PHYSICS OF FLUIDS, 55 Tin films flowing down inverted substrates: Two dimensional flow Te-Seng Lin and Lou Kondic Department of Matematical Sciences and Center for Applied Matematics and Statistics, New Jersey Institute of Tecnology, Newark, New Jersey 7, USA Received 6 December 9; accepted April ; publised online 4 May We consider free surface instabilities of films flowing on inverted substrates witin te framework of lubrication approximation. We allow for te presence of fronts and related contact lines and explore te role wic tey play in instability development. It is found tat a contact line, modeled by a commonly used precursor film model, leads to free surface instabilities witout any additional natural or excited perturbations. A single parameter D= Ca / cot, were Ca is te capillary number and is te inclination angle, is identified as a governing parameter in te problem. Tis parameter may be interpreted to reflect te combined effect of inclination angle, film tickness, Reynolds number, and fluid flux. Variation of D leads to cange in te wavelike properties of te instabilities, allowing us to observe traveling wave beavior, mixed waves, and te waves resembling solitary ones. American Institute of Pysics. doi:.6/.4875 I. INTRODUCTION Tere as been significant amount of teoretical, computational, and experimental work on te dynamics of tin liquid films flowing under gravity or oter body or surface forces in a variety of settings. Te continuous and extensive researc efforts are understandable recalling large number of applications wic in one way or anoter involve dynamics of tin films on substrates. Tese applications range from nanoscale assembly, to a variety of coating applications, or flow on fibers, to mention just a few. Te researc activities evolved in a few rater disjoined directions. One of tem is flow down an incline of te films caracterized by te presence of fronts contact lines. Tese flows are known to be unstable wit respect to transverse instability, leading to formation of fingerlike or triangular patterns. 5 One may also consider flow of a continuous stream of fluid down an incline. Experimentally, tis configuration was analyzed first by Kapitsa and Kapitsa 6 and more recently and in muc more detail in a number of works, in particular by Gollub and co-workers. 7 9 Te reader is also referred to Refs. and for relatively recent reviews. Linear stability analysis LSA sows tat tese films are unstable wit respect to long-wave instability wen te Reynolds number, Re, is larger tan te critical value, Re c =5 cot /6, were is te inclination angle., As te waves amplitude increases, LSA cannot describe tem anymore as nonlinear effects become dominant. Terefore, nonlinear models ave been developed to analyze tis problem, including te Kuramoto Sivasinsky equation, 4,5 Benney equation, 6 and Kapitsa Skadov system. 7,8 Typically, film flows exibit convective instability, suggesting tat te sape and amplitude of te waves are strongly affected by external noise at te source. Tere as been a significant amount of researc exploring te consequences of imposed perturbations of controlled forcing frequencies at te inlet. 9,9 It is found tat solitary waves appear at low frequencies, wile saturated sinelike waves occur at ig ones. Moreover, it as been demonstrated tat, furter downstream, te film flow is dominated by solitary waves weter tey result from imposed perturbations or are natural. 9 It is known tat te amplitude and velocity of tese waves are linearly proportional to eac oter 9, and te slope of te amplitudevelocity relation in te case of falling and inclined film ad been examined by several autors. 9, 5 Anoter interesting feature is wave interaction. Since te waves caracterized by larger amplitude move faster, tey overtake and absorb te smaller ones. Furtermore, tis merge causes te peak eigt and velocity to grow significantly. 9, In general, te works in tis direction concentrate on a continuous stream of fluid and do not consider issues introduced by te presence of a contact line. In a different direction, tere is some work on flow of fluids on inverted substrates. Tese works involve eiter te matematical/computational analysis of te situation leading to finite time singularity, i.e., detacment of te fluid from te surface under gravity, 6 or experimental works involving te so-called tea-pot effect, 7,8 wic includes te development of streams and drops tat occurs as a liquid film or parts of it detaced from an inverted surface, or bot. 9 Tese considerations typically do not include contact line treatment; te fluid film is assumed to completely cover te considered domain. In tis paper we concentrate on te flow down an inverted inclined substrate of films wit fronts. We will see tat tis problem includes aspects of all of te rater disjoined problems considered above. For clarity and simplicity, we will concentrate ere only on two dimensional D flow; terefore we are not going to be concerned wit te tree dimensional D contact line instabilities. In addition, we assume tat te flow is slow so tat te inertial effects can be ignored, and furtermore tat te lubrication approximation is valid, requiring tat te gradients of te computed solutions are small. Te final simplification is te one of complete wetting, i.e., vanising contact angle. Tis final simplification can easily be avoided by including te possibility of 7-66// 5 /55//$., 55- American Institute of Pysics
