Journal of Colloid and Interface Science 355 (2011) 243–251

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Journal of Colloid and Interface Science

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Stability of a volatile liquid film spreading along a heterogeneously-heated substrate

Naveen Tiwari, Jeffrey M. Davis ⇑Department of Chemical Engineering, University of Massachusetts, 686 North Pleasant Street, Amherst, MA 01003, USA

a r t i c l e i n f o a b s t r a c t

Article history:Received 17 September 2010Accepted 23 November 2010Available online 30 November 2010

Keywords:Thin liquid filmEvaporationFingering instabilityMarangoni effectMicrofluidicsContact line

0021-9797/$ - see front matter � 2010 Elsevier Inc. Adoi:10.1016/j.jcis.2010.11.071

⇑ Corresponding author.E-mail address: jmdavis@ecs.umass.edu (J.M. DaviURL: http://www.ecs.umass.edu/che/faculty/davis

The dynamics and stability of a thin, viscous film of volatile liquid flowing under the influence of gravityover a non-uniformly heated substrate are investigated using lubrication theory. Attention is focused onthe regime in which evaporation balances the flow due to gravity. The film terminates above the heater atan apparent contact line, with a microscopically thin precursor film adsorbed due to the disjoining pres-sure. The film develops a weak thermocapillary ridge due to the Marangoni stress at the upstream edge ofthe heated region. As for spreading films, a more significant ridge is formed near the apparent contactline. For weak Marangoni effects, the film evolves to a steady profile. For stronger Marangoni effects,the film evolves to a time-periodic state. Results of a linear stability analysis reveal that the steady filmis unstable to transverse perturbations above a critical value of the Marangoni parameter, leading to fin-ger formation at the contact line. The streamwise extent of the fingers is limited by evaporation. Thetime-periodic profiles are always unstable, leading to the formation of periodically-oscillating fingers.For rectangular heaters, the film profiles after instability onset are consistent with images from publishedexperimental studies.

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1. Introduction

Microscale flows with free surfaces have attracted considerableattention due to their widespread technological applications [1,2].Many of these applications involve flow over heated surfaces,including materials processing flows, thermal management of elec-tronic devices, micro-electro-mechanical systems and microfluidicdevices, and separation processes. The variation in temperature ofa liquid–gas interface causes a gradient in the surface tension, orMarangoni stress [3]. In thin liquid layers, the thermocapillary flowassociated with these tangential stresses can lead to significantinterfacial deformation, instability, and pattern formation [4,5].Furthermore, evaporation of a volatile liquid can significantlyaffect its motion and stability [6–8].

The dynamics of a volatile liquid film spreading under theinfluence of gravity along an inclined plane with a semi-infiniteheater was studied by Ajaev [8], who extended the model of Ajaevand Homsy [9]. The stress singularity at the moving contact linewas removed through the presence of a microscopic precursor filmwith a thickness determined from the condition of zero mass flux,as the disjoining pressure causes vapor molecules to adhere to thesubstrate, creating a thin, adsorbed film. For weak evaporation, theapparent contact line was found to move at a constant speed as thefilm advances. For very strong evaporation, steady profiles

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s)..html (J.M. Davis).

(i.e., stationary contact lines) were obtained as the liquid’s flowrate is balanced by the evaporative mass flux above the heater.

These (steady) profiles appear similar to the traveling waves forspreading liquid films with moving contact lines when viewed in amoving reference frame [10–12]. In particular, the treatment of thecontact line by Ajaev [8] is phenomenologically similar to thewidely-used precursor film model [11,13] for a perfectly wettingliquid, particularly when Van der Waals interactions are included[14,15]. The key difference is that for a volatile film, the thicknessof the precursor film is not arbitrary but determined from the con-dition of zero mass flux. In most instances, the specific model forthe contact line does not significantly alter the film dynamicsand stability, and quantitative agreement has been shown betweenthe precursor film model and a model based on slip at the contactline, which allows independent specification of the contact slope[16–18]. The emphasis of most studies has been on the fingeringinstability of the moving contact line.

There have also been many studies of fingering and rivulet insta-bilities in flows of volatile liquid films. For example, Lyushnin et al.[19] studied via linear stability analysis the fingering instability ofisothermal, ultrathin evaporating liquid films as dry spots grow.Motivated by experiments and simulations revealing rivulet forma-tion in liquid films flowing over a rectangular heater [20–22], Tiwariand Davis [7,23] studied the stability of a volatile film flowing overan inclined substrate with localized heating. The heater wassufficiently narrow and evaporation sufficiently weak that the filmdid not rupture to form a contact line. It has been shown that thisrivulet instability can be suppressed for a range of values of the

244 N. Tiwari, J.M. Davis / Journal of Colloid and Interface Science 355 (2011) 243–251

Marangoni parameter and Biot number by modifications to the sub-strate temperature profile via feedback control [24] or by the intro-duction of suitable topographical features on the substrate [25].Klentzman and Ajaev [26] recently studied the two-dimensionalevolution of spreading, volatile films by numerically integratingthe governing evolution equation in a rectangular domain. Theiremphasis was on the effect of evaporation on fingering instabilitiesfor weak thermocapillary effects. A linear stability analysis of theone-dimensional film profiles has not yet been performed.

