Hydrodynamics of Bounded Vertical Film with Nonlinear Surface Properties

Transcription

1 Under consideration for publication in Journal of Colloids and Interface Science Hydrodynamics of Bounded Vertical Film wit Nonlinear Surface Properties A. Homayoun Heidari 1,, Ricard J. Braun,, Amir H. Hirsa 1, Steven A. Snow 3 and Sailes Naire 1 Department of Mecanical Engineering, Aeronautical Engineering and Mecanics, Rensselaer Polytecnic Institute, Troy, NY 118; Department of Matematical Science, University of Delaware, Newark, DE 197; 3 Interfacial Expertise Center, Dow Corning Incorporation, Midland, MI ; and Matematical Science Department, Worcester Polytecnic Institute, Worcester, MA 9 ABSTRACT Te drainage of a tin liquid film wit an insoluble monolayer down a vertical wall is studied. Lubrication teory is used to develop a model were te film is pinned at te top wit a given tickness and te film drains into a bat at te bottom. A nonlinear equation of state is used for te surface tension and te surface viscosity is a nonlinear function of te surfactant concentration; tese are appropriate for some aqueous systems. Te tree partial differential equations are solved via discretiation in space and ten solving te resulting differential algebraic system. Results are described for a wide range of parameters, and te conditions under wic te free surface is immobilied are discussed. 1. INTRODUCTION Te drainage of a tin liquid film due to gravity as been studied in a variety of contexts; recent reviews ave been given by Oron, Davis and Bankoff (1) and Cang (). We will use te terminology from Oron et. al (1) to describe a film on a solid substrate as "bounded" and a film bounded only by its own free surfaces as free. In tis work we are interested in te effect of surfactants on bounded film evolution. Te flow down a vertical wall witout eating as been studied, among oters, by Joo, Davis and Bankoff (3), Joo and Davis () and Cang, Demekin, and Kopelevic (). Two-dimensional solutions are found to become unstable to tree-dimensional waves in te films of infinite extent studied in tose papers. Te vertical film flow we study in two dimensions ere is too sort to allow te long wave solutions tat tey studied; owever, it is possible tat te solutions studied in tis work will be unstable in tree dimensions and we will study tis in te near future. Present address: Computational Hydrodynamics Laboratory, Department of Civil & Environmental Engineering, University of Houston, Houston, TX To wom correspondence sould be addressed. braun@mat.udel.edu 1

2 Modified by numerous applications, including foam stabiliation, we ave been interested in te drainage of vertically oriented free films wit an insoluble surfactant present on te film's surface. In tose papers, we studied twodimensional films wit a tangentially-immobile surface (6); wit constant or linearied surface properties were te average surface tension is important (7) or is negligible (8); and wit nonlinear surface properties (9). In tose papers, lubrication teory is used to derive evolution equations for te film, and boundary conditions are applied at te top and bottom of te film. Te bottom boundary conditions enforce te conditions for a static meniscus at a single point; tis is a model for connecting te film to te bat. Te models ave te advantage of computing te evolution of te entire film wile not computing te flow in te bat at te bottom. However, te bottom boundary conditions err on te side of enancing te surfactant effects tere under some conditions. Te enancement of any surfactant effects at te bottom of te film is minimied for te pysically common case wit nonlinear surface properties in te situations we ave studied (9). Vertical free films draining into a bat ave also been studied by Stein and coworkers in an effort to measure and explain te film sapes and te complex flows bot in te middle and at te edge of a vertical film in a finite widt window (1). Tey developed quasistatic equations for te free surface sape in teir model. Nierstras and Frens (11,1) developed a model tat focuses on te region tat connects te film and te bat, and tey do not compute a solution for te wole film. Tey also ave discontinuities in te dependent variables' slopes witin te computational domain. Tey speculate tat competition between gravity and te Marangoni effect is responsible for complex flows observed at te film-bat junction, wic may be called "peacock featers" and occur only in vertical films (13). Bounded films on a vertical wall entering a bat ave been studied by a number of workers; for example, in te context of coating flows, see Kesgi, Kistler and Scriven (1) and te review by Smit (1). In Kesgi et. al., falling films are studied wit different end conditions; tese end conditions represent different coating processes and te infinite bat surface tat we consider is a special case of wat tey studied. Tey did not consider surfactant transport, owever. Te drag-out problem wit surfactant was analyed using matced asymptotic expansions by Park (); e studied te case were te film is dragged vertically out of te bat wereas we are interested in te case were te film flows down a stationary wall into te bat. Te case were a film flows down te sligtly inclined wall into a bat wit evaporation was studied by Hosoi and Bus (17). Tey found tat tere was an instability transverse to te flow into te bat tat was due to evaporation.