55- T.-S. Lin and L. Kondic Pys. Fluids, 55 partial wetting, but we do not consider tis ere. Contact line itself is modeled using precursor film approac wic is particularly appropriate for te complete wetting case on wic we concentrate. As discussed elsewere e.g., Ref., te coice of regularizing mecanism at a contact line does not influence in any significant way te large scale features of te flow; models based on any of te slip models will produce results very similar to te ones presented ere. Te main goal of tis work is to understand te role of contact line on te formation of surface waves. Tis connection will ten set a stage for analysis of more involved problems, involving contact line stability wit respect to transverse perturbations, and te interconnection between tese instabilities. In addition, te presented researc will also allow connecting to te detacment problem of a fluid anging on an inverted substrate. We also note tat altoug we concentrate ere on gravity driven flow, our findings wit appropriate modifications may be relevant to te flows driven by oter forces suc as electrical or termal and across te scales varying from nano to macro. II. PROBLEM FORMULATION Consider a gravity driven flow of incompressible Newtonian film down a planar surface enclosing an angle wit orizontal could be larger tan /. Assume tat te film is perfectly wetting te surface suc as commonly used silicone oil polydimetylsiloxane on glass substrate. Furter, assume tat lubrication approximation is appropriate, as discussed in Ref.. Witin tis approac, one finds te following result for te dept averaged velocity v : v = g cos + g sin i, were is te viscosity, is te surface tension, is te density, g is te gravity, = x,ȳ,t is te fluid tickness, and = x, ȳ x points downwards and ȳ is in orizontal transverse direction. By using tis expression in te mass conservation equation, / t + v =, we obtain te following dimensionless PDE:, t + D + x =. Here, tickness and coordinates x, y are expressed in units of te fluid tickness far beind te front, and = Ca /, respectively. Te capillary number, Ca = U/, is defined in terms of te flow velocity U far beind te front. Te time scale is cosen as /U. Single dimensionless parameter D= Ca / cot measures te size of te normal component of gravity. In addition, one sould be aware lubrication approximation is strictly valid for /, corresponding to te Ca limit. Equation requires appropriate boundary and initial conditions, wic are formulated below. We concentrate on te pysical problem were uniform stream of fluid is flowing down an incline and terefore far beind te fluid front we will assume te fluid tickness to be constant constant flux configuration. At te contact line itself, we will assume precursor film model discussed in some detail elsewere. Te initial condition is put togeter wit te idea of modeling incoming stream of te fluid and te only requirement is tat it is consistent wit te boundary conditions. For te flow down an inclined surface wit / and correspondingly D, te solutions of Eq. are fairly well understood bot in te D setting were = x,t, and in te D one, were = x,y,t. Te solution is caracterized by a capillary ridge wic forms just beind te front. 4 Tis solution, wile stable in D, is known to be unstable to perturbations in te transverse, y, direction. In tis work, we will concentrate on te D setup only, but for D, terefore analyzing te flow down an inverted surface anging film. A. Initial and boundary conditions Consider D flow, terefore is y-independent. Equation can be rewritten as t + xxx D x + x =. Te numerical simulation of Eq. is performed via a finitedifference metod. More specifically, we implemented implicit second-order Crank Nicolson metod in time, secondorder discretization in space and Newton s metod to solve te nonlinear system in eac time step, as described in detail in, e.g., Ref.. Te boundary conditions are suc tat constant flux at te inlet is maintained. Te coice implemented ere is,t =, xxx,t D x,t =. 4 At x=l, we assume tat te film tickness is equal to te precursor, so tat L,t = b, x L,t =, 5 were L is te domain size and b is te precursor film tickness, b. Typically, we set b=.. Te initial condition is cosen as a yperbolic tangent to connect smootly = and =b at x=x f ; it as been verified tat te results are independent of te details of tis procedure. III. COMPUTATIONAL RESULTS It is known tat for flow down a vertical plane, a capillary ridge forms immediately beind te fluid front. Tis capillary ridge can be tougt of as a strongly damped wave in te streamwise direction. As we will see below, tis wave is crucial for understanding te instability tat develops for a flow down an inverted surface. Here, we first outline te results obtained for various D and ten discuss teir main features in some more details in te following section. We use x f =5 for all te simulations presented in tis section. Type :. D. For tese values of D, we still observe existence of a dominant capillary ridge; tis ridge becomes more pronounced as te magnitude of D is increased. In addition, we also observe secondary, strongly damped oscillation beind te main ridge. Figure sows an example of time evolution profile for D=.. For longer times, traveling wave solution is reaced, and te wave speed reaces a constant value equal to U=+b+b, as dis-
