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Colloids and Surfaces

A: Physicochemical and Engineering Aspects 206 (2002) 87–104

Sliding drops on an inclined plane

Uwe Thiele a,*, Kai Neuffer a,b, Michael Bestehorn b, Yves Pomeau a,c,Manuel G. Velarde a

a Instituto Pluridisciplinar, Uni�ersidad Complutense, Paseo Juan XXIII, 1, 28040 Madrid, Spainb Lehrstuhl fur Statistische Physik und Nichtlineare Dynamik, BTU-Cottbus, Erich-Weinert-Strasse 1, 03046 Cottbus, Germany

c Laboratoire de Physique Statistique de l’Ecole Normale Superieure, associe au CNRS, 24 Rue Lhomond,75231 Paris Cedex 05, France

Abstract

An evolution equation for the film thickness was derived recently combining diffuse interface theory and long-waveapproximation (Phys. Rev. E 62 (2000) 2480). Based on results for the structure formation in a thin liquid film ona horizontal plane, we study one-dimensional periodic drop profiles sliding down an inclined plane. The analysis ofthe dependence of their amplitude, velocity, advancing and receding dynamic contact angles on period and theinteraction parameters reveals an universal regime of flat drops. The main properties of the flat drops do not dependon their volume. Both types of drops—the universal flat drops and the non-universal drops—are analyzed in detail,especially the dependence of their properties on inclination angle. Finally, an outlook is given on two-dimensionaldrops and front instabilities. © 2002 Elsevier Science B.V. All rights reserved.

Keywords: Thin film structure and morphology; Interfacial instability; Diffuse interface theory; Sliding drops

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1. Introduction

The investigation of structure formation in thinliquid films comprises many different aspects oftheir behavior that are important for both, theimprovement of the basic understanding of theunderlying physics and the applications in a widefield of areas. Two well studied problems are therupture and dewetting of a very thin liquid filmon a solid substrate and the formation of surface

waves in thin films flowing down an inclined plane[1].

If a solid surface is covered by a liquid and thecombination of substrate, liquid and surroundinggas is such, that the situation is energeticallyunfavorable (non- or partially wetting), aftersome transient the liquid is collected in drops onthe surface, assuming that gravity is acting verti-cally and in downwards direction. As a result thesolid–liquid and the liquid–gas interface form anequilibrium contact angle determined by the vari-ous molecular interactions. For relatively largedrops the angle is given by a combination of thevarious interface tensions (Young–Laplace rela-tion) [2]. However, the interface tensions onlyaccount for the ‘mismatch’ of the molecular inter-

* Corresponding author. Present address: Department ofPhysics, University of California, Berkeley, CA 94720-7300,USA. http://www.uwethiele.de.

E-mail address: [email protected] (U. Thiele).

0927-7757/02/$ - see front matter © 2002 Elsevier Science B.V. All rights reserved.

PII: S0 927 -7757 (02 )00082 -1

http://www.uwethiele.de

mailto:[email protected]

U. Thiele et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 206 (2002) 87–10488

actions at the interface of two semi-infinite halfspaces filled with different substances. Therefore,for very small thicknesses (�100 nm) of the thinliquid film between substrate and surrounding gasone has to correct the error made by using thebulk interface tensions by explicitly introducingdistance dependent molecular interactions in thehydrodynamic equations. Normally, this is doneby using an additional pressure term, the disjoin-ing or conjoining pressure introduced first byDerjaguin [3,4]. This pressure may comprise long-range van der Waals and/or various types ofshort-range interaction terms [5–8].

If the liquid forms a very thin film of finiteextension on the substrate because of viscosity thedynamics of retraction at the outer edges of theliquid film is very slow and the transition towardsdrops eventually starts at many points with spon-taneous film rupture due to a surface instability,called spinodal dewetting [9], or due to nucleationat defects [10]. Lateral growth of the resultingholes yields a network of liquid rims that maydecay later into small drops [11]. The process andthe resulting structures are in principle well under-stood. For a short review see the introduction of[12] in this volume. If however, the liquid is notdeposited as a thin film but already in form of adrop, it relaxes towards its equilibrium shape byspreading or retracting [1,2,13,14]. Both, thegrowth of a hole and the spreading of a dropinvolve the movement of a three phase contactline. If this line is seen as a material line itsmovement is made impossible by the no-slipboundary condition at the liquid–solid interface.The contact line can only move if a very thinprecursor film is introduced on the ‘dry’ parts ofthe substrate or if the no-slip condition is replacedby some sort of slip condition near the contactline [2,15,16].

Independently of the behavior of a very thinfilm on a substrate discussed above, many groupsstudied the behavior of a thin liquid film flowingdown an inclined plane, sometimes called ‘fallingfilms’. First experiments on falling films were per-formed to study the formation of surface waves[17]. Later the interest also focused on the occur-rence of localized structures and their interaction[18]. Falling films and later also falling films on

heated substrates were analyzed using linear sta-bility analysis [19,20], weakly non-linear analysis[21–25] and integration of the time dependentequations [26,27]. However, all these studies focusmainly on the surface instability caused by inertia(measured by the Reynolds number). Because themolecular interactions between film and substrateare normally neglected the obtained results cannotbe applied to very thin films or to drops on verythin films.

This is also the case for most works that focuson the evolution of falling sheets or ridges [28,29],i.e. the advancing edge of a fluid film or longone-dimensional drops on an inclined ‘dry’ sub-strate. As for the spreading drop a precursor filmor slip at the substrate [15,30,31] helps to avoiddivergence problems at the contact line, but intro-duces new ad-hoc parameters into the model. Theadditional parameters, namely the slip length orthe precursor film thickness have an influence onthe base state profiles and on the transverse frontinstability (growth rate and fastest growingwavenumber) [31–34]. Also the equilibrium anddynamic contact angles remain to be fixed inde-pendently in this kind of theory [15,35,36]. Analternative approach [37] treats either the vapor–liquid or fluid–solid interface, or both, as a sepa-rate phase whose properties differ from the bulkproperties. In a study of a droplet of non wettingviscous liquid rolling down an inclined plane[38,39] it was shown that in this case the classicalstress singularity at the contact line is alleviated.Recent experiments on drops on an inclined planefound stationary drops of different, angle-depen-dent shape that slide down the plane withoutchanging shape [40].