In this paper, the linear stability analysis of liquid films flowingover a nonuniformly-heated wall is extended to a volatile liquid thatterminates in a contact line. A long-wave, lubrication analysis isused to reduce the momentum and energy equations to a nonlinearpartial differential equation that governs the spatial and temporalevolution of the local film thickness in Section 2. A one-sided modelfor evaporation is employed [6,7]. This study is focused on theregime in which evaporation balances the flow rate of liquid downthe plane, leading to steady and time-periodic one-dimensionalprofiles that are presented in Section 3. The stability of these baseprofiles to transverse perturbations is studied in Section 4, and astate-space map is constructed to delineate regimes that producestable (steady), unstable, and time-periodic profiles. The two-dimensional evolution of the film after instability onset is trackedusing numerical simulations, with results presented in Section 5.Conclusions are presented in Section 6.

2. Problem formulation

Shown schematically in Fig. 1 is a thin film of volatile liquidflowing due to gravity over a heterogeneously-heated plate in-clined from horizontal by an angle h. The liquid has density q, vis-cosity l, and surface tension c. The film flows in the x-directionwith a constant film thickness h1 far upstream of the heater,x ? �1. The coordinate z is directed normal the plate, and y isthe transverse coordinate in the plane of the substrate. For a verti-cal plate, the mean fluid velocity upstream of the heater isU ¼ qgh2

0=ð3lÞ, where g is the gravitational acceleration. Thenon-uniformity in the heating of the substrate is modeled hereas a prescribed temperature variation along the surface of theplate, T(x,y,z = 0) � T0(x). The interfacial temperature of the filmincreases as the upstream edge of the heater is approached, andthe resulting Marangoni stress opposes the flow due to gravity,which induces a thermocapillary ridge. This ridge has been shownto evolve into an array of rivulets aligned with the direction of flowfor a sufficiently large temperature increase or to exhibit time-periodic behavior [7]. Another capillary ridge forms just upstreamof the apparent contact line, and similar capillary ridges in spread-ing films are susceptible to a fingering instability [11,26].

Using long-wave theory to simplify the governing momentumand energy equations, a lubrication equation has been derivedfor a volatile liquid film flowing over a non-uniformly heated

Fig. 1. Schematic diagram of a thin film of volatile liquid flowing over a heater. Athermocapillary ridge forms at the upstream edge of the heater, and a capillaryridge forms near the contact line. A thin precursor film is adsorbed on the substrateabove the heater due to the disjoining pressure.

surface by Tiwari and Davis [7], with slightly different scalingsthan those used by Ajaev[8,26]. In dimensional variables, thepartial differential equation that governs the evolution of the localfilm thickness, hðx; y; tÞ, is

@h@tþ 1

lr �

"ðqg sin hex þ rðc0r2hþ bP

�qgh cos hÞÞ h3

3� cT

h2

2rbT i

#¼ �

bJq: ð1Þ

In Eq. (1), dimensional variables are denoted by �. The evaporativemass flux at the interface is bJ , and bP is the disjoining pressure term.To account for the change in surface tension with temperature, alinear variation is assumed,[27] cðbT Þ ¼ c0 � cTðbT � bT1Þ wherecT > 0 and cT ¼ @c=@bT . Here c0 is the surface tension at the ambienttemperature, bT1, which is the equilibrium saturation temperature.Changes of q and l with T are neglected, and the variation of c isrestricted to the Marangoni term [27]. Eq. (1) is valid forCa1/3� 1 and Ca1/3Re� 1, where Ca � lU/c0 is the capillarynumber and Re = qU h1=l is the Reynolds number.

To non-dimensionalize Eq. (1), the constant, upstream filmthickness, h1, and the dynamic capillary length, lc ¼ h1Ca�1=3, areused as the appropriate length scales in the z and x-directions,respectively. The velocity scale in the x-direction is U. The temper-ature is non-dimensionalized as T ¼ ðbT � bT1Þ=MbT , where MbT is themaximum temperature increase at the heater relative to the iso-thermal film far upstream. The evaporative mass flux is scaled asJ ¼ bJMHvaph1=ðkMbT Þ, where MHvap is the latent heat of vaporizationand k is the thermal conductivity of the liquid.

The dimensionless form of Eq. (1) is

@h@tþr � ðsin hex þrðr2hþPÞÞh

3

3�M

h2

2rTi

" #¼ �EJ: ð2Þ

The disjoining pressure term P is small for the bulk film but is retainedas an O(1) quantity because it dominates as the ultrathin, adsorbedfilm is approached. The parameter E ¼ ðkMbT Þ=ðqUh1MHvapÞ is adimensionless evaporation number that is the ratio of themaximum heat flux through the film, kMbT=h1, to the heat fluxrequired to vaporize the liquid at the rate it is supplied by gravity,qUMHvap. The Marangoni parameter M = cTMT/(c0Ca2/3) quantifiesthe magnitude of the surface-tension gradient from the nonuniformtemperature profile at the heater.

