates how the liquid flow rate and plate inclinationangle affect the various rivulet flow regimes for a system consisting of water ona stainless steel plate. An increase in plate inclination angle causes the criticalflow for transition from one regime to another to decrease. It is found that theeffect of plate inclination on regime transition can be accounted for by the simpleformula [3,50]

αα sinmmV = , (23)

where Vm is the critical flow rate for regime transition for the system measuredfor a vertical plate and αm is the corresponding critical flow rate for the samesystem with the plate inclined at an angle α from the horizontal. This expression



Figure 20: Flow-regime mapping showing the effects of liquid flow rate andplate inclination angle on rivulet flow regimes for water on astainless steel plate. (Filled symbols are based on data of Schmukiand Laso [50].)

is based on the assumption that the various regimes of rivulet flow aredependent only on the component of gravity tangential to the surface.Equation (23) appears to be in good agreement with the data for moderateand large values of α, but of course this assumption would be expected tobreak down at small values of α as the ratio of the normal to tangentialcomponents of gravity becomes large.We caution that for certain substrate surfaces and test liquids, one or more of therivulet flow regimes described above may not be observed or additional regimes maybe observed beyond what is described here. For instance, in experiments with wateron a glassy hydrophobic surface, Nakagawa [43] reports that the liquid transitionsdirectly from a droplet regime to a meandering-rivulet regime, with no straight-rivulet regime in-between. This author further reports that for high flow rates(particularly at low surface inclination values), the rivulet enters a restable regime inwhich it adopts a straight, stable form with periodic variation in width. As shown inthe sketch in Fig. 21, in this restable regime the rivulet exhibits symmetric,periodic oscillations of the contact line along its length with corresponding periodicvariations in the layer-thickness profile.

Various theoretical models for the flow field in the different rivulet regimesand for the transition between these flow fields have been presented in theliterature. The classical analysis of the straight-rivulet regime is given by Towelland Rothfeld [55], which provides an analytical expression for the width and



Figure 21. Diamond-shaped waveform on rivulet in the restable regime, alongwith sketch of changes in rivulet-thickness profile (Nakagawa andNakagawa [44]).

maximum height of a straight rivulet flowing down an inclined slope at a givenflow rate. The analysis assumes that the velocity field is one-dimensional, fullydeveloped, and steady to reduce the governing equations (using the samecoordinate system as in Section 2) to

0=∂∂

yp , αρ cosg

zp

−=∂∂

,ναsin

2

2

2

2 gz

uy

u−=

∂∂

+∂∂ , (24)

where α again represents the plate inclination angle relative to thehorizontal. The first part of eqn (24) requires that pressure be constant acrossthe rivulet width at a given elevation above the plate. The second part of eqn(24) indicates that the pressure field in the direction normal to the platevaries hydrostatically (using the reduced gravity). The third part of eqn (24)gives a Poisson equation that must be solved for the velocity field ),( zyu .



Combining the first two parts of eqn (24) with the no-slip boundary conditionon the plate, the free-shear boundary condition on the liquid-gas interface,and the kinematic condition of the interface, Towell and Rothfeld [55] derivea nonlinear ordinary differential equation for the cross-stream variation ofrivulet thickness )(yh as

Hba

HH

+=′+′′

2/32 )1(, (25)

where )(/)( 0 YHahhH =−≡ is the dimensionless liquid-sheet thickness,ayY /≡ is the dimensionless length in the cross-stream direction, )0(0 hh = ,

2/1)sin/( αρσ ga ≡ is the capillary length scale, and b is the radius ofcurvature of the rivulet at the centerline 0=y . Equation (25) yields a familyof rivulet shapes )(YH with the single parameter being the constant ba / . Theconstant ba / is in turn determined by choice of the stream width and contactangle.