V V 55- Tin films flowing down inverted substrates Pys. Fluids, 55 4 8 x 6 FIG.. Te flow down an inverted substrate D=.. From top to bottom, t=,4,8,,6. cussed, e.g., in Ref. 5. Appendix gives more details regarding tis traveling wave solution, including discussion of te influence of precursor film tickness on te results, see Fig. below. Type :.9 D.. Te capillary ridge is still observed; owever, ere it is followed by a wave train. Figure sows as an example of te evolution for D=.5. Waves keep forming beind te front, and, furtermore, tey move faster tan te front itself. Terefore te first wave beind te front catces up wit te ridge, interacts, and merges wit it. Te oter important feature of te results is tat tere are tree different states observed beind te capillary ridge: two types of waves and a constant state. Tese states can be clearly seen in te last frame of Fig.. Immediately beind te front, tere is a range caracterized by waves resembling solitary ones discussed in some more detail below. Tis range is followed by anoter one wit sinusoidal sape waves. Finally tere is a constant state beind. Suc mixedwave feature remains present even for very long time. To illustrate tis, Fig. sows te result at muc later time, t = 4, using an increased domain size. Also, Fig. 4, wic includes typical results from te type regime discussed below, imply tat type corresponds to a transitional regime between te types and. Additional simulations not sown ere suggest tat te regions were 8 6 x 4 4 FIG.. Te flow down an inverted substrate D=.5 at t=4. te waves are present witin type regime become more and more extended as te magnitude of D is increased. Future insigt regarding te nature of wave formation in type regime is discussed in te following section. Here we note tat te available animations of wave evolution are very elpful to illustrate te complexity of wave interaction in type and type regime discussed next. Type :. D.9. Tis is a nonlinear steady traveling wave regime. Tere is no damping of surface oscillations tat we observed, e.g., in Fig.. Figure 5 sows an example obtained using D=.. Here, a wave train forms beind te first still dominant capillary ridge. Similarly as before, since tis wave train travels faster tan te fluid front, tere is an interaction between te first of tese waves and te capillary ridge. On te oter end of te domain, tese waves also interact wit te inlet at x=; te role of tis interaction is discussed in more details later in Sec. IV C. We find tat te type includes two subtypes. For smaller absolute values of D, suc as D=., one finds sinusoidal waves as sown in Fig. 5. For larger magnitudes of D, we find solitary type waves, te structures sometimes referred to as solitary umps, suc tat caracteristic dimension of a ump is muc smaller tan te distance between tem. Bot types of waves are illustrated in Fig. 4, wic sows te results for D=. and D=., and in Fig. 6 sowing te typical wave profiles for D =., D=.5, and D=.. Te wave profiles tat we find are very similar to te ones observed for continuous films exposed to periodic forcing. 9,, For te flow considered ere, 4 8 x 6 FIG.. Te flow down an inverted substrate D=.5. From top to bottom, t=,4,8,,6 enanced online. URL: ttp://dx.doi.org/.6/.4875. D=-. D=-. D=-.7 D=-.5 D=-. D=-. 5 x 5 FIG. 4. Comparison of te results for different D at t=5.
V 55-4 T.-S. Lin and L. Kondic Pys. Fluids, 55 4 8 x 6 FIG. 5. Te flow down an inverted substrate D=.. From top to bottom, t=,4,8,,6. Note tat tere is a continuous interaction of te surface waves and te front, since te surface waves travel faster tan te front itself enanced online. URL: ttp://dx.doi.org/.6/.4875. x r FIG. 6. Wave profile for different D. a D=., b D=.5, and c D =.. Te wave profile as been sifted to illustrate te difference in wave number, i.e., x r =x x, were x is an arbitrary sift. (a) (b) (c) V 4 8 time FIG. 7. Velocity profile of te leading capillary ridge for different D. D=. solid line, D=. dased line, and D=. dotted line. te governing parameter is D, in contrast wit te forcing frequency in te works referenced above. In te next section, we will discuss in more detail some features of te results presented ere. Here, we only note tat it may be surprising tat all te waves discussed are found using numerical simulations, remembering tat we do not impose forcing on te inlet region, and furtermore we do not include inertial effects in our formulation. Instead, we ave a anging film wit a contact line in te front. Terefore, it appears tat te presence of fronts and corresponding contact lines plays an important role in instability development. We note tat it is possible in principle to carry out te computations also for more negative D. We find tat, as absolute value of D is increased, te amplitude of te waves, including te capillary ridge, increases, and furtermore te periodicity of te wave train following te capillary ridge is lost. However, since te observed structures are caracterized by relatively large spatial gradients wic at least locally are not consistent wit te lubrication approximation, we do not sow tem ere. It would be of interest to consider tis flow configuration outside of lubrication approximation and analyze in more detail te waves in tis regime. In addition, tis regime sould also include te transition from flow to detacment, te configuration related to te so-called tea-pot effect. 7 9 IV. DISCUSSION OF THE RESULTS In tis section, we discuss in some more detail te main features of te numerical results and compare tem wit te ones tat can be found in te literature. We consider in particular te difference between various regimes discussed above. In Sec. IV A we give te main results for te velocities of te film front and te propagating waves. In Sec. IV B we discuss te main features of te instability tat forms and sow tat te presence of contact line is important in determining te properties of te waves, including teir typical wavelengt. Ten we finally discuss one question tat was not considered explicitly so far: Wat is te source of instability? As we already suggested, contact line appears to play a role ere. However, it is appropriate to also discuss te influence of numerical noise on instability development, sown in Sec. IV C. As we will see, bot aspects are important to gain better understanding of te problem. A. Front speed and wave speed Figure 7 compares te velocity of te leading capillary ridge for different D. Te speed of te traveling wave solution, U, is +b+b, and is exactly te front speed for D=., as discussed in Appendix in connection wit Fig.. For all oter cases sown, te velocity of te leading capillary ridge oscillates around U, due to te interaction between te leading capillary ridge and te upcoming waves. Table I sows te speed of waves in te type regime. TABLE I. Wave amplitude and wave speed for different D in te type regime. Note tat te wave speed is always larger tan te capillary ridge speed, sown in Fig. 7. D Wave amplitude Wave speed..5.88.5.88...8.65