Energy transport by convection in the liquid is negligible if PeCa1/3� 1, where Pe ¼ Uh1=ath is the Peclet number and ath is thethermal diffusivity. The term h1=lc � Ca1=3 appears in this con-straint due to the thinness of the film, which motivates differentscalings for the x- and z-directions. The interfacial temperature,Ti, is then found from the simplified energy balance, Tzz = 0, subjectto boundary conditions at the solid substrate and free surface. Thetemperature is specified along the substrate, T(x,y,z = 0) = T0(x). Atthe free surface, z = h(x,y, t), the dimensionless non-equilibriumcondition is [8]

KJ ¼ �dðr2hþPÞ þ Ti; ð3Þ

where K ¼ ðkbT 3=21 ð2pRg=MwÞ1=2

=ah1qvMH2

vapÞ behaves as an inverseBiot number that quantifies the resistance to heat transfer from thefree surface to the surrounding gas relative to the thermal resis-tance from conduction across the liquid film [7,28]. In the definitionof K, Rg is the universal gas constant, Mw is the molecular weight ofthe liquid, and qv is the vapor density at bT1. The parameter a < 1 isthe ratio of the observed evaporation rate to the rate predicted bykinetic theory and is typically much smaller than unity, especiallyfor polar liquids or in the presence of trace impurities at the inter-face [29]. Another condition is needed to relate interfacial temper-ature Ti(x) to the temperature at the solid substrate, T0(x), and can

N. Tiwari, J.M. Davis / Journal of Colloid and Interface Science 355 (2011) 243–251 245

be obtained by equating heat conduction through the liquid film tothe latent heat required by the evaporative mass loss. In dimension-less form, this constraint is

J ¼ �@T@z

����z¼h

: ð4Þ

The temperature of the liquid-gas interface is then

TiðxÞ ¼ KT0ðxÞ þ dhðr2hþPÞðK þ hÞ : ð5Þ

The dimensionless rate of mass loss due to evaporation is foundfrom Eqs. (3) and (5),

J ¼ T0ðxÞ � dðr2hþPÞðK þ hÞ : ð6Þ

The scaled disjoining pressure term is assumed to be P = e/h3 toaccount for Van der Waals interactions, where e� 1 is a material-dependent property.

The boundary conditions for Eq. (2) as x ? �1 are

h! 1; hx ! 0: ð7Þ

At the downstream end of the domain (corresponding to the heatedregion of the substrate), the solid surface is assumed to be macro-scopically dry, and the evaporative mass flux is suppressed by theattractive disjoining pressure. Using this formulation, the precursorfilm thickness, b, is obtained by setting J = 0 in Eq. (6), which givesb = (ed)1/3. The boundary conditions as x ?1 are therefore

h! b; hx ! 0: ð8Þ

Fig. 2. Steady base profiles for a volatile liquid film flowing over a semi-infiniteheater for several values of the Marangoni number, M, with the temperature profileT0(x) superimposed. The dimensionless thickness of the precursor film is b = 0.004.

3. Base profiles

The one-dimensional base profile is found by substitutingh(x,y, t) = h0(x, t) into Eq. (2), which yields

@h0

@tþ EJ0 þ

@

@xðsin hþ h0xxx þP0xÞ

h30

3�MTi

0xh2

0

2

" #¼ 0; ð9Þ

where the mass flux is

J0 ¼T0 � dðh0xx þP0ÞðK þ h0Þ

; ð10Þ

the disjoining pressure is

P0 ¼ e=h30; ð11Þ

and the interfacial temperature is

Ti0ðxÞ ¼

KT0 þ dðh0xx þP0ÞðK þ h0Þ

: ð12Þ

The base states were computed for a finite-domain, x 2 [�L1,L2]with h0(x =�L1) = 1, h0x(x =�L1) = 0, h0(x = L2) = b, and h0x(x = L2) = 0.Typically, L1 = L2 = 40. The numerical solution was found using themethod of lines, with fourth-order, centered finite differences usedfor the spatial discretization. Time stepping was provided by theMATLAB solver ode15 s. Convergence was verified by successive gridrefinement and variation of L1 and L2. One-dimensional base stateswere determined for different values of M, E, K, and h with aprescribed temperature field T0(x) along the substrate.

The stability of a volatile film spreading along a heated surfacehas recently been studied by evolving the analog of Eq. (2) in a rect-angular domain with periodic boundary conditions in the transversedirection [26]. For weak evaporation, the fingering instability wasfound to be similar to that for nonvolatile films [16,30,31]. Becausemass loss diminishes the amplitude of the capillary ridge near theadvancing contact line, evaporation was found to diminish the rate

of finger growth and even suppress the instability for large evapora-tion numbers. Attention in the present work is therefore focused onthe regimes that give rise to steady or time-periodic base states,which have not been studied previously. A linear stability analysisis used to assess the stability of these base states to transverse per-turbations of arbitrary wave number.