Towell and Rothfeld [55] consider two limiting cases in order to get closed-form solutions. In the first limiting case, the rivulet is so small that we can write

0/ Hba >> , for which case the last term in eqn (25) is negligible. In this case eqn(25) reduces to the equation of a circle with radius ab / , so that the rivulet willhave the shape of a section of the circle. The ratio ab / is then related to thedimensionless rivulet width aL /≡ and the contact angle θ on the rivulet side by

θsin2L

ab= . (26)

The liquid flow rate Q for this case is solved by solution of the Poisson equation(24), with use of the approximation 2222 // yuzu ∂∂>>∂∂ , and integrationacross the rivulet cross-section to get

192)(

cos4

θαρ

µ fg

Q= , (27)

where the function )(θf is given by

θθθθθθθθθ

4

32

sinsincos13sincos23cos12)( −−+

=f . (28)

For a given flow rate Q, the rivulet width obtained from eqn (27) is highlysensitive to contact angle. It is also curious that the rivulet width in this limit isindependent of surface tension.



The other limiting case is that of a wide, flat rivulet for which the centerlineradius of curvature is very large, implying 0/ Hba << . For this case the centerlinerivulet thickness approaches the Rayleigh limiting value

)2/sin(20 θ=H , (29)

such that with greater flow rate the rivulet widens without change in maximumthickness. The flow rate is given by

)2/(sin38sintan 3 θ

σαρ

σαµ

=gQ

. (30)

The rivulet width is again found to be sensitive to variation in contact angle.Notable later work on stable rivulet flows includes Allan and Biggin [3],

who numerically solve the Poisson equation (24) for the rivulet-velocity fieldand compare the results to the approximate solutions of Towell and Rothfeld[55]. An analysis that accounts for the effect of contact-angle hysteresis onstable rivulet flows is given by Nawrocki and Chuang [45].

The straight rivulet exhibits several different types of waves and instabilities,including Kelvin−Helmholtz-type waves on the liquid-gas interface and variousmodes of waves involving motion of the contact line. A linear stability analysis ofthese various types of waves on a straight rivulet is given by Davis [12] and Weilandand Davis [59]. A particularly interesting aspect of these studies is that the authorsexamine the sensitivity of the stability results using three different ways for treatingthe contact line, including fixed contact lines, moving contact lines with fixedcontact angles, and moving contact lines with angles determined by the contact-linespeed. The authors report that for the case of fixed contact lines, the rivulet is stablebelow a certain Reynolds number. For the more realistic case of moving contactlines with fixed contact angles, the rivulet is unconditionally unstable, leading togrowth of meandering and symmetric waves on the rivulet. In the yet more realisticapproach where the contact angles vary smoothly with contact-line speed (with nohysteresis), the change in contact angle is found to dissipate waves on the rivulet andto stabilize the rivulet under certain conditions.

Stable meandering rivulets have been examined theoretically by Mizumura andYamasaka [41] and Mizumura [40], who employ depth-averaged momentumequations and a perturbation method analogous to that used to study river meandering[28,47]. The analysis predicts the shape of stable, periodic rivulet meanders that appearto compare well to experimental results.

An analysis predicting transition from a stable meandering rivulet to anoscillatory rivulet is given by Schmuki and Laso [50] for the case of a rivuletwhose cross-section forms the arc of a circle. This theory is based on thehypothesis that the number of rivulets will adjust itself such that the total energyassociated with kinetic energy and surface tension will be a minimum. Theauthors evaluate the energy of a system of n parallel rivulets that transport agiven liquid flow rate. They then speculate that when the energy is a minimum



for 1=n the stable meandering-rivulet regime will be observed, whereas whenthe energy is a minimum for 1>n an oscillating-rivulet regime will occurtypified by periodic shedding of sub-rivulets by the main rivulet. Estimatesusing this minimal-energy approach to predict both the transition point from themeandering to the oscillating-rivulet regimes and the shedding frequency (calledthe “decay frequency”) in the oscillating-rivulet regime yield excellentagreement with the experimental data of Schmuki and Laso [50].