Recently, the long-wave approximation for thinfilms [1] and a diffuse interface description for theliquid–gas interface [41] were combined to derivea film thickness equation that incorporates a dis-joining pressure term [42]. There, first the verticaldensity profile for the liquid in a flat horizontallayer of fluid is determined taking into accountthe smooth but nevertheless relatively sharp den-sity transition between fluid and gas, and thedensity variation close to the solid substrate. Thelatter variation is due to the molecular interac-tions and is incorporated into the calculation

U. Thiele et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 206 (2002) 87–104 89

through the boundary condition for the fluid den-sity at the substrate. However, note that, thenotation interaction is after all rather vague: theeffective exponential interaction between substrateand liquid–gas interface represented by the dis-joining pressure is derived from thick interfacetheory, but it is not obviously present as a molec-ular interaction in the local van der Waals equa-tion for the density of the liquid. This difficulty,namely the difference between molecular interac-tion and effective interaction between mesoscaleobjects like interfaces and surfaces, is often over-looked in works on the topic. To account fordynamical situations the obtained density profileis then combined with the Stokes equation in thelong wave approximation. The film thicknessequation derived in this way has the usual form ofa thin film equation with disjoining pressure [1],where the disjoining pressure now results fromdiffuse interface theory. This theory is fully con-sistent with Stokes equation of fluid mechanicsand with Young–Laplace equilibrium theory inits van der Waals formulation. The structure for-mation in an unstable liquid film on a horizontalsolid substrate was studied using this evolutionequation in Ref. [43] and is reviewed and extendedin this volume [12]. Based on this results a studyof structure formation in a flowing film on aninclined plane was performed in Ref. [44], whichis here reviewed and extended toward parametervalues corresponding to a broader range of physi-cal situations. The crucial difference to the contri-butions on falling films mentioned above is thathere the surface instability is caused by the molec-ular interaction between film and substrate andnot by inertia. However, the resulting structureformation results from an interplay of the molecu-lar interactions and the viscous flow due to theinclination of the plane.

In Section 2 the evolution equation for the filmthickness is discussed, the equation is non-dimen-sionalized by an appropriate scaling and the ordi-nary differential equation describing stationarysurface profiles is derived. In Section 3 flat filmsolutions and their stability are discussed beforestudying the families of stationary periodic solu-tions in Section 4. The different family typesfound in Ref. [44] are shortly revised before em-

barking on a detailed study of the dependence ofthe properties of the periodic solutions on period,inclination angle, wetting properties and ratio ofthe strength of gravitation and molecularinteraction.

Flat drops are identified as solutions with uni-versal behavior, i.e. drops whose amplitude, ve-locity, and dynamic contact angles do not dependon mean film thickness. The dependence of theirproperties on the inclination angle is studied inSection 4.1. Finally, in Section 5 we give anoutlook towards the time evolution of two-dimen-sional drop profiles.

In the conclusion we discuss the results in thecontext of other approaches and explain the rela-tion of the study of sliding drops performed hereand work on moving concentration profiles de-scribed by the convective Cahn–Hilliard equation[45].

2. Film thickness equation

We study the film evolution equation derivedby Pismen and Pomeau [42] for a thin liquid filmflowing down an inclined plane. For a two-dimen-sional geometry as sketched in Fig. 1, we have

�th= −�x(Q(h){�x [� �xxh−�h f(h, a)]+ ��g}).(1)

h(x, t) denotes the x-dependent film thickness,Q(h)=h3/3� is the mobility factor arising forPoiseuille flow; �, � and � are the respective den-sity, surface tension and viscosity of the liquid, gis the gravitational acceleration and � is the smallinclination angle measured between the inclined

Fig. 1. Sketch of the geometry.


plane and the horizontal. x is the coordinatealong the inclined plane increasing downwards.The subscripts t, x and h denote the correspond-ing partial derivatives. Furthermore,

�h f(h, a)=�M(h, a)+�gh

=2�

ae−h/l�1−

1a

e−h/l�+�gh, (2)

is the derivative of the free energy, f(h, a), and hasthe dimension of a pressure. �(h)= −�M(h, a)is the disjoining pressure derived from diffuseinterface theory [42], a is a small positive parame-ter describing the wetting properties in the regimeof partial wetting, l is the length scale of thediffuse interface, and � is the strength of themolecular interaction having the dimension of aspreading coefficient per length. Eq. (1) has theform of a mass conservation law at �th= −�x �(x, t), where �(x, t) is the flow composed of:(i) the flow due to the gradient of the laplace orcurvature pressure, −��xxh ; (ii) the flow due togradients of additional pressure terms, �hf(h, a),(resulting from molecular interactions, �M(h, a),and hydrostatic pressure, �gh); and (iii) the gravi-tational driven flow down the inclined plane, ��g.

We introduce suitable new scales to obtaindimensionless quantities (with tilde):

x=�l�

�x,

h= lh� ,

t=3��

�2lt� . (3)

After dropping the tildes the dimensionless filmevolution equation is:

�th= −�x{h3[�x(�xxh−M(h, a)−Gh)+�G ]},(4)

�= ��

�l�1/2

and G=l�g�

. (5)

For the slightly non-wetting situation, spreadingcoefficients are very small, implying in the diffuseinterface approach that the ratio �l/� is O(a2)with a being a small parameter [42]. Then thescale in x-direction is O(l/a) and �= �/O(a) isnot any longer a small parameter. G expresses the

ratio of gravitation and molecular interactionswith its value taken to be always positive in thiswork. Negative values would arise for a liquidfilm below an inclined plane that also partiallywets the substrate. From here on we will only usethe introduced dimensionless quantities to studythe qualitative behavior of a liquid layer on aninclined plane. However, let us roughly estimatethe values for G that are to be expected in realphysical situations. Assuming �l/��a2 as dis-cussed above, G can be written as l2/l c

2a2, where lc,is the ‘normal’ capillary length ��/�g. Takingnow lc�10−3 m, l in the nanometer range and ato lie between 10−4 and 0.1 gives a range for Gbetween 10−2 and 10−10. The main scope of thiscontribution is beside reviewing results obtainedin Ref. [44] for relatively large G between 0.01 and0.1 to extend the calculations toward very small Gdown to 10−7 to show, that the qualitative resultsregarding solution types and their behavior aremainly unchanged.