The substrate temperature is modeled as

T0 ¼ 0:5ð1þ tanhðxÞÞ: ð13Þ

The base profiles were computed by evolving Eq. (9) in time until asteady or time-periodic state is attained, with the initial condition

h0ðx; t ¼ 0Þ ¼ bþ 0:5ð1� bÞð1þ tanhð�xÞÞ; ð14Þ

which satisfies the boundary conditions far upstream and down-stream. For most of the results, e = 10�2 and d = 10�3, givingb ’ 0.02. The values of the parameters M, K, and E correspond to afilm thickness of O(10 lm).

Shown in Fig. 2 are steady base profiles for K = 0.1, E = 0.1, h = p/4,b = 0.004, and several Marangoni numbers. The dotted curve is thetemperature profile at the solid substrate. The base profile forM = 0 has a small capillary ridge near the contact line, which issimilar to the ridge that develops for the spreading of a non-volatilefilm [11], although in the present case the contact line is stationary.As the Marangoni number is increased, the height of the ridgeincreases, and the depression upstream of the ridge becomes morepronounced. As seen from Eq. (5), thicker regions of the film arecooler than thinner regions, and Marangoni stresses promoteaccumulation of liquid in the ridge. A smaller thermocapillary ridgealso forms at the upstream edge of the heater near x = 0 because thethermocapillary flow opposes the flow due to gravity, diminishingthe mean velocity and causing the film to thicken locally.

An example of an oscillating, time-periodic base profile isshown in Fig. 3 for M = 4, K = 2, and E = 0.1. The period is Tper � 45,and the profile is plotted at an interval of Mt = 11. For this case ofstrong evaporation, the film terminates at a contact line close tothe upstream edge of the heater, and the ridge induced by theMarangoni stress at the upstream edge of the heater and the ridgedue to the contact line interact, as seen from the profile at t = 155.At t = 166, the two ridges are well separated and of comparableheight. Because of the strong evaporation above the heater, thefilm ruptures at the depression between the two ridges byt = 177, and the forward ridge is shed as a two-dimensional dropletthat continues to propagate downstream and evaporate. Theremaining ridge increases in height, as shown for t = 188, and theincreased mobility of this thicker ridge causes it to evolve underthe influence of gravity back to the structure shown for t = 155,and the oscillations continue. Similar oscillatory behavior has also

Fig. 3. Oscillating base profiles for M = 4, K = 2, and E = 0.1. The dimensionlessthickness of the precursor film is b = 0.02.

246 N. Tiwari, J.M. Davis / Journal of Colloid and Interface Science 355 (2011) 243–251

been observed for falling liquid films without contact lines thatflow over a finite-width heater, with this behavior attributable tothe interaction between gravity, capillarity, Marangoni stresses,and evaporation [7].

4. Linear stability

In order to determine the linear stability of the base profiles toperturbations that vary sinusoidally in the transverse direction, theone-dimensional base state, h0(x, t), is perturbed ash(x,y, t) = h0(x, t) + �G(x, t)exp(iqy), where �� 1. Due to the pertur-bation of the film thickness in Eq. (2), P = P0 + �G(x, t)P1, where

P1 ¼ �3e=h40: ð15Þ

Also, Ti ¼ Ti0 þ �Gðx; tÞT

i1 þ �GxxðxÞTi

2, where

Ti1 ¼

dP1h0 þ dðh0xx þP0Þ � q2dh0 � Ti0

ðK þ h0Þ;

Ti2 ¼

dh0

ðK þ h0Þ:

ð16Þ

Similarly, the mass flux becomes J = J0 + �G(x, t)J1 + �Gxx(x)J2, where

J1 ¼ðq2 �P1Þd� J0

ðK þ h0Þ;

J2 ¼ �d

ðK þ h0Þ:

ð17Þ

The parameters with a subscript ‘‘0’’ correspond to the base stateand are given by Eqs. (10)–(12).

Collecting O(�) terms from the perturbation of the base state,the linearized equation governing the evolution of small perturba-tions is obtained,

@G@t¼Xi¼4

i¼0

Li@iG@xi

; ð18Þ

where

L0¼� h20ðsinhþh0xxxþP0xxxÞ

� �xþ

h30P1x

� �x

3�M Ti

0xh0

� �x�

M Ti1xh0

� �x

2

24þq2MTi

1h20

2þq4h3

0

3þEJ1

#;

L1¼� h20ðsinhþh0xxxþP0xxxÞ�q2h2

0h0x�MTi0xh0�

MTi1xh2

0

2�MðTi

1h20Þx

2

"

�P0xh20þðh3

0P1Þx3

þh30P1x

3

#;

L2 ¼ � �2q2h3

0

3�MTi

1h20

2�MTi

2xh20

2þ q2MTi

2h20

2þ h3

0P1

3þ EJ2

" #;

L3 ¼ � h20h0x �

MTi2xh2

0

2�MTi

2h0h0x

2

" #;

L4 ¼ �h3

0

3�MTi

2h20

2

" #: ð19Þ

Eq. (18) is solved subject to the boundary conditions

Gðx! �1; tÞ ! 0; Gxðx! �1; tÞ ! 0 ð20Þ

to focus on perturbations that are localized near the contact line.When discretized in space using fourth-order, centered finite

differences, Eqs. (18) and (20) yield a linear system of equationsthat can be written in vector form as

@G@t¼ AG; ð21Þ

where A is a matrix and G is the discretized form of G. Assuming expo-nential time dependence for G, G(x, t) = G(x)exp(bt), yields the eigen-value problem, bG = AG. For stationary base states, h0(x, t) = h0(x), theleading eigenvalue of A was found using the MATLAB 6.1 function eig

for different values of the wave number q. For unsteady base states,Eq. (18) was integrated in time in parallel with Eq. (9), and thelong-time growth rate of G was determined. Convergence was veri-fied for the linear stability results by successive refinement of thecomputational grid and variation of the spatial domain.