6 Wind-driven rivulet break-up and droplet flows

Rivulets and droplets exhibit very different flow regimes for wind-shear-driven flowsthan they do for gravity-driven flows. In particular, wind-driven rivulet and dropletflows tend to be dominated by the parts of the flow that experience significantaerodynamic-form drag, whereas in gravity-driven flows the body force is uniformlyapplied throughout the fluid.

A recent series of experiments on wind-driven rivulet break-up and dropletflows on a horizontal plate subject to a variety of gravitational states (rangingfrom 1g to 0g) has recently been performed by the authors and their students[38]. The experiments at 1g were performed in the laboratory. The 0gexperiments, as well as a limited number of partial-gravity experiments, wereperformed on the KC-135 parabolic flights run by NASA Glenn. Theexperiments examined a stream of water ejected from a 2-mm hole on a smoothPlexiglas surface at a variety of water and air flow rates. Briefly described hereare the main differences and commonalties observed between shear-driven andgravity-driven rivulets flowing from a point source. An immediate commonaltyis that in the absence of a driving force (gravity or wind), a small flow of watersimply forms a puddle that spreads radially, as shown in Fig. 22. The descriptionbegins with illustrations of gravity-driven rivulets, and then proceeds to discusswind-driven rivulets under different normal-gravity conditions.

Figure 22: A puddle of water forming from a source on a flat slope in still air.



6.1 Gravity-driven rivulets

As soon as the surface slope is increased from a horizontal state, flow fromthe source forms as a rivulet that advances down-slope, as shown in Fig.23. When the slope is small, the rivulet maintains a more-or-less straight,linear form as it progresses down-slope (Fig. 24). The rate of rivuletadvance down-slope is directly related to the flow rate of water. On asteeper slope, in accordance with the rivulet regimes indicated in Fig. 20,the rivulet assumes a meandering course (Fig. 25). The rivulets shown inFigs 24 and 25 form for the same flow rate.

When the flow rate is reduced to 5% of the flow rate associated with therivulets in Figs. 23 and 24, a relatively large, pendant droplet of water formsimmediately downstream of the source. The droplet enlarges until its massexceeds the critical value given by

ALKwmg γα =/)sin( , (31)

in which m is droplet mass, α is surface slope, w is droplet width, γAL issurface tension of droplet water, and K is a term accounting for the dropletgeometry.

The droplet moves down-slope, initially trailing a short, fairly straightrivulet back to the water source (Fig. 26). During this development, the dropletdiffers slightly in form compared to the approximately rectangular plan form ofan individual droplet moving on under gravity on a slightly inclined surface[17,18,23]. The rear of the droplet tapers more than the front, and merges withthe thin trailing rivulet. By virtue of its larger mass per unit length, the dropletaccelerates and detaches from the rivulet. The rivulet then forms a new droplet,and the cycle repeats.

6.2 Wind-driven rivulets

It is of interest to compare wind-driven rivulets formed for the same water-flowrates used in Figs 23–26. Our description of wind-driven rivulets is limited to thedroplet and straight-rivulet regimes. The presence of wind shear and pressureforces along the flanks of wind-driven rivulets inhibits lateral movement ofrivulets in a uniform wind field. A consequence of restrained lateral movement isthe heightened importance of Kelvin−Helmholtz-type waves on the water-airinterface, and the major importance of air pressure in shaping rivulets, especiallytheir downstream ends. Furthermore, these influences accentuate the dropletregime of rivulets. The ensuing descriptions describe rivulet formation for arange of gravity situations: 1g, 0g, and intermediate values of gravityacceleration. Applications involving rivulet flow where 0g and partial-g valuespertain include rivulets on aircraft with sharply curved trajectories and variousindustrial and mechanical processes used during space flight on potential futurelunar and Martian space stations.



Figure 23: A finger of water begins advancing down-slope from the pointwater source.

Figure 24: Rivulet in linear regime source.

Figure 25: Rivulet in meandering regime.



Figure 26. Droplet, with thin trailing rivulet, moves down-slope. Eventually,the droplet detaches from the rivulet, which then develops anotherdroplet, and so forth.

For a wind speed of 0 m/s, there will be no rivulet, and a circular puddle willspread laterally from the inlet due to hydrostatic pressure until colliding withsides of the test section (Fig. 22). As the wind speed is increased, a puddle stillspreads from the inlet, but the puddle shape is increasingly distorted from a circleto an ellipse that is shifted downstream of the inlet. Additionally, the puddlebecomes wavy and oscillates irregularly. For higher wind speeds, the puddle isshifted completely downstream of the inlet. When this occurs, water spreads asan unusual fan-shaped puddle extending from the water source (Fig. 28b). Theapex angle of the fan decreases as wind speed increases.

For higher wind speeds, a rivulet forms from the water ejected from thehole, which is initially straight with a nearly uniform width but with a somewhatthicker and wider “head” at the downstream end (Fig. 22a). After the rivuletprogresses a short distance, the forward progression of this head is observed tostop and grow to form a large droplet (Fig. 29b), fed by the rivulet, that continuesto increase in size with time. This large droplet spreads laterally as it increases involume, forming a raised ridge just behind the downstream contact line, which issubject to significant form drag. The aerodynamic drag on the droplet increaseswith time as its volume grows until a critical point is reached at which thedownstream aerodynamic force balances the upstream surface-tension force. Atthis point the droplet breaks off the rivulet and advects downstream, and thecycle repeats itself.

The break-off of the droplet at the end of the rivulet is resisted by surface-tension forces acting at the contact lines. This same surface-tension force hasbeen studied by several investigators to examine the critical condition for adroplet to stick to an inclined surface and to find the velocity of droplets flowingdown an inclined surface [17,18,23]. The surface-tension force arises from thedifference in contact angle between the front and the back of a droplet, typicallycalled the advancing and receding angles, respectively. These two angles areshown schematically in Fig. 27a, and a diagram indicating how these two anglesvary with velocity of the contact line is given in Fig. 27b. As the aerodynamicdrag (or gravitational force for rivulets on a slope) attempting o drive the



V

θ

θA

R

V0

θ

θA

θR

(a) (b)

Figure 27: (a) Schematic illustrating the advancing and receding contactangles for a droplet flowing down an inclined plane; (b) typicalvariation of the contact angles with contact-line velocity, showinghysteresis at zero velocity.

(a) (b)

Figure 28: (a) At very low wind speeds, the puddle shown in Fig. 22 spreadsdownwind; (b) increasing wind speed causes the puddle to becomefan-shaped. The surface is horizontal in both cases.

stationary droplet forward is increased, the receding and advancing contact anglesapproach their limiting values at zero velocity as indicated on a plot similar to Fig.27b. The surface-tension force thus has a maximum value that occurs whenthe two contact angles have reached their limiting values, and if the driving



(a) (b)

Figure 29: Photographs showing break-up of a wind-driven rivulet to form adroplet in 1g, including (a) the rivulet at a time where it has stoppedprogressing and is beginning to form a large droplet, and (b) close-up of the droplet about to detach from the rivulet. Wind flow isfrom top to bottom (McAlister et al [38]).

aerodynamic or gravitational force exceeds this maximum value the droplet willbegin moving forward. This condition for forward motion of a droplet is essentiallythe same as that for break-off of the droplet from the end of the rivulet, and canthus be used to determine the critical volume of the droplet and hence the rivuletbreak-off frequency. As shown by Dussan and Chow [18] and Dussan [17], thedroplet speed can be predicted by balancing the resisting force of surface tension(with the advancing and receding contact angles at the appropriate value from Fig.27b for the droplet speed V) with the driving gravitational or aerodynamic force,after subtracting the internal frictional force within the droplet.