3 Te nonlinear surface properties used in Naire et. al. (9) were based on measurements by Lope and Hirsa (18) for emicyanine dye, wic forms an insoluble monolayer on water. Te surface tension is essentially constant for low surface concentration of surfactant, and ten it rapidly decreases about 1% in magnitude near te concentration were a pase transition occurs. Te surface sear viscosity as an exponential dependence on te surface concentration in te same interval. Te incorporation of tose properties causes more localied variation in te surface velocity and film sape tan would oterwise be seen. Tis localiation sould be observable in experiment. Toward tat end, we now wis to combine te matematics of tese previous works wit te detailed interfacial property measurements reported in Lope and Hirsa (18). We develop a model tat is general enoug to andle many equations of state, including tose of te type mentioned above. Lubrication teory is used to derive equations tat govern te film evolution in two dimensions; te equations will determine te location of te film surface, te surface concentration of te insoluble surfactant, and te surface velocity along te film. Te parameters tat are representative for a class of surfactants like emicyanine will be used, and te effect of varying some of te parameters studied.. FORMULATION.1. Dimensional Problem A two-dimensional tin liquid film draining down a rigid vertical wall is modeled (see Figure 1). Te Cartesian coordinate system is considered suc tat x is along te film tickness and is along te gravity vector, so g = gk, were k is te unit vector along and g is te gravity constant (te bar over variables indicates dimensional). Te free surface of te film is defined by x = (, t ) and it is assumed a constant at te top of te film, (, ) t =. Te ydrodynamics of te film is governed by te incompressible Navier-Stokes equations v =, [1] v ρ + v v = µ v p+ ρ gk, [] t were [1] is te continuity equation and [] is te vector momentum equation. Here v = ( u, w) is te velocity vector, ρ and µ are te density and dynamic sear viscosity of te bulk liquid, respectively, and p is te pressure. Te boundary conditions applied on te above equations is te no-slip and no-flux condition at te wall defined by 3

4 u = w=, at x = [3] Also te kinematic boundary condition on te free surface defines te instantaneous location of te surface as u = + w t [] Based on te Boussinesq-Scriven surface stress model (19,) te normal stress boundary condition at te free surface, for a one-dimensional interface, is given by Edwards et al. (1) and Slattery () s s ( ) n T n = Hσ + H κ + µ v. [] Here, n is te unit outward surface normal vector, H = I : b is te mean curvature of te free surface, s s Is = I nn is te spatial idem factor, = s b n is te surface curvature dyadic, s denotes te surface gradient s operator, σ is te surface tension, κ and s µ are te surface dilatational and sear viscosity, respectively, and T = Ta T denotes te jump in te stress tensor across te free surface, were T a is te stress tensor in te gas (air) pase and is assumed to be ero in tis model. T is defined by t T = Ip + µ ( v+ v ). [6] Te tangential stress boundary condition for a one-dimensional interface wit variable surface viscosities, based on te above model, is defined as follows (1,) s s s s ( ) ( ) t T n = t σ + κ + µ t v+ t κ + µ s v, [7] s s s s were t is te unit tangent vector along te free surface. Te last term in [7] accounts for te possibly strong variability of te surface viscosities wit surfactant surface concentration. Te surface differentiation operators follow Scriven (), Stone (3) and Wong et al. (). Te concentration of te insoluble surfactant on te free surface is governed by te following advectiondiffusion equation (,3) were Γ=Γ (, t ) is te surface concentration of te insoluble monolayer of surfactant; and D is te surface diffusivity. Γ + s Γ = s Γ t ( ) v D s, [8] s