The thermocapillary ridges induced by localized heating and thecapillary ridges near moving contact lines have been shown to beunstable to transverse perturbations, leading to the formation riv-ulets or fingers [7,11,17,23,32,33]. The stationary film profilesexamined here are also unstable to transverse perturbations abovea critical value of M. Shown in Fig. 4 are the base profiles and dis-persion curves for K = 0.1, E = 0.1, h = p/4, b = 0.02, and several M.Because evaporation is strong, the steady base state terminatesnear the upstream edge of the heater, and the amplitude of theridge increases with M. By contrast with Fig. 2 with weaker evap-oration, there is only one capillary ridge because the film extendsonly slightly onto the heater. In the absence of Marangoni stresses(M = 0), the film is stable, as seen from Fig. 4(b). As M is increased,the base profile becomes unstable, and the dominant eigenvaluefrom the linear stability problem, bmax �maxqb, increases withM. This behavior is linked to the increasing height of the capillaryridge of the base profiles with increasing M, which is similar to thebehavior of isothermal films with moving contact lines as the influ-ence of hydrostatic pressure is diminished [34].

The dispersion curves are different than those for the long-waveinstability typical of spreading films. For the perturbations that de-cay as x ? ±1, b(q = 0) = 0 is not an eigenvalue, as the nonuniformtemperature profile T0(x) breaks the translational invariance of Eq.(2), and perturbations to the downstream edge of the capillaryridge decay due to evaporation, so b(q = 0) < 0. Similar behavior isfound for falling films (without contact lines) that flow over a lo-cally-heated surface, although discrete eigenfunctions are foundonly for particular ranges of q [7,35]. In that problem the continu-ous spectrum (with eigenfunctions that attain constant, nonzerovalues away from the heater) given by b = �q4 does pass throughb(q = 0) = 0, but the corresponding spectrum is not shown here be-cause discrete modes are found for all q. For spreading films, bycontrast, the base profile is translationally invariant, and G = h0x

is therefore a neutrally-stable solution to the eigenvalue problemcorresponding to Eq. (18) for q = 0 that satisfies the decay bound-ary conditions far upstream and downstream of the ridge.

In Fig. 5, base states and dispersion curves are shown for weakerevaporation, E = 0.01, and several Marangoni numbers, with other

(a)

(b)

Fig. 4. Steady base profiles and dispersion curves for K = 0.1, E = 0.1, h = p/4,b = 0.02, and several values of the Marangoni number.

(a)

(b)

Fig. 5. Steady base profiles and dispersion curves for K = 0.1, E = 0.01, h = p/4,b = 0.02, and several values of the Marangoni number, M.

(a)

(b)

Fig. 6. Steady base profiles and dispersion curves for M = 4, E = 0.1, h = p/4, b = 0.02,and several values of the volatility parameter, K.

N. Tiwari, J.M. Davis / Journal of Colloid and Interface Science 355 (2011) 243–251 247

parameters the same as in Fig. 4. The profile without Marangoni ef-fects, M = 0, is again stable to transverse perturbations of any wavenumber. The film becomes unstable above a critical value of M. Bycomparing of the dispersion curves for M = 4 in Figs. 4 and 5, it canbe seen that E has very little influence on the maximum eigenvaluefor these parameter values, but the wavelength of the most unsta-ble mode decreases as E decreases. The dependence of bmax and itscorresponding wave number on E varies with different M and K(not shown).

In the evaporation model, K is an important parameter thatquantifies the volatility of the liquid, with a larger value of K corre-sponding to a less volatile liquid. This parameter plays an analo-gous role in its influence on Ti as the inverse of the Biot numberfor non-volatile liquids. [7] It is shown in Fig. 6(a) that the variationin the value of K has a non-monotonic effect on the height of thecapillary ridge. The highest ridge in the figure appears for K = 0.5because the thermocapillary ridge induced by the Marangoni stressoptimally reinforces the capillary ridge near the contact line. As Kis decreased, more liquid evaporates at the upstream edge of theheater, and the film does not extend as far in the streamwise direc-tion. If K is increased, the height of the ridge decreases again be-cause the ridges due to the Marangoni stress and the contact linebegin to separate, as seen from the curves for K = 1 and 1.1. Thisseparation between the two ridges increases for a smaller valueof the evaporation number, E. For K P 1.2, a steady profile wasnot obtained for the parameter values in Fig. 6.

The dispersion curves that correspond to the base profiles inFig. 6(a) are shown in Fig. 6(b). The maximum growth rate of theperturbations increases as K is increased from 0.1 to 0.5, corre-sponding to a less volatile film and a larger capillary ridge. Thisbehavior is consistent with the earlier studies that show that theperturbation growth rate is larger for a ridge of larger amplitude

248 N. Tiwari, J.M. Davis / Journal of Colloid and Interface Science 355 (2011) 243–251

for spreading films.[11,34] As K is increased from 0.5 to 1.1, corre-sponding to a separation between the capillary ridge at the contactline and the thermocapillary ridge, the maximum eigenvalue isnearly constant, but the most unstable wave number increases.

The liquid flux in the x-direction depends on the inclination an-gle, as the mean velocity of the film upstream of the heater isUsinh. The gravitational driving force therefore decreases relativeto the Marangoni stress as the substrate becomes more horizontal,and the steady film shape changes. The effect of h is shown inFig. 7(a) for M = 4, K = 0.1, and E = 0.1. As the angle of inclinationis decreased from h = p/2 to h = p/4, the Marangoni stress increasesrelative to gravity, which results in a larger thermocapillary ridge.In addition, evaporation above the heated portion of the substratebecomes relatively more significant. The film extends a smallerdistance over the heater, and the amplitude of the capillary ridgedecreases once h falls below a critical value because of the dimin-ished interaction of the Marangoni stress with the capillary ridgecaused by the contact line, as shown for h = p/6.

The dispersion curves corresponding to Fig. 7(a) are shown inFig. 7(b). As h is decreased, the maximum growth rate of perturba-tions decreases, and the film becomes linearly stable for suffi-ciently small inclination angles. This behavior is consistent withthe results of Klentzman and Ajaev [26], who found that evapora-tion stabilizes spreading films of volatile liquid, with the criticalevaporation number required to stabilize the film decreasing asthe inclination angle decreases. Because the hydrostatic compo-nent of the pressure is neglected for these thin films, this stabiliza-tion is the result of more significant evaporation relative to gravity,which is destabilizing due to mobility differences experienced byperturbations to the film thickness [16].

For some parameter values, the film evolves to a time-periodicbase state, such as that shown in Fig. 3 for M = 4, K = 2, E = 0.1, andh = p/4. The stability analysis of this oscillating profile was

(a)

(b)

Fig. 7. Steady base profiles and dispersion curves for M = 4, K = 0.1, E = 0.1, b = 0.02,and several values of the inclination angle, h.

performed by evolving Eq. (9) in time until the time-periodic stateis attained at t = t�, and then solving Eq. (21) simultaneously withan initial Gaussian perturbation. The growth of the perturbationwas quantified from the amplification ratio, kG(t)k/kG(t = 0)k. Ascan be observed from Fig. 8(a), for sufficiently long times a constantgrowth rate with periodic oscillations is observed. For comparisonwith the dispersion curves presented earlier, the asymptotic growthrate is quantified by

b0 ¼ limt!1

sup t�1 lnkGðtÞkkGðt ¼ 0Þk ; ð22Þ

which is similar to the first Lyapunov exponent. The growth rate b0

was calculated for various wave numbers and plotted as a disper-sion curve, as shown in Fig. 8(b). The asymptotic growth rate andmost unstable wave number are comparable to those for the steadybase states shown earlier.

The shape of the evolved perturbation is shown in Fig. 9 at fourtimes separated by an interval Dt = 12 for a perturbation of wavenumber q = 0.5. To elucidate the regions of the film that are mostaffected by the perturbation, the film profiles have been superim-posed as dotted curves. The perturbation is localized near the con-tact line and forward portion of the capillary ridge, as found inearlier studies on the stability of thin, spreading films with movingcontact lines [16,17,34]. The thermocapillary ridge that remains atthe upstream edge of the heater is minimally affected by the grow-ing perturbation, indicating that the instability is associated withthe ridge formed because of the contact line. The instability is thusdistinct from that of the thermocapillary ridge for a continuousfilm flowing over a narrow heater investigated by Tiwari and Davis[7,23].

(a)

(b)

Fig. 8. (a) Temporal evolution of the perturbation amplification ratio for thetime-periodic base profile for M = 4, K = 2, E = 0.1, h = p/4, and b = 0.2. Theone-dimensional, time-periodic base state is attained at time t⁄. (b) Dispersion curveshowing the long-time, exponential growth rate of perturbations corresponding tothe evolution in (a).

(a) (b)

(c) (d)

Fig. 9. Oscillating base profiles (dashed) and evolved perturbations (solid) for M = 4, K = 2, E = 0.1, h = p/4, and b = 0.02 at time (a) t = 200, (b) t = 212, (c) t = 224, and(d) t = 236.

N. Tiwari, J.M. Davis / Journal of Colloid and Interface Science 355 (2011) 243–251 249

The parameter space involving the Marangoni number and theevaporation parameters K and E can be divided into regions ofstable, unstable, and oscillatory profiles, as shown in Fig. 10 forK = 0.1 and h = p/4. As M is increased for constant K and E, the filmtransitions from linearly stable to linearly unstable. A further in-crease in M leads to an oscillatory, time-periodic profile. As oscilla-tory profiles develop for values of M above the critical Marangoninumber at which the film becomes unstable, the two-dimensional,time-periodic profiles are always unstable with respect totransverse perturbations. This instability was confirmed by a linearstability analysis and by direct simulations of the nonlinear, two-dimensional film evolution (h(x,y, t)).