Now the remaining constant in the disjoiningpressure, a, is transfered into the mobility factorQ by the transformation h� =h+ ln a to facilitatethe comparison with results obtained for patternformation on a horizontal substrate [43,12].

After dropping the bar, in Eq. (1) becomes

�th= −�x{Q(h, a)[�x(�xxh−�h f(h))+�G ]}.(6)

with the mobility factor Q(h, a)= (h− ln a)3. Thederivative of the free energy now only containsthe parameter G :

�h f(h)=2e−h(1−e−h)+Gh. (7)

To study stationary solutions of Eq. (6), i.e.film thickness profiles, h(x, t), that move with aconstant dimensionless velocity, �, we write Eq.(6) in the comoving coordinate system xco=x−�t, implying �th= −��xh, when setting the timederivative in the comoving system to zero. Thenan integration with respect to x gives,

0=Q(h, a)(�xxxh− fhh �xh)+�GQ(h, a)−�h

+C0. (8)

C0 is a constant of integration that here, in con-trast to the reflection symmetric problem on a


horizontal plane [12], cannot be set to zero. C0

can be identified as the flow in the comovingsystem. C0�0 implies that a second integrationis not possible. We choose

C0= − (�0−�h0)= −Q(h0, a)�G+�h0, (9)

to introduce a flat film or homogeneous solutionof thickness h0 in a natural way. The corre-sponding flow in the laboratory system, constantwith respect to x, is �0= −Q(h0, a)�G.

3. Flat film solutions and their linear stability

Beside the flat film solution given by thechoice of h0 other film thicknesses, hi, may existthat yield the same flow in the comoving frame,i.e. that are solutions of Eq. (8) with identicalC0. They correspond to additional fixed points(beside h0) of Eq. (8), if seen as a dynamicalsystem. Given the flow �0 by prescribing h0,there exist two such solutions for flat film thick-nesses:

h1/2= (h0− ln a)�

−12

��

G�(h0− ln a)2−34�

+ ln a. (10)

The physical film thickness h− ln a has to bepositive. Thus only the positive sign in Eq. (10)can give a physically meaningful result. Takingthis choice a second flat film solution exists for�/G�(h0− ln a)2�1. It corresponds to the socalled conjugate solution discussed also in fallingfilm problems as, for instance, in Ref. [46].

For �/G�(h0− ln a)2�3 the second film thick-ness is larger than the imposed one, h0− ln a.Note, that the second flat film thickness doesnot directly depend on the molecular interactionM(h). However, it is known that for the limitingcase �=0 the molecular interaction selects anupper and an lower plateau thickness (i.e. thefixed points) and therefore has also to determinethe behavior for ��0 as will be discussed inmore detail below in Section 4.1. Note, that Eq.(10) does not serve to calculate the fixed pointsbecause the velocity, �, is not an independent

variable, but has to be determined together withthe solution profiles.

The linear stability of the flat film solutions isdetermined by using the ansatz h(x)=h0+exp(t+ ikx) to linearize the time dependentEq. (6) yielding the dispersion relation:

= − (h0− ln a)3k2(k2+ fhh �h 0)

− i3�Gk(h0− ln a)2. (11)

The growth rate of the harmonic mode withwavenumber k is given by the real part of (k),whereas its imaginary part determines the down-wards phase velocity of the mode to be �= −Im /k=3�G(h0− ln a)2. This corresponds tothe fluid velocity at the surface of the unper-turbed flat film.

The onset of the instability occurs with k=0for Re =0. Using Eq. (7) yields linear instabil-ity for,

fhh(h0)= −2e−h0(1−2e−h0)+G�0, (12)

i.e. for the intermediate thickness range,

h− �h0�h+ (13)

where,

h� = − ln�1

2�1

2��1

4−G

�n. (14)

For G�1 one finds h− � − ln(G/2) and h+ �ln 2+G. There is a critical point at (Gc=1/4,hc,= ln 4) where the lower and the upper linearinstability border meet. For G�Gc, flat films arealways linearly stable.

Given a film thickness in the linearly unstablerange, the critical wavenumber, kc, as deter-mined by Re =0, is kc=�− fhh(h0), whereasthe fastest growing mode has the wavenumberkm=kc/�2 and the growth rate, m= (h0−ln a)3fhh(h0)2/4. Note, that the stability of flatfilm solutions does not at all depend on thedynamical aspect of the problem (inclination andvelocity), but only on the molecular interactions,fhh(h). The stability condition is exactly as for aflat film on a horizontal substrate [12,43].


4. Periodic solutions

A linearly unstable flat film will start to evolvein time resulting in the rupture of the film or— ifthe evolution saturates— in a stationary filmprofile of finite amplitude. The stationary filmprofiles have to be solutions of Eq. (8) and can befound using continuation methods [47] alreadyapplied in the case of structure formation on thehorizontal plane ([12] in this volume). Denotingthe stationary solutions obtained in this way byh0(x) their linear stability is calculated using theansatz h(x)=h0(x)+h1(x) et in Eq. (6). Lin-earizing in gives after transforming into thecomoving frame an eigenvalue problem for thegrowth rate, , and disturbance, h1(x):

h1={[3q2(h0x fhh−h0xxx)]x+ (q3h0x fhhh)x}h1

+ [2q3h0x fhhh+3q2(2h0x fhh−h0xxx)]h1x

+q3fhhh1xx−3q2h0xh1xxx−q3h1xxxx

− (3�Gq2h1)x+�h1x, (15)

where q=h0(x)− ln a and all derivatives of f arefunctions of the stationary profile h0(x). For de-tails of the numerical methods, see Ref. [44]. Here,we study the linear stability of the periodic solu-tions taking one period of the solution as the unitof the stability analysis, i.e. the stability to smalldisturbances when the period is fixed. The solu-tions may also be unstable with respect to coars-ening as in the case on the horizontal plane [43],but this is not investigated here.