55-5 Tin films flowing down inverted substrates Pys. Fluids, 55 TABLE II. D num is te value of D used in simulations, te position x is taken from Fig. 4 t=5 and used to calculate te speed x/t ; D lin is calculated from Eq. 6 using te negative sign. D num x x/t D lin time. 5..5..66.8.5..44.7..5 5 x 45 FIG. 8. Wave interaction wit te capillary ridge. Time evolution is from te bottom to te top, D=.. As we can see, te wave speed in all cases is greater tan U. A simple explanation on wy te waves move faster tan te fluid front itself is tat te motion of te front is resisted by te precursor film recall tat, in te limit b, tere is infinite resistance to te fluid motion witin te formalism implemented ere. Te surface waves, owever, travel wit a different, larger speed. Terefore, te upcoming waves eventually catc up wit te front, interact, and merge into a new capillary ridge. Figure 8 illustrates tis process. Due to te conservation of mass, te eigt of te leading capillary ridge increases strongly rigt after te merge. Also te speed of te capillary ridge increases. As te leading capillary ridge moves forward, its eigt decreases until te next wave arrives. Tat is te reason wy we see suc pulselike velocity profiles in Fig. 7; eac pulse is a sign of a wave reacing te front. In te type regime, te velocity of te front sows similar oscillatory beavior, altoug te approximate periodicity of te oscillations is lost due to more irregular structure of te surface waves. Going back to type and Table I, we see tat wave amplitude and speed are bot increasing wit D, consistently wit te beavior of continuous vertically falling films. 9, B. Absolute versus convective instability Here we analyze some features of te results in particular from type and type regimes using LSA. Let us ignore for a moment te contact line and analyze stability of a flat film. Te basic framework is given in Appendix. We realize tat Eq. A4 can be reduced to a linear Kuramoto Sivasinsky equation in te reference frame moving wit te nondimensional speed equal to. Consider ten te evolution of a localized disturbance imposed on te flat film at t =. Tis disturbance will transform into an expanding wave packet wit two boundaries moving wit te velocities x/t and x/t +. 5 In te laboratory frame, tese velocities are given by 6 x t.6 D /. 6 Te rigt going boundary moves faster tan te capillary ridge and can be ignored. Considering now te left boundary, we see tat tere is a range D c D D c suc tat te speed of tis boundary is positive and smaller tan U. Alternatively, one can use te approac from Ref. 7, wic is based on studying te beavior of te curve i = in te complex k plane, wit te same result. Using eiter approac, one finds D c.5 and D c.5. Tis result explains te boundary between te type and type regimes since for type, D D c and te left boundary moves faster tan te front itself. For D D c, te speed of te left boundary is positive, and terefore te instability is of convective type. Tis can be seen from Fig. 4 and is illustrated in detail in Table II. In tis table, we sow te value of D lin, predicted by Eq. 6, using te position of te left boundary x obtained numerically. Wile te agreement between D num and D lin is generally very good, we notice some discrepancy for D num =.7; tis can be explained by te fact tat for tis D num tere is already some interaction wit te boundary at x=. Tese results suggest tat we sould split our type regime into two parts: type a, for wic te speed of te left boundary is positive D D c, and type b, for wic te speed of te boundary is negative and te instability is of absolute type. In type a regime, a flat film always exists and expands to te rigt wit time. In type b regime, flat film disappears after sufficiently long time. As an illustration, we note tat D=.5, sown in Fig., lies approximately at te boundary of tese two regimes, since ere te lengt of te flat film is almost time independent. We also note tat in te type b regime, we always observe two types of waves, in contrast wit type ; tat is, te structure sown, e.g., in Fig. 4 for D=.7 persists for a long time. To allow for better understanding of te properties of te waves tat form, in te results tat follow we modified our initial condition put x f =5 to allow for longer wave evolution witout interaction of te wave structure wit te domain boundary x=. Figure 9 sows tat for D=., te waves form immediately beind te leading capillary ridge; see also te animation attaced to tis figure. For longer time t 4 in Fig. 9, te disturbed region covers te wole domain as expected based on te material discussed in Sec. IV B. Note tat even for t= we still see transient beavior: te long time solution for tis D consists of uniform stream of waves and is sown in Fig. 6 a. Tis long time solution is independent of te initial condition. However, te time period needed for tis uniform stream of waves to be reaced depends on te initial film lengt and is muc longer for larger x f used ere. Figure 9 suggests tat contact line plays a role in wave formation te oter candidate, numerical noise, is discussed below. One may tink of contact line as a local disturbance.