As the evaporation number is increased, the critical Marangoninumber decreases slightly for the parameters of Fig. 10. For largerE, the evaporative mass flux is larger relative to the flow rate of the

Fig. 10. Parameter space that delineates the two-dimensional base profiles that aresteady and stable, steady and unstable to transverse perturbations, and time-periodic (and unstable to transverse perturbations) for K = 0.1, h = p/4, and b = 0.02.

film, and the film terminates at a contact line further upstream.The Marangoni stress at the upstream edge of the heater enhancesthe capillary ridge near the contact line, which destabilizes thefilm. This behavior is analogous to that shown in Fig. 6 as K is var-ied, as the position of the capillary ridge relative to the upstreamedge of the heater has a significant influence on the film’s stability.By contrast, Klentzman and Ajaev [26] found that for volatile filmsspreading on heated surfaces, increasing the evaporation numberstabilizes the film because the amplitude of the capillary ridge isdiminished. These results are therefore consistent with those ofthe present study, as increased stabilization in each case is linkedto a diminished capillary ridge near the contact line.

5. Non-linear simulations

Non-linear simulations were performed by solving Eq. (2) alongwith the boundary conditions given by Eqs. (7) and (8). The bound-ary conditions in the transverse direction are hy = hyyy = 0, and peri-odic boundary conditions were also used in some simulations. Thefinite difference method described in Section 3 was extended tocompute the temporal evolution of the two-dimensional film pro-files, h(x,y, t). Second-order, centered finite differences were usedfor the spatial discretization, with time stepping again providedby the MATLAB solver ode15 s. In addition to the finite differencemethod, a finite element approach was used and implementedusing COMSOL 3.3a and MATLAB with quadratic basis functions. Theinitial condition was the one-dimensional profile given by Eq.(14). The two methods yielded identical results for several testcases, but the implementation in COMSOL required significantly lesscomputational time and was therefore used to generate the resultspresented in Figs. 11 and 12. Convergence was verified by succes-sive grid refinement and variation of the computational domain.

Fig. 11. Rendered, two-dimensional film profile from above that shows the steadyprofile after instability onset for a film spreading from left to right. (a) Semi-infiniteheater (right of dashed yellow line) for M = 4, K = 0.2, E = 0.01, h = p/4, and b = 0.02.(b)Rectangular heater (between dashed yellow lines) for M = 5, K = 0.2, E = 0.01,h = p/4, and b = 0.02.

250 N. Tiwari, J.M. Davis / Journal of Colloid and Interface Science 355 (2011) 243–251

While qualitatively similar to fingers that form in spreading,non-volatile films [30,31] because of the dominance of surface ten-sion for these flows at small capillary number, the fingers for thenon-volatile film are typically narrower because of the combinedeffects of evaporation and the disjoining pressure. A detailed com-parison to the profiles shown in Eres et al. [30] reveals that the fin-gers for the non-volatile film have a nearly constant thickness inthe x-direction with a thickened tip, while the fingers for the vola-tile film decrease in thickness in the x-direction (with only aslightly thickened tip) because of the mass loss to evaporation.

In the early stages of finger formation, small droplets also formbetween the fingers and flow down the plate as they evaporate. Fort > 250, steady fingers are formed, as the flow into the fingers isbalanced by evaporation above the heater. Such a steady profileis shown in Fig. 11(a) for M = 4, K = 0.2, and E = 10�2 at timet = 300. The periodic fingers have a wave number of q = 0.56, whichis the wave number corresponding to bmax found from a linearstability analysis.

For parameters that yield time-periodic base states, the finger-ing instability can lead to either unsteady (time-periodic fingers) or

t=50

t=60

t=70

Fig. 12. Contour plot illustrating the onset of instability for the film with M = 4, K = 2, Ebase profile. Contours are shown for h = 0.1,0.4, 0.7, . . . ,2.8.

steady two-dimensional configurations. As an example, the finger-ing instability is illustrated in Fig. 12 for the film with M = 4, K = 2,E = 0.1, h = p/4, and b = 0.02. The one-dimensional film (h(x, t))is evolved until the oscillatory behavior begins at t = 110, andthis evolved profile is used as the initial condition for the two-dimensional evolution (corresponding to t = 0 in Fig. 12). Theone-dimensional base state is time-periodic, as illustrated inFig. 3, and the linear stability analysis suggests that in the earlystages after instability onset, the film has oscillating rivulets thatshed droplets. This prediction is consistent with the initial stagesof the nonlinear film evolution. As shown in Fig. 12, a finger beginsto form around t = 50 and grows by t = 60. Because of the strongevaporation and Marangoni effects, the thinner region behind thecapillary ridge of the advancing finger begins to thin significantlyby t = 70. This thinning causes a droplet to break away from thetip of the finger and continue to flow down the substrate andevaporate, as shown at t = 80. At t = 100 the oscillations dueto the loss of the droplet are damped, and by t = 200 a steadyfinger has formed. The two-dimensional, nonlinear computationstherefore reveal interesting dynamics not anticipated from theone-dimensional evolution.