First, we study periodic solutions in their de-pendence on period at fixed inclination angle, �,and interaction parameters, G and a. To do thiswe start from small amplitude stationary solutionsthat have the form h(x)=h0+ eikx as can beseen by linearizing Eq. (8). The resulting condi-tion 0=k3+ fhh(h0)k− i(3�Gk(h0− ln a)2−�k)determines the wave number and the velocity tobe k=kc=�− fhh(h0) and �=3�G(h0− ln a)2,respectively. The small amplitude stationary solu-tions correspond to the neutrally stable modesobtained in the linear stability analysis in Section3 and in consequence exist only in the linearlyunstable film thickness range.

Fig. 2. Characterization of stationary periodic solutions forrelatively large G. Shown are the dependencies of: (a) thevelocity, �, on period; and of (b) the amplitude on period fordifferent mean film thicknesses, h� , as given in the legend of (a).G=0.1, �=0.1 and a=0.1.

Solution families are continued through theparameter space starting at these stationary solu-tions. Fixing, for instance, h� , G and a, solutionswith different periods are obtained. Depending onmean film thickness, h� , families with qualitativelydifferent behavior are found, as shown for arelatively large G in Fig. 2(a) (velocity-perioddependence) and Fig. 2(b) (amplitude-period de-pendence). We distinguish the following familytypes (listed with increasing mean film thickness).1. For very small h� only the trivial solution

h(x)=h0 exists.2. Two branches exist, both continuing towards

L�. This implies the existence of two solu-tions with different velocity, �, and amplitude,A, for every period larger than a minimalperiod. Both branches show a monotonic de-pendence of � and A on period, L. The small-amplitude branch consists of linearly unstablesolutions and the large-amplitude branch oflinearly stable ones (example: h� =0.7).


3. Two branches exist, but only the branch withlarger velocity continues towards L�. Itcontains the linearly stable solutions, whereasthe terminating branch consist of linearly un-stable solutions. Both branches show amonotonic dependence of � on L. Consideringthe period as a bifurcation parameter, theendpoint of the terminating branch corre-sponds to the locus of a subcritical bifurcationfrom uniform solutions to solutions oscillatingin space (example: h� =0.82).

4. One, linearly stable branch exists continuingtowards L�. � and A increasemonotonously with L. The endpoint of thebranch corresponds to the locus of a supercrit-ical bifurcation (h� =1.0).

5. One, linearly stable branch exists continuingtowards L�, A increases monotonously but� depends non-monotonically on period (su-percritical bifurcation, example: h� =1.5).

6. Two branches exist. The low velocity branch islinearly stable, continues towards L� andits velocity depends non-monotonically on pe-riod. The other branch terminates at finiteperiod, is linearly unstable and shows amonotonic dependence of � on L. (subcriticalbifurcation, example: h� =2.5).

7. As 2, the only difference is that the branchwith lower velocity corresponds to larger am-plitude (example: h� =3.5).

8. For very large h� the only solution is the trivialh(x)=h0.

All the families described up to now resemblefamilies of profiles found on the horizontal sub-strate, that are here varied by the inclination ofthe plane. They are not any more reflection sym-metric with respect to the extrema of the profilesand move with a certain velocity. The linearlyunstable branches represent nucleation solutionsas for �=0 [43]. This was shown by direct numer-ical integration of Eq. (6) in Ref. [44]. The nucle-ation solutions give the critical disturbance forlinearly stable but not absolutely stable (i.e.metastable) situations as flat films outside thelinearly unstable thickness range or linearly un-stable flat films with an imposed period smallerthan the smallest linearly unstable wavelength. Atconstant shape, disturbances with smaller ampli-

tude than the critical disturbance will relax to-wards the flat film, whereas an amplitude largerthan critical leads to growth of the disturbance.The system will then reach the solution on theother, the stable branch.

From Fig. 2 one finds that one branch of everyfamily converges to a line common to all families.This implies a certain ‘universality’ of the solu-tions on this line because their velocity and ampli-tude are independent of the mean film thickness.The convergence to the common line occurs atlower period for large mean film thickness thanfor small film thickness. Also the film profileconverges to a common shape: a flat drop with anupper plateau of thickness, hu, sitting on a verythin (precursor) film of thickness hd. At the down-stream front of the drop a capillary ridge forms.For the flat drops only the length of the upperplateau depends on the mean film thickness.

The phenomenology or qualitative behavior ofthe solutions is for G smaller then the criticalvalue at Gc=1/4 nearly independent of G. Thismade possible to discuss in Ref. [44] all the occur-ring processes for a relatively large G between0.05 and 0.2. However, physically realistic valuesfor G are much smaller implying a separation ofthe length scales for the thicknesses of the upperand lower plateau, i.e. of the precursor film thick-ness and drop height. For small � the thicknessesof the upper and lower plateau can be estimatedto be of order �1/G and �G, respectively, imply-ing an order 1/G difference between the twoplateau thicknesses. In the remaining part of thiswork we extend the results of Ref. [44] towardsvery small, more realistic values of G between10−4 and 10−7. Thereby, it is determined whetherin the future it will be possible to describe experi-mental results with the numerical methods usedhere.

Fig. 3 gives the equivalence of a part of Fig. 2for G=10−5 and Fig. 4 shows the correspondingadvancing and receding contact angles defined asthe absolute values of the slope of the profile atthe respective inflection points. For small periodsthey increase strongly from their value for flatfilms that is zero. At larger periods the advancingangle increases slowly with period, whereas thereceding angle decreases. At a period of L=106


Fig. 3. Characterization of stationary periodic solutions.Shown is the dependence of: (a) the amplitude, A; and (b) thevelocity, �, on period, L, for different mean film thicknesses, h�(given in the legend) for G=10−5, �=0.2 and a=0.1.

Fig. 5. Profiles for different period, L, for mean film thicknessh� =10. G=10−5, �=0.2 and a=0.1. Part (a) and (b) showdifferent ranges of the period as given in the legend. Theemergence of flat drops is shown in part (c), where the periodis fixed at L=100 000 and the mean film thickness is increased(values given in legend).

some of the curves in Fig. 2 have already con-verged onto the common line, but for the smallermean film thicknesses still larger periods are nec-essary. The change of the drop profiles with in-creasing period can be seen for h� =10 in Fig. 5(aand b). For relatively small periods the dropskeep their shape, only their height and in conse-quence their velocity (Fig. 3(b)) increase withperiod. Then, at about L=50 000 the shape startsto change: the height increases only slowly but thedrop starts to develop a hunchback that prolongs

Fig. 4. Characterization of stationary periodic solutions.Shown is the dependence of: (a) the advancing dynamic con-tact angle, �a; and (b) the receding dynamic contact angle, �r,on period, L, for different mean film thicknesses, h� (given inthe legend) for G=10−5, �=0.2 and a=0.1.

with increasing period. To see this process moreclearly the period is fixed at L=106 and moreliquid volume is fed into the system by increasingthe mean film thickness as shown in Fig. 5(c).When the profiles are on the common branch,feeding more liquid or prolonging the period onlychanges the length of the upper plateau, whereasthe shape and height of the capillary ridge and theback front remain unchanged.


To underline the resemblance of the behaviorfor small and relatively large G in Fig. 6, a zoomof Fig. 3 is plotted showing the existence ofbranches of nucleation solutions, non-monotonicdependence of velocity on period and, conse-quently, the transition between family types (1–8)as described above (the figure only shows thetransition between families (3) and (4). Although,we will not dwell further on this subject here, takenote, that beside the solution branches havingdirect equivalents in the horizontal case, �=0[12,43], also new surface wave branches appearfor certain film thicknesses that are intrinsic fea-tures of the flowing film. In [44] this type ofsolutions is discussed in connection with a dynam-ical wetting transition with hysteresis occurringwhen changing the inclination. Here, we show theoccurrence of the new branch in Fig. 7, an evenmore detailed zoom of Fig. 3 for a small range offilm thicknesses. There it can be seen how thenucleation branches shown in Fig. 6 bulge outgiving rise to an additional branch consisting oflinearly stable surface waves.

Now, we study the influence of the interactionparameters describing the wetting properties, a,and the ratio of gravitation and molecular inter-action, G. To this purpose again the dependencesof the properties of the profiles on period arecalculated, but now comparing different parame-ter values of a or G. Fig. 8(a and b), Fig. 9(a andb) show the respective change of amplitude, veloc-ity, advancing and receding contact angle with

Fig. 7. Strong zoom of Fig. 3 for small amplitudes showing thesolution branch of linearly stable, non-linear surface wavesthat have no counterpart for �=0.

period for different a. The curves for different Gare shown in Fig. 10(a and b), Fig. 11(a and b),respectively. The wetting properties in the studiedrange of small partial wettability measured by ado not have an influence on the amplitude of theprofiles (Fig. 8(a)), but influence both, velocityand contact angles. The latter can be easily under-stood remembering that in the used scaling aremains only in the mobility factor Q(h, a)=h−ln a and that the performed shift in h (Section 2)implies the existence of a physical precursor filmof at least (− ln a) thickness facilitating the move-

Fig. 8. Dependence of the stationary periodic solutions on theparameter a. Shown is the change of: (a) the amplitude, A ; and(b) the velocity, � with period, L, for different a as given in thelegend for h� =10.0, G=10−5 and �=0.2.Fig. 6. Zoom of Fig. 3 for small amplitudes.


Fig. 9. Dependence of: (a) the advancing dynamic contactangle, �r; and (b) the receding dynamic contact angle, �a, onperiod, L, for different parameter a (given in the legend) forh� =10.0, G=10−5 and �=0.2.

Fig. 11. Dependence of: (a) the advancing dynamic contactangle, �a; and (b) the receding dynamic contact angle,�r onperiod, L, for different parameter G (given in the legend) forh� =10.0, �=0.2 and a=0.1.

ment of the profiles. Fig. 9 shows that in morewettable situations (smaller a) the advancing frontof the drop is less steep, whereas the back frontflattens with decreasing wettability.

The ratio of gravitation and molecular interac-tion, G, influences crucially the dependence of allprofile properties on period (Figs. 10 and 11). Ghas a strong impact on both, the plateau thick-nesses as discussed above for the limit ��0 andthe driving force (last term of Eq. (6)). Thesmaller G the larger becomes the upper plateauthickness implying a slower convergence towardsthe universal flat drop solutions. The curves for

G=10−4 have already converged at L=20 000,whereas the curve for G=10−6 and identicalmean film thickness shows no sign of convergenceeven at L=100 000. Take note, that the velocityis not determined by the amplitude, i.e. it is notlarger for larger amplitude but smaller G as onecould expect giving the mobility factor to muchimportance. It seems to be only determined by thedriving force that is proportional to G. However,the behavior of the dynamic contact angles (Fig.11) is less clear. Taking only the already con-verged curves for G=10−4 and 10−5 into ac-count, both angles seem to increase withdecreasing G.

4.1. Dependence on inclination angle

The dependence of the properties of the flatdrops on the system parameters was already stud-ied for relatively large G [44]. There the velocityincreases monotonously with � and the flat dropbecomes thinner by decreasing its upper plateauthickness and increasing the lower plateau orprecursor film thickness. The relative height of thecapillary ridge as compared to the upper plateauthickness increases with increasing inclination.For larger inclination angles, i.e. larger velocitiesof the drop, secondary oscillations or ‘wiggles’evolve behind the capillary ridge and in the pre-cursor film in front of the drop.

Fig. 10. Dependence of the stationary periodic solutions on theparameter G. Shown are: (a) the amplitude, A ; and (b) thevelocity, � as a function of the period, L, for different G asgiven in the legend for h� =0.9, �=0.2and a=0.1.


In the limit �, ��0 for G�1 the slope of thefunction �(�) at �=0 can be estimated becausethe plateau thicknesses have to converge to thevalues obtained for a drop on a non-inclinedplane (�=0) [43]. These depend only on the prop-erties of the free energy f(h), and are for G�1

hd=�G/2+38

G+O(G3/2) and

hu=�2/G−�216

�G+21

128G+O(G3/2). (16)

Inserting relations Eq. (16) in the equation deter-mining the flat film thicknesses Eq. (10), gives thecondition

�

�

��0

tending to

2−3�2G ln a+�3

4+3(ln a)2�G+O(G3/2).

(17)

This was shown to be a good estimation for theslope of the curve �(�) at �=0 [43].

As described above, fixing the period, the ratioof the length of the upper and the lower plateau isdetermined by the given liquid volume imposinglimits for the existence of the flat drops. Theycannot exist: (i) if there is not enough matter inthe system to form it including the capillary ridgeat its front; and (ii) if there is to much mattercausing the downstream end of the drop to ap-proach the upstream end of its predecessor. Inconsequence, the ends start to interact transform-ing the profile into a moving hole in an otherwiseflat film or into small non-linear waves on anotherwise flat film. In other words, when theupper plateau thickness approaches the mean filmthickness the flat drops cannot longer exist andthe solution families have to depart from theuniversal branch as shown in Fig. 12 for relativelylarge G. With decreasing h� the point of departureshifts towards larger inclination angles. The prop-erties of the non-universal solutions dependstrongly on the mean film thickness (further dis-cussed in [43]). The S-shaped curve for the depen-dency of velocity on inclination angle for some h�in Fig. 12(a) implies the existence of two linearlystable branches that are separated by an linearly

unstable branch acting as nucleation solution: ithas to be overcome from either side in order tojump to the other branch. This has two importanteffects: (I) an apparent stability of the flat film;and (II) the existence of a dynamic wetting transi-tion with hysteresis.

(I) Because the surface waves on the large-ve-locity stable branch have very small ampli-tudes (Fig. 12(b)) they can easily be missedin an experiment that concentrates on dropformation.

(II) The existence of two linearly stablebranches between two inclination angles, �d

and �u, gives rise to hysteresis: changing �

the system jumps between the linearly sta-ble branches: from the large to the smallamplitude branch at �u and from the smallto the large amplitude branch at �d. Thetransition between the two branches maybe called a dynamical wetting transition withhysteresis.

Concentrating again on small G we discuss asan example the dependence of the drop propertieson inclination angle for different mean film thick-

Fig. 12. The dependence of: (a) velocity, � ; and (b) amplitude,A, on inclination angle, �, for different mean film thicknessesas given in the legend. a=0.1, G=0.1 and L=2000.


Fig. 13. The dependence of: (a) amplitude, A ; and (b) velocity,�, on inclination angle, �, for different mean film thickness h� asgiven in the legend. G=10−5, a=0.001 and L=20 000.

Fig. 15. Drop profiles for different inclination angle, �, asgiven in the legend for mean film thickness h� =50. G=10−5,a=0.001 and L=20 000.

than for the flat drops. For the non-universaldrops it decreases with decreasing mean filmthickness due to the lower mobility of smallerdrops. The dynamic contact angles show a veryinteresting behavior (Fig. 14). The receding angledecreases nearly linearly in both regimes with �

(smaller slope �r/� for the flat drops), but theadvancing angle increases with � for non-univer-sal drops and decreases slowly and linearly for theflat drops. This implies the counter intuitive de-crease of the advancing angle with increasingvelocity in the flat drop regime as can be seen forthe larger velocities in Fig. 16. A non-monotonicdependence of the advancing angle on velocitywas already noted in Ref. [44]. Take note, that inthe transition region between non-universal anduniversal drops the overshooting of the velocity(Fig. 13) and the advancing contact angle (Fig.14) over the respective straight lines for the flatdrops results in a dependence of �a on � that isnot one to one as shown in Fig. 16.

nesses, but fixing G=10−5, a=0.001 and theperiod L=20 000. Fig. 13(a and b), Fig. 14(a andb) show the respective dependences of amplitude,velocity, advancing and receding contact angle oninclination angle. Some profiles for different � areshown in Fig. 15. All plots show the two clearlydistinguishable regimes of non-universal drops atsmall � and universal flat drops at larger �. Theinclination angle where the transition occurs de-pends strongly on mean film thickness. For h� =10it is not yet reached at �=10. The velocity de-pends in both regimes linearly on �. However, theslope �/� is larger for the non-universal drops

Fig. 14. The dependence of: (a) the advancing dynamic contactangle, �a; and (b) the receding dynamic contact angle, �r, oninclination angle, �, for different mean film thickness h� asgiven in the legend. G=10−5, a=0.001 and L=20 000.

Fig. 16. The dependence of the advancing dynamic contactangle, �a on velocity, �. For mean film thicknesses, h� , as givenin the legend. G=10−5, a=0.001 and L=20 000.


In the future, the results obtained here forone-dimensional non-universal and flat dropshave to be compared to results for two spatialdimensions. In the following Section 5, we give ashort outlook on the numerical techniques thathave to be used in this case and show some firstresults.