V V 55-6 T.-S. Lin and L. Kondic Pys. Fluids, 55 4 8 x 6 FIG. 9. D=.. From top to bottom, t=,,,4,. For early times, te contact line induced instability propagates to te left. For longer times, sinelike and solitarylike waves are observed, covering te wole domain by t=4 enanced online. URL: ttp://dx.doi.org/.6/.4875. 4 8 x 6 FIG.. D=.. From top to bottom, t=5,7,9, double precision. Te initial condition for tis simulation is cosen to be a yperbolic tangent wit contact line located at x=5. Also note te comparison between contact line induced wave x 9 and error-induced wave 5 x enanced online. URL: ttp://dx.doi.org/.6/.4875.4 It generates an expanding wave packet as we ave just sown, and te velocities of te two boundaries are given by Eq. 6. In particular, for D D c, te left boundary moves slower tan te capillary ridge. Te wave number, k l, along tis boundary is defined by k k=k l = x, 7 t and it sould be compared wit te sinelike waves tat form due to contact line presence suc as te waves sown in Fig. 9 for early times. Table III gives tis comparison: te values of k l for a given D are sown in te second column, followed by te numerical results for te wave number, q n. We find close agreement, suggesting tat k l captures very well te basic features of te waves tat form due to contact line presence. Furtermore, bot q n and k l are muc larger tan k m, te most unstable wave number expected from te LSA described below viz., te last column in Table III. Tis difference allows to clearly distinguis between te contact line induced waves and te noise induces ones, discussed in wat follows. TABLE III. Te second column sows teoretical results for te wave number of te left moving boundary k l, in te limit of small oscillations, see Eq. 7. Te tird column sows te wave number resulting from simulations for different D in type and type regimes in te contact line induced part e.g., te waves sown in Fig. t=6 for 4 x 8 for type, or in Fig. 9 t= about x=4. Te last column sows te wave number of maximum growt, k m, resulting from te LSA. D k l q n k m....8.5.8.4.87.7.5..9..6.5..5.5.48. C. Noise induced waves Te results of LSA of a flat film see Appendix for te most unstable wave number sown in Table III confirm tat a flat film is unstable to long wave perturbations for negative D. Altoug our base state is not a flat film, tere is clearly a possibility tat numerical noise, wic includes long wave component, could grow in time and influence te results. As an example, we consider again D=.. Similar results and conclusions can be reaced for oter values of D. Let us first discuss expected influence of numerical noise. For D=., te LSA sows tat it takes time units for te noise of initial amplitude of 6 typical for double precision computer aritmetic to grow to. LSA also sows tat waves wit small amplitude sould move wit te speed. Tat is, natural noise, wic is initially at x=, sould arrive to x=9 after time units. Figure illustrates tis penomenon. As t approaces, we see tat te noise appears at about x = 9. Noise manifests itself troug te formation of waves beind te contact line induced waves wic were already present for earlier times; see also te animation attaced to Fig.. To furter confirm tat tis new type of waves is indeed due to numerical noise, we also performed simulations using quadruple precision computer aritmetic. Figure sows te outcome: wit iger precision, te noise induced waves are absent, as expected. We note tat in order to be able to clearly identify various regimes, we take x f =5 in Figs. and, so tat no influence of te boundary condition at x= is expected. In Fig. t=, we can clearly distinguis between te waves induced by contact line x 9, and te natural waves induced by noise 5 x. Te main difference is te wavelengt. Te contact line induced waves ave teir specific wavelengt, /k l, wile te noise induced ones are caracterized by a wavelengt,, corresponding very closely to te mode of maximum growt, /k m, obtained using LSA. Tis can be clearly seen by comparing te numerical results sown in Fig. t= wit te LSA results given in Table III. To summarize, te evolution of te wave structure in te
V 55-7 Tin films flowing down inverted substrates Pys. Fluids, 55 4 8 x 6 FIG.. D=.. From top to bottom, t=5,7,9, quadruple precision. Te initial condition is te same as in Fig.. type and type regimes proceeds as follows. First one sees formation of contact line induced waves, caracterized by relatively sort wavelengts compared to wat would be expected based on te LSA of a flat film. Depending on te value of D, one may also see formation of solitary-looking waves immediately following te capillary ridge. At some later time, tese waves are followed by noise-induced one. Tese tree types of waves are all presented in Fig.. Ten, at even later times, wen te waves cover te wole domain and interact wit te x= boundary, te final wave pattern forms, as illustrated for D=. by Fig. 4. In te conclusions we discuss briefly under wic conditions tese waves may be expected to be seen in pysical experiments. Remark I. Finally, one may wonder wy te traveling wave solution for D =. sown in Fig. remains stable for suc a long time. Recall tat te LSA predicts tat natural noise wit amplitude 6 sould grow to in time units, wile our numerical result sows tat flat film is preserved even for t=6. Te reason is te domain size. It takes approximately 66 time units for noise to travel across te domain moving wit te speed equal to, and te noise can only grow from 6 to 9 during tis time period for D=. Tis is wy we do not see te effect of noise for small D. Remark II. We used LSA of a flat film terefore, ignoring contact line presence to predict te evolution of te size of te region covered by waves in Sec. IV B. However, in order to understand te properties of te waves tat form in tis region, one as to account for te presence of a front, as discussed in Sec. IV C. In our simulations, we are able to tune te influence of noise on te results viz., Fig. versus Fig.. In