When the droplet detaches and progresses downstream under the influence ofthe wind drag, it quickly reshapes itself in accordance with the airflow-pressuredistribution, the surface-tension and bottom-shear forces. The droplet evolves intoan elongated, double-lobed shape as shown in Fig. 30a. The elongation of thedroplets is in the cross-stream (rather than streamwise) direction, and occurs inresponse to the low pressure at the droplet sides. Fig. 30 shows several examples ofdownstream-propagating droplets of a flat surface. The liquid layer is significantlythicker within the lobes at the ends of the elongated droplet than at the center.Droplets are observed to gradually spread in the lateral direction, attainingprogressively greater aspect ratio until they suddenly bifurcate into two droplets.These offspring droplets themselves subsequently elongate as they continue theirdownstream progression, although at a smaller speed than the original parentdroplet. Since in general, larger droplets travel faster than slower droplets,there exists sufficiently far downstream a large diversity of droplet sizes andspeeds. This situation gives rise to the occurrence of droplet collisions, in which



(a) (b) (c)

Figure 30: Examples of downstream-propagating droplets on a horizontalsurface in 1g: (a) droplet shortly after break-off from rivulet, (b)close-up view of the elongated, double-lobe structure of the droplet,and (c) a larger droplet overtaking two smaller droplets. Wind flowis from top to bottom (McAlister et al [38]).

a larger droplet will overtake and collide with a smaller droplet. Dropletscoalesce upon collision forming a single larger droplet [4,19,39] that continuesmoving downstream at a yet more rapid speed. This larger droplet graduallyelongates and bifurcates, repeating the cycle. The resulting droplet flowsufficiently far downstream of the rivulet-injection point is dominated by seriesof droplet collision, elongation and bifurcation events that appear to occur in arandom manner. Throughout the foregoing process the rivulet progresses slowlydownstream.

6.3 Wind-driven rivulets in micro-gravity

Our experiments on rivulets in micro-gravity conditions reveal very significantdifferences between wind-driven rivulets under different normal-gravityconditions, as well as between gravity-driven and wind-driven rivulets ingeneral. The micro-gravity conditions examined include 0g as well as Martianand Lunar gravities (0.16g and 0.38g, respectively). The experiments werecarried out on NASA’s KC-135 aircraft.

The main differences observed for rivulets formed at 0g, compared to those at1g, are the approximate hemispheric shape of the water puddle initially formed atthe point water source (Fig. 31). The puddle balloons in size until wind dragpushes it downstream as a pendant droplet trailing a rivulet. The principalmechanism leading to formation of the rivulet is wind drag on the pendant droplet.The rate of downstream movement increases with wind speed and water flow rate.Eventually drag on the pendant droplet causes the droplet to detach from therivulet. Because the drag coefficient on a hemisphere for the 0g case is muchlarger than that for a flattened droplet, the critical volume of the pendantdroplet required for rivulet break-up is correspondingly less. In fact, the measured



Figure 31: A small, almost hemispherical puddle (or large droplet) formsdirectly from the water source on a horizontal surface. The puddlemoves downstream when wind drag exceeds surface tension andbottom shear, and it becomes a pendant droplet trailing a rivulet.

rivulet break-up frequency in our experiments with wind-driven rivulets on ahorizontal surface is nearly an order of magnitude greater in 0g cases than it is incorresponding 1g cases. The high frequency of the rivulet break-off at 0g alsoresults in relatively small droplets that retain an approximately hemisphericalshape, except for near the contact lines, as the droplets advect downstream. Thedroplets are much smaller than those formed at 1g. All the droplets progressdownstream with nearly the same velocity and without the lateral elongation orbifurcation observed in the 1g experiments (Fig. 30). As is also the case for rivuletsat 1g, rivulets at 0g progress further downstream and shed droplets less frequentlywhen the water-flow rate increases, as increased water flow enables the rivulet tokeep extending as the pendant droplet moves downstream. Commensurate with thegreater frequency of droplet shedding, the downstream advance of the rivulet at 0gis about an order of magnitude slower in 0g than in 1g conditions.