5. Two-dimensional outlook

For the first calculations in two spatial dimen-sions we focused our interest on the evolution ofdifferent initial states on an inclined plane. Wediscuss a liquid layer on a plane that is inclined inx-direction as before. The two-dimensional ver-sion of Eq. (6) reads now for h=h(x, y, t):

�th= −�{(h− ln a)3[�(�h−�h f(h))+�Gex ]},(18)

with ex being the unit vector in x-direction. Tosolve Eq. (18) numerically, the time derivative hasto be discretized, in the simplest case by an Euler-forward scheme. Space can be discretized by finitedifferences. A fully explicit time scheme, however,would lead to severe stability restrictions on thetime step, �t, since on the l.h.s of Eq. (18) one hasterms of the orders

h3�h and h3�2h,

whose absolute values increase rapidly with h.Therefore, it is recommended to take as manyterms of this kind as possible implicitly. This canbe done by introducing the mean film thickness

h� =�V

dx dy h,

which is a conserved quantity and can be consid-ered as a control parameter. Using thetransformation,

u(x, y, t)=h(x, y, t)−h� ,

one can write Eq. (18) as,

�tu=D(h� ) �u−h� 3 �2u+FNL(u, �u, ...) (19)

where the ‘diffusion coefficient’ D turns out to be,

D(h� )=h� 3�−2a

e−h� +4a2 e−2h� +G

�,

and FNL denotes a non-linear function of u and itsspatial derivatives up to fourth order. The firsttwo terms in Eq. (19) are linear and can be takenimplicitly, leading to the scheme

L(�) u(x, y, t+�t)=u(x, y, t)

�t+FNL(t)

= (x, y, t), (20)

with the linear differential operator,

L(�)=1�t

−D(h� )�+h� 3�2 (21)

At each time step, L has to be inverted which canbe done most easily in Fourier space:

u(kx, ky, t+�t)=1

L(−k2) � (kx, ky, t) (22)

where u and �� denote the Fourier transform. Thenon-linearities are treated explicitly. To avoidtime consuming convolution sums they are com-puted in real space (pseudo-spectral method). Thetwo Fourier transforms needed at each step aredone by a standard FFT routine.

The code is implemented on an alpha-worksta-tion and allows there for a spatial resolution up to1048×1048 mesh points. Due to the semi-implicittime integration, the time step can be fixed atvalues of order one. This allows computations forrather large domains in a reasonable time (somehours). For all runs we use fixed parameters G=0.2 and a=0.1 but different initial conditions.1. Deformation and separation of drops. We

take three drops with different size as initialpattern. Feeling the tangential component ofgravity, the drops move and change their formto long ovals. Later-on cusps are formed attheir trailing ends for not too large values of �.For larger values, the cusps get sharper andsmall drops are pinched off (Fig. 17). This isclearly a feature that lies outside the range ofa one-dimensional model. The speed of thedrops increases with their size as in the one-di-mensional case. On smaller drops the cusps aregenerated faster and the separation of sec-ondary drops is seen earlier.


Fig. 17. Time series found by numerical integration of Eq. (18). The layer is inclined in the vertical direction in the figure. The filmfalls downwards. Initial condition are three drops of different volume. After inclining to �=0.2 larger drops travel faster. Theinclination angle shown is large enough to observe secondary drops pinch off from the cusp of the primary drops. We fixed h� =3.2in the metastable region to avoid instability of the flat domains. 256×256 points. G=0.2, a=0.1 and domain size equal to 950.

2. Formation of rivulets. Starting the numericalintegration with an initial pattern of 200 ran-domly distributed small drops, the evolutionshown in Fig. 18 takes place. Drops coalesceto narrow rivulets meandering downwards. Atthe trailing end of a rivulet, drops are pinchedoff from time to time and are left behind. Incontrast to the one-dimensional situation, sta-tionary patterns were not found.

3. Fronts and fingers. Next we study the instabil-ity of a one-dimensional flat drop that is con-tinued homogeneously in the transversaly-direction, i.e. of a liquid ridge that is transla-tion invariant in y-direction. For �=0.2 theadvancing front of the ridge gets transverselyunstable to periodic disturbances (Fig. 19).From these disturbances, more or less equallyspaced fingers grow. The transverse wavenumber of the finger pattern in Fig. 19 isk�0.05, which is about half of the criticalwave number of the flat film instability calcu-lated in Section 3. Qualitatively, the same pic-ture was obtained in experiments with fallingfilms of silicone oil or glycerine [48,28], andnumerically in a very recent work with a dif-ferent disjoining pressure [49].

Finally, Fig. 20 shows an evolution startingfrom a similar initial condition as in Eq. (3),where both fronts, the advancing and the recedingone, are unstable to transverse periodic distur-bances. It is interesting to see that the wavelengthsof the growing inhomogeneities are different forthe two fronts. Growth rate as well as wavelengthdepend also on the inclination of the plane. Fi-nally, rivulets arise as obtained in Fig. 18.

6. Conclusion

We have revised and extended the analysis ofthe evolution of a thin liquid film flowing downan inclined plane. The model of interaction withthe substrate used is based on a disjoining pres-sure arising in a diffuse interface theory couplingvan der Waals equations and thin film hydrody-namics [42]. We have studied flat film solutionsand their linear stability and have found thatalthough the thickness of the flat film solutionsdepends on velocity and inclination angle, theirlinear stability depends only on the first derivativeof the used disjoining pressure as for film ruptureon a horizontal substrate [43], but not on thedynamics of the flow.

Numerical continuation techniques were usedto calculate families of one-dimensional finite am-plitude stationary solutions. Depending on meanfilm thickness different types of solution familieswere found and their linear stability was analyzedleading to the identification of linearly stable solu-tions and linearly unstable nucleation solutions.The solution families found for small inclinationangle correspond to the family types found with-out inclination [43]. However, here the solutionsare asymmetric and slide down the inclined planewith a velocity depending on period and meanfilm thickness. This motion has clearly no coun-terpart in the non-inclined case. Beside thefamilies that have a counterpart in the case �=0there exist stationary non-linear surface wavesthat are specific for ��0. They do only exist ifboth, molecular interaction and viscous flowdriven by gravitation are present and depend


strongly on the ratio of mean film thickness andlength scale of the diffuse interface.

The study of the solutions as a function of theperiod showed the existence of universal flat dropsolutions whose velocity and plateau thicknessesare independent of period and mean film thick-ness. They have the form of flat or pancake dropsof different length with a capillary ridge at theirdownstream end. The dependence of the proper-ties of the non-universal and universal flat dropson period, inclination angle and interactionparameters was determined for both, relativelylarge but unrealistic and very small ratios ofgravitation and molecular interaction. The influ-ence of the parameter a describing the wettingproperties was also determined.