pysical experiments, tese two effects will quite possibly appear togeter. D. Relation of pysical quantities wit nondimensional parameter D It is useful to discuss te relation between te nondimensional parameter D in our model, Eq., and pysical quantities. In particular, we recall tat tere are two quantities, film tickness and inclination angle, wic can be adjusted in an experiment, and ere we discuss ow variation TABLE IV. Relation of te parameter D to oter parameters for fixed contact angle, left, and for fixed film tickness, rigt. Te up arrow means an increase and down arrow means a decrease. fixed of eac of tese modifies our governing parameter and te results. We also relate D to te fluid flux and te Reynolds number. Te velocity scaling in Eq. can be expressed as U = g sin. Terefore te parameter D can be written as D = g / cos / sin /. In our simulations, te flux Q in te x-direction as been kept constant and equals to. Te dimensional flux is Q = U = g sin. Reynolds number can be expressed as fixed D D D D D D Q Q Re Re U U Re = U = g sin. We note tat tere is no contradiction in considering Re, altoug inertial effects were neglected in deriving te formulation tat we use. Te present formulation is valid for Re=o / or smaller, were is te ratio of te lengt scales in te out-of-plane and in-plane directions. 8 Considering te influence of Re for tis range is permissible. Te relation between D and relevant pysical quantities is sown in Table IV. For fixed inclination angle an increase in te magnitude of D is equivalent to an increase in te film tickness, flux, and Reynolds number. On te oter and, for fixed film tickness, i.e., =const, raising te magnitude of D leads to lower flux and Reynolds number, and te inclination angle approaces orizontal. We can use tis connection to relate to te experimental results of Alekseenko et al. see Fig. in Ref. 9. Tey performed te experiments wit fixed inclination angle and increasing te flux, wic corresponds to an increase in te magnitude of D in our case. Our Fig. 6 sows tat te trend of our results is te same as in te above experiments. In addition, te results in Ref. 9 suggest tat furter increasing of te flux leads to pinc-off, consistently wit our results, since for D., numerics suggest tat lubrication assumption is not valid. 8 9
55-8 T.-S. Lin and L. Kondic Pys. Fluids, 55 TABLE V. c is te inclination angle at wic lubrication teory ceases to be formally valid. D c deg. 5.5 47. 54. 6 TABLE VI. is te film tickness defined by Eq. 8. deg D mm 5..9.5.7 7..7..75..4 Finally, one sould recall tat lubrication approximation is derived under te condition of small slopes, wic translates to sin a /, if nondimensional slopes are O, see, e.g., Ref. ; ere a= / g is te capillary lengt. In addition, by combining te above lubrication limit wit Eq. 8, one gets te following condition see also Ref. 4 : D cot. Terefore, for a given D, tere exists a range of inclination angle for wic te tin film model, Eq., is valid. Table V sows tis range for some values of D. V. CONCLUSIONS In tis paper we report numerical simulation of tin film Eq. on inverted substrate. It is found tat by canging a single parameter D, one can find tree different regimes of instability. Eac regime is caracterized by a different type of waves. Some of tese waves sow similar properties as te ones observed in tin liquid films wit periodic forcing. However, in contrast wit tose waves produced by perturbations at te inlet region, our instability comes from te front. We find tat te presence of a contact line leads to free surface instability witout any additional perturbation. According to LSA, we know tat for negative D, te model problem, Eq., is unstable in te sense tat any numerical disturbance grows exponentially in time. However, we can also take advantage of te stability analysis to separate te instability caused by noise and any oter sources. Finally, we may ask about experimental conditions for wic te waves discussed ere can be observed. As an example, consider polydimetylsiloxane, also known as silicone oil surface tension: dyn/cm; density:.96 g/cm, and discuss te experimental parameters for wic te condition D. is satisfied. For =7 te value used in Ref. 9, te tickness sould be less tan.4 mm. Table VI gives te values for tis, as well as for some oter D. However, one sould recall tat Eq. sows tat our model is formally valid only up to a certain D for a given inclination angle. In addition, one sould be aware tat te use of lubrication approximation is easier to justify for inclination angles furter away from te vertical. ACKNOWLEDGMENTS Te autors tank Linda J. Cummings and Burt Tilley for useful comments. Tey also acknowledge very useful input from anonymous referee leading to te material presented in Sec. IV B. Tis work was partially supported by NSF Grant No. DMS-9858. APPENDIX: EVOLUTION OF SMALL PERTURBATIONS Equation is a strongly nonlinear PDE and, to our knowledge, as no analytical solutions. In tis appendix, we present two analytical approaces wic consider evolution of small perturbations from a base state witin linear approximation. Wile tese results are useful for te purpose of verifying numerical results, tey also provide a very useful insigt into formation and evolution of various instabilities discussed in tis work.. Traveling wave solution Setting s=x Ut in Eq., a traveling wave H s = x,t must satisfy UH + H H DH + = c. A Imposing te conditions H ass, and H b as s, wefindu=+b+b, c= b b. 5,4 Te traveling wave speed, U, is useful for verifying weter or not te numerical result is a traveling wave solution. Figure sows a typical profile of te traveling wave solution for D=. A capillary ridge forms beind te fluid..99 x..9999 8 x 4 x 4 5 FIG.. Traveling wave solution of D= case at tree different scales wit precursor tickness b=. solid line and b=. dased line. Note tat te precursor tickness only canges amplitude of te traveling wave profile but not te wavelengt. Te arrows point to te zoomed-in regions.