Because of the more pronounced evaporation and Marangonistresses, the fingers illustrated in Fig. 12 have a less pronouncedridge near the tip of the finger than is typical of spreading, non-volatile films [30,31]. The fingers shown for a volatile liquid byKlentzman and Ajaev [26] also have a more pronounced ridgebecause they were computed for weaker evaporation, negligibleMarangoni stresses (M = 0), and conditions for which the filmcontinues to advance along the substrate.

While this study has been focused on a heater with semi-infiniteextent in the streamwise direction, heaters in practical applica-tions can be narrow [20,22,36,37]. The dynamics and stability ofa nonvolatile film flowing over a heater were shown to be similarfor heaters with finite and semi-infinite extend [35]. Volatile filmscan behave differently, as the film terminates at a contact line for a

t=80

t=100

t=200

= 0.1, h = p/4, and b = 0.02, which corresponds to a time-periodic, one-dimensional

N. Tiwari, J.M. Davis / Journal of Colloid and Interface Science 355 (2011) 243–251 251

semi-infinite heater, and evaporation can promote film ruptureeven for a relatively narrow heater [7]. To consider the non-linearevolution of a film of volatile liquid flowing over a finite-widthheater, the boundary condition as x ?1 is changed to h isbounded, which is implemented numerically as hx = hxxx = 0 for alarge domain. The finite-width heater is modeled as

T0ðxÞ ¼ 0:5ðtanhðxþ 4Þ � tanhðx� 4ÞÞ: ð23Þ

The result of the non-linear simulation for flow over a finite widthheater is shown in Fig. 11(b) for M = 5, K = 0.2, and E = 0.01 withh = p/4. The fingers again form at the upstream edge of the heater,and macroscopic dry patches appear over much of the heater, illus-trating the potentially detrimental impact of instability on heattransfer from the substrate. Downstream of the heater the rivuletscoalesce and ultimately form a film of uniform thickness far down-stream. Similar structures were found in experiments [20] and incomputations by Frank [37] based on the unsimplified momentumand energy equations that were solved using the method of parti-cles for a non-volatile liquid. Frank and Kabov [22] extended thesecomputations to determine the variation of the critical Marangoninumber and instability wavelength with Reynolds number andfound strong agreement with their presented experimental results.The similarity of the experimental structures and those computedwith both direct numerical simulation of a nonvolatile liquid andwith the lubrication model of the present work for a volatile liquidfurther reveals the dominant influence of capillary and thermocap-illary effects on the instability.

6. Conclusions

A lubrication-type analysis was used to investigate the dynam-ics and stability of a thin, viscous film of volatile liquid flowing un-der the influence of gravity over a non-uniformly heated substrate.This model includes the effects of capillarity, disjoining pressure,evaporation, gravity, and Marangoni stresses. Previous work re-vealed that a volatile film can rupture above a relatively narrowheater for a sufficiently large evaporative mass flux [7], and atten-tion is focused on wide heaters and stronger evaporation, such thatthe film terminates above the heater at a three-phase contact line.An adsorbed precursor film is introduced to remove the stress sin-gularity at the contact line, and the thickness of this film is deter-mined from thermodynamic equilibrium with zero net mass flux.

For these conditions the film evolves to either a steady (station-ary) or time-periodic one-dimensional profile with a thermocapil-lary ridge at the upstream edge of the heater and a capillary ridgenear the contact line. The film’s response to transverse perturba-tions is studied through a linear stability analysis to determinethe influence of the substrate inclination angle, evaporation,Marangoni stresses, and a volatility parameter that acts as an in-verse Biot number. The steady film is unstable above a critical va-lue of the Marangoni parameter, leading to finger formation at thecontact line. This instability is due to the increasing amplitude ofthe capillary ridge with the Marangoni parameter, as perturbationsto the ridge induce mobility differences (under the influence ofgravity) that lead to fingers. Evaporation is generally stabilizingfor spreading films [26] because it diminishes the amplitude of

the capillary ridge near the contact line. For steady films and smallvalues of the Marangoni parameter, however, the film can bedestabilized by increasing the evaporation number. As evaporationis increased, the film extends a smaller distance over the heater,and the destabilization is due to the interaction between thethermocapillary ridge at the upstream edge of the heater and thecapillary ridge near the contact line. The film can be stabilized bydecreasing the inclination of the substrate from horizontal.

The time-periodic profiles are always unstable to transverse per-turbations, and linear stability analysis suggests that the resultingfingers oscillate and periodically shed droplets that flow along thesubstrate as they evaporate. By contrast, nonlinear simulations re-veal that the one-dimensional, time-periodic state can evolve to asteady, two-dimensional configuration after instability onset. Com-puted film profiles for rectangular heaters are consistent with thosefound in previous experimental and computational studies [22].

Acknowledgements

This work was supported by NSF Award CBET-0644777 and theCamille Dreyfus Teacher–Scholar Awards Program.

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