Studying the flat drops we have found that withincreasing inclination angle the upper (lower)plateau thickness decreases (increases), the veloc-ity increases, and the receding dynamic contactangle decreases. For relatively large G the advanc-ing dynamic contact angle shows a non-monotonic behavior as a function of the velocity,it increases first then decreases even below itsstatic equilibrium value [44]. For very small G westudied the transition between non-universal andflat drops when increasing the inclination angleand found in the flat drop regime a monotonicdecrease of the advancing contact angle with in-creasing velocity.

The universal regime breaks down if one of theplateau thicknesses approaches the mean filmthickness. The study of the non-universal familieshas revealed an hysteresis effect, when jumps be-

tween a small and a large amplitude solutionoccur that both exist for a certain range of incli-nation angles for some mean film thicknesses. Theexistence of the branch of stable small amplitudesolutions may in an experiment lead to the im-pression of stable flat film flow in the linearlyunstable mean thickness range, because no largeamplitude solutions can be seen without overcom-ing a nucleation solution. The occurring surfacewave may go unnoticed because of their verysmall amplitude.

A first outlook towards two-dimensional calcu-lations showed the transition from oval stationarydrops, to stationary drops with a cusp at its back,and further to drops that periodically pinch offsmall satellite drops at the cusp. This three typesof drops are exactly the ones found recently in anexperiment [40]. In the study of the stability offronts we have found both, transversal instabili-ties of advancing and receding fronts. To ourknowledge, this is the first report on recedingfronts found in theoretical models for the move-ment of liquid fronts. The findings may be appli-cable to instabilities found for receding fronts indewetting [50,51].

Most work done on liquid sheets or ridgesflowing down an inclined plane [28,35,52] takes atotally different approach than we have takenhere. Based on the fact that they study situationsthat are a superposition of spreading and slidinggoing on forever, they analyze separately the threeregions: (1) upstream end; (2) central part; and (3)downstream end of the ridge and scale themdifferently in time assuming respective power-law

Fig. 18. The same as Fig. 17 but for a larger domain and 200 initial drops. After a long time, rivulets are formed streamingdownwards. 512×512 points for a domain size of 1900.


Fig. 19. Falling film, where the initial condition is a liquid ridge as explained in the main text. The advancing front gets unstableto a periodic transverse instability and fingering can be observed. 256×256 points. G=0.2, �=0.1, a=0.2 and domain size 950.

dependence. Then they determine similarity solu-tions in the regions (1), (2) and (3) and match themtogether. An additional problem of most models inthe literature is that in slip models the dynamiccontact angle is explicitly fixed at the contact line[31,36] or varies in a prescribed way with velocity[35]. In a precursor film model, however, thecontact angle is zero without motion although withmotion a mesoscale dynamic contact angle de-pends on the imposed velocity.

On the contrary, here we do not study thesuperposition of spreading and sliding, but themovement of sliding drops of constant shape in amoving frame as observed by Podgorski [40].When the inclination angle vanishes, the shape ofour droplets converges to the equilibrium shape ofa drop of a partially wetting liquid on a horizontalplane. All the ad-hoc parameters in the aboveapproach (static and dynamic contact angle, dropvelocity, drop and precursor film thickness) arededuced within our theory from the two parame-ters describing wetting properties and ratio ofgravitation to molecular interaction. If the liquidvolume in an individual drop is small, also thevolume influences the drop properties.

Because the rupture of a thin film on a horizon-tal substrate due to molecular interactions isclosely related to the decomposition of a binarymixture as described by the Cahn–Hilliard equa-tion [53,9,43], the sliding drops treated here havea close relationship to moving concentration profi-les in a convective Cahn–Hilliard model [45]. Theconvective Cahn–Hilliard equation can be seen asa special case of our film evolution Eq. (6). Taking and g to be small deviations defined by h(x)=hc+(x) and G=Gc+g, close to the critical point

(Gc=1/4, hc= ln 4) (see Section 3) (�hf can beexpressed as a cubic [43] and Eq. (6) writes

�t= −�x

(hc− ln a+)3

×�

�x(�xx−f4

63−g)+�g

n, (23)

where f4 is short for �hhhh f �h c. Assuming �1,

g=O(), f4=O(−1), x=O(−1/2), �=O(−1/2)and t=O(−2), in lowest order of Eq. (23) takesthe form:

�t= −�xx�

�xx−f4

63−g

n+

6�gh c

2 �x. (24)

Another scaling of t, x and h transforms thisequation into,

�t= −�xx [�xx−3− ]+��x, (25)

with � the only remaining parameter. This is theconvective Cahn–Hilliard equation studied in [45]and itself related to the Kuramoto–Sivashinskyequation used to describe formation of small am-plitude structures on falling liquid films [1]. Thisanalogy has some consequences that should befurther exploited in the future. So it can be ex-pected that something similar to the dynamicalwetting transition with hysteresis described in Ref.[44], i.e. the existence of both, linearly stable smalland large amplitude concentration profiles foridentical driving �, can be found in the convectiveCahn–Hilliard model. Also the results for theconvective Cahn–Hilliard model [45] on thecrossover between concentration profiles unstableand stable to coarsening lead to the hypothesis ofan analogue to this transition for drop profiles onan inclined plane at least for certain values ofmean film thickness and inclination angle.


Fig. 20. The same as Fig. 19 but for a larger geometry and smaller angle �. Also the trailing edge of the wall is now unstable.512×512 points, �=0.1 and domain size 1900.

Acknowledgements

This research was supported by the EuropeanUnion under ICOPAC grant HPRN-CT-2000-00136, by the German Academic Exchange Board(DAAD) under grant D/98/14745, by theDeutsche Forschungsgemeinschaft under grantTH781/1-1 and by the Spanish Ministry of Educa-tion and Culture under grant PB 96-599. Wewould like to thank U. Bahr, G. Diener, E.Knobloch, L. Pismen and V. Starov for helpfuldiscussions.

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