55-9 Tin films flowing down inverted substrates Pys. Fluids, 55 TABLE VII. Properties of te damped oscillatory region tail beind te capillary ridge. front, similarly as for te flow down a vertical or inclined D substrate. We also find tat tere exists a long oscillatory region beind te capillary ridge. To analyze tis tail, we expand Eq. A around te base state, H, and consider te evolution of a small perturbation of te form exp qs, were q=q r +iq i. We find 8q r +Dq r + b b =, A q i + D =q r. A Table VII sows te only positive root for q r, for a set of D. Te positivity of tis root signifies tat te amplitude of te tail decays exponentially in te x direction, as also suggested by te insets of Fig.. Furtermore, as sown in Table VII, q r decreases for more negative D, meaning tat te tail is longer for tese D. Tis table also sows te imaginary part of q; we see an increase in its magnitude as D becomes more negative, suggesting sorter and sorter wavelengts in te tail. Tail beavior is very useful for computational reasons. For example, to solve Eq. A by sooting metod, one can evaluate suitable sooting parameters troug Eqs. A and A. Figure also sows te effect of precursor tickness on traveling wave solution. It is found tat wile te precursor tickness canges te eigt of te capillary ridge, it as almost no effect on te wavelengt of te tail.. Linear stability analysis Anoter approac to analyze te stability of a flat film is classical LSA. Assume =+ were. Equation can be simplified to te leading order as t + xxxx D xx + x =. A4 By putting exp i kx t, were = r +i i, we obtain te dispersion relation i r + i i + k 4 + Dk +ik =, A5 ence r =k, D q r q i.6.9..5...8.56...8 i = k 4 + Dk = k + D/ + D /4. A6 As a result, for non-negative D, a flat film is stable under small perturbations. For negative D, it is unstable for te perturbations caracterized by sufficiently large wavelengts. Te critical wave number k c = D and te perturbation wit wave number k m = D/ as te largest growt rate. Besides, te speed of a linear wave is, and it is significantly larger tan te traveling wave speed, U. As discussed in te main body of te text, tis speed is very important to identify te waves induced by natural noise. In addition, one sould note tat te maximum growt rate increases as inclination angle goes from / to. Particularly, in te limiting case anging film, te growt rate is exactly te same as for tin film Rayleig Taylor instability e.g., Ref. 4. Note tat te scaling used in present work is not appropriate for ; to establis te result for tis case, one sould consider a different scaling, or te dimensional formulation of te problem. H. Huppert, Flow and instability of a viscous current down a slope, Nature London, 47 98. N. Silvi and E. B. Dussan V, On te rewetting of an inclined solid surface by a liquid, Pys. Fluids 8, 5 985. A. M. Cazabat, F. Heslot, S. M. Troian, and P. Carles, Fingering instability of tin spreading films driven by temperature gradients, Nature London 46, 84 99. 4 J. R. de Bruyn, Growt of fingers at a driven tree-pase contact line, Pys. Rev. A 46, R45 99. 5 A. L. Bertozzi and M. P. Brenner, Linear stability and transient growt in driven contact lines, Pys. Fluids 9, 5 997. 6 P. L. Kapitsa and S. P. Kapitsa, Wave flow of tin fluid layers of liquid, Z. Eksp. Teor. Fiz. 9, 5 949. 7 J. Liu and J. P. Gollub, Onset of spatially caotic waves on flowing films, Pys. Rev. Lett. 7, 89 99. 8 J. Liu, J. D. Paul, and J. P. Gollub, Measurement of te primary instabilities of film flows, J. Fluid Mec. 5, 69 99. 9 J. Liu and J. P. Gollub, Solitary wave dynamics of film flows, Pys. Fluids 6, 7 994. H.-C. Cang and E. A. Demekin, Complex Wave Dynamics on Tin Films Elsevier, New York,. S. V. Alekseenko, V. E. Nakoryakov, and B. G. Pokusaev, Wave Flow of Liquid Films Begell, New York, 994. T. B. Benjamin, Wave formation in laminar flow down an inclined plane, J. Fluid Mec., 554 957. C. S. Yi, Stability of liquid flow down an inclined plane, Pys. Fluids 6, 96. 4 H.-C. Cang, Wave evolution on a falling film, Annu. Rev. Fluid Mec. 6, 994. 5 S. Saprykin, E. A. Demekin, and S. Kalliadasis, Two-dimensional wave dynamics in tin films. I. Stationary solitary pulses, Pys. Fluids 7, 75 5. 6 A. Oron, S. H. Davis, and S. G. Bankoff, Long-scale evolution of tin liquid films, Rev. Mod. Pys. 69, 9 997. 7 Y. Y. Trifonov and O. Y. Tsvelodub, Nonlinear waves on te surface of a falling liquid film. Part. Waves of te first family and teir stability, J. Fluid Mec. 9, 5 99. 8 H.-C. Cang, E. A. Demekin, and S. S. Saprikin, Noise-driven wave transitions on a vertically falling film, J. Fluid Mec. 46, 55. 9 S. V. Alekseenko, V. E. Nakoryakov, and B. G. Pokusaev, Wave formation on a vertically falling liquid film, AICE J., 446 985. N. A. Malamataris, M. Vlacogiannis, and V. Bontozoglou, Solitary waves on inclined films: Flow structure and binary interactions, Pys. Fluids 4, 8. T. Nosoko and A. Miyara, Te evolution and subsequent dynamics of waves on a vertically falling liquid film, Pys. Fluids 6, 8 4. H.-C. Cang, E. A. Demekin, and D. I. Kopelevic, Nonlinear evolution of waves on a vertically falling film, J. Fluid Mec. 5, 4 99. H. S. Kesgi and L. E. Scriven, Disturbed film flow on a vertical plate, Pys. Fluids, 99 987. 4 R. R. Mudunuri and V. Balakotaia, Solitary waves on tin falling films in te very low forcing frequency limit, AICE J. 5, 995 6. 5 J. Tion, K. Serifi, K. Argyriadi, and V. Bontozoglou, Solitary waves on inclined films: Teir caracteristics and te effects on wall sear stress, Exp. Fluids 4, 79 6. 6 V. Sikin and E. Lebedeva, Anti-gravitational instability of te neutral elium film, J. Low Temp. Pys. 9, 469. 7 W. G. Pritcard, Instability and caotic beaviour in a free-surface flow, J. Fluid Mec. 65, 986. 8 S. F. Kistler and L. E. Scriven, Te teapot effect: Seet-forming flows
55- T.-S. Lin and L. Kondic Pys. Fluids, 55 wit deflection, wetting and ysteresis, J. Fluid Mec. 6, 9 994. 9 A. Indeikina, I. Veretennikov, and H.-C. Cang, Drop fall-off from pendent rivulets, J. Fluid Mec. 8, 7 997. J. A. Diez, L. Kondic, and A. L. Bertozzi, Global models for moving contact lines, Pys. Rev. E 6, 8. L. Kondic, Instabilities in gravity driven flow of tin liquid films, SIAM Rev. 45, 95. L. W. Scwartz, Viscous flows down an inclined plane: Instability and finger formation, Pys. Fluids A, 44 989. A. Oron and P. Rosenau, Formation of patterns induced by termocapillarity and gravity, J. Pys. II, 99. 4 L. Kondic and J. A. Diez, Flow of tin films on patterned surfaces: Controlling te instability, Pys. Rev. E 65, 45. 5 P. Huerre and P. A. Monkewitz, Local and global instabilities in spatially developing flows, Annu. Rev. Fluid Mec., 47 99. 6 H.-C. Cang, E. A. Demekin, and D. I. Kopelevic, Stability of a solitary pulse against wave packet disturbances in an active medium, Pys. Rev. Lett. 75, 747 995. 7 A. S. Fokas and D. T. Papageorgiou, Absolute and convective instability for evolution PDEs on te alf-line, Stud. Appl. Mat. 4, 95 5. 8 D. J. Aceson, Elementary Fluid Dynamics Clarendon, Oxford, 99. 9 S. V. Alekseenko, D. M. Markovic, and S. I. Stork, Wave flow of rivulets on te outer surface of an inclined cylinder, Pys. Fluids 8, 88 996. 4 R. Goodwin and G. M. Homsy, Viscous flow down a slope in te vicinity of a contact line, Pys. Fluids A, 55 99. 4 S. M. Troian, E. Herbolzeimer, S. A. Safran, and J. F. Joanny, Fingering instabilities of driven spreading films, Europys. Lett., 5 989. 4 J. M. Burgess, A. Juel, W. D. McCormick, J. B. Swift, and H. L. Swinney, Suppression of dripping from a ceiling, Pys. Rev. Lett. 86,.