At the point of separation from the rivulet (Fig. 32), the droplet is quitehemispherical in appearance. This is expected because, when the gravity force isabsent, surface tension becomes the dominant parameter, and seeks to achieve minimaloverall potential energy by keeping the water mass in a spherical shape [10]. Thedroplet alters in shape once it begins sliding downstream. Airflow pressure around thedroplet and shear stress along the droplet’s base reduce the angle of the droplet’s rearface, and slightly increases the angle of the droplet’s front contact line (Fig. 33). Thedroplet assumes a fairly complex and oscillatory equilibrium form as it progressesdownstream. As the droplet advects downstream, air pressure around the droplet, andshear resistance along the droplet’s base, cause the droplet to undergo a cyclic bobbingmotion, that gives the moving droplet a galloping gait. Figure. 34 shows the droplet atone point in its cyclic motion. Airflow and base shear cause the droplet to develop ahead and a semi-cylindrical body that tapers to a short tail, which stays with the droplet.The form of the head and the body of the droplet are remarkably similar to thehead and body of the gravity-driven, finger-like rivulets illustrated in Figs 4 and5. However, in airflow at 0g, the droplet form is not stable; airflow depresses



Figure 32: Pendant droplet at point of separation from wind-driven rivulet; 0g,horizontal surface.

Figure 33: Droplet separating from wind-driven rivulet; 0g, horizontal surface.

Figure 34: Droplet moving downstream undergoes a cyclic bobbing motion.Water in the droplet sloshes backwards and forwards as itpropagates downstream in an inch-worm fashion.



the elevated head, momentarily pushing water back into the droplet’s body.Airflow around the droplet body promptly regurgitates the water forward to thehead, which bobs up. This process repeats as the droplet advects downstream.There is no behavior in 0g analogous to the roller phenomenon seen for dropletsformed from wind-driven rivulets at 1g. Similarly, the droplet at 0g remainsintact throughout its downstream progression, and does not undergo anybifurcation or collision process.

Rivulet behavior at partial gravity (0.16g and 0.38g) is a mix of thebehaviors observed at 0g and 1g, though closer to those for 0g. The sameprocess of pendant-droplet formation and droplet shedding occurs forpartial-g as noted above for 0g. The frequency of droplet shedding decreasesas gravity increases from 0g through 0.38g to 1g. There is a correspondingincrease in droplet size as gravitational acceleration increases, as well as anoticeable flattening of droplet profile, though in the partial-gravity statesexamined the shed droplet does not deform laterally and bifurcate asobserved at 1g. The foregoing observations of wind-driven rivulets weremade during the past year, and we presently are further analyzing andquantifying them.

7 Conclusions

Problems involving driven liquid sheets and films on a solid substrate continueto be an active field of research with a wide variety of important applications.Certain aspects of film-flow mechanics, such as the fingering instability, havereceived a great deal of attention in the literature, in part because these problemsare amenable to existing analytical and computational methods. Other importantaspects of driven contact-line problems have received surprisingly little attentionto date. Among the latter category are the interesting effects of inertia and ofvariation of contact angle on contact-line instabilities. Problems involvingsurface-attached droplets, in general, have received relatively little attention,particularly for wind-driven flows dominated by the effects of form drag onraised regions on the interface. Several studies have found that the component ofgravity normal to the substrate has an important effect on contact-lineinstabilities and rivulet development, which suggests that contact-line problemsfor conditions of microgravity and reduced gravity might present furtherinteresting research.

Acknowledgements

This research was supported by the NASA Office of Biological and PhysicalResearch under grant number NAG3-2368. Computer time was provided by agrant from the National Partnership for Advanced Computational Infrastructure,San Diego, California. The assistance of our students, Shufang Wang, GeoffMcAlister, and Yongli Zhao, and of our project monitor at NASA, Dr. CharlesNiederhaus, is greatly appreciated.



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Contact-line instabilities of driven liquid films · 2014-05-10 · solvent film, which is used, for instance, in the manufacture of circuit boards [32]. Rivulet formation limits