5 .. Surface Properties Te surface tension, for a fluid containing a dilute insoluble surfactant, is no longer a constant value. Tere are a number of matematical models presented for te surface tension in te presence of different surfactants. Lope and Hirsa (18) introduced te following expression for emicyanine, an insoluble monolayer used in several experiments, ( ) F (, ) σ Γ = σ + β α Γ, [9] m F ( α, Γ ) = tan α( Γ 1 ), ( α < ), [1] were σ m is a caracteristic surface tension, Γ=Γ Γm is te nondimensional surfactant concentration, and Γ is m te reference surfactant concentration, and β and α are a dimensional and a nondimensional fitting parameter, respectively. For te same surfactant, an analytical fit for te surface sear viscosity is also given by Lope and Hirsa (18) were s µ m s s µ = µ m ( 1+ exp( γγ )), [11] is te reference surface sear viscosity, and γ is a nondimensional parameter. Te surface dilatational viscosity is not yet been measure consistently wit any two different experimental tecniques (1) and it is usually s s modeled as a multiple of surface sear viscosity. A wide range for te ratio of θ = κ µ ave been suggested in various works, e.g. :1 () to :1 (6). However, we will consider tis ratio as a variable and will discuss its effect in te parameter study of te results. Taking tis ratio into account, equation [11] may be written as follows for te total surface viscosity s s s ( ) ( 1) m ( 1 exp( κ + µ = θ + µ + γγ )). [1].3. Nondimensional Problem Te lengt scales for te nondimensionaliation are based on te tangentially immobile unbounded film (7,8). Tree scales are introduced W 1,,and =d 3 3 d = µ σ D ρg = ρg l D. [13]

6 Wic are, respectively, te average initial film tickness, equilibrium meniscus radius, and te intermediate scale, and are also depicted in te scematic of te model (FIGURE 1). W is te vertical velocity scale wic is defined from [13], aving te value d. Separation of scales occurs wen te ratio of lengt scales is very small, i.e. δ = d l 1. Using te above lengt and velocity scales, te following nondimensional variables are introduced into te problem x W u x =,, t t, u, w d = l = l = δ W = w δ l, and p = p. [1] W µ W Based on te above nondimensionaliation, te motion equations inside te film are furnised as follows u + w =, x 8 R ( t x ) ( xx ) δ u + uu + wu = δ u + δ u p, x [1] [] ( ) δ R w + uw + ww = w + δ w p + 1, [17] t x xx tese te continuity equation and te momentum equation for x and directions, respectively, and R is te Reynolds number, ρw l R =. [18] µ Te nondimensional surfactant transport equation [8] will be given by 1 t N δ ( u) ( w) Γ + A Γ + A Γ = ( δ N A ) + ( N A ) Γ P, [19] were te Peclet number is defined by P W l = D. [] s Also, te following definitions are used in [19] 1 N =, A +. [1] 1+ δ x Te nondimensional kinematic boundary condition becomes u+ w=. [] t 6

7 For te sake of generality, te surface property functions will be rewritten based on [1] as follows in te interfacial stress components nondimensional formulation ( ) ( ) ( κ s + µ s = 1+ θ µ s m Ω Γ), [3] were Ω( Γ) is a nondimensional function of surfactant concentration. Te nondimensional normal component of te interfacial stress, is given by were te following nondimensional groups are used ( ) p+ N δ ux δ u wx δ w + + = δ M N F N u w C α δ + δ SΩ Γ δ + ( )( ) [], σ Γm Γ Γ µ W m M = δ, C=, and µ W σ m s ( 1 ) m +θ µ S = δ, [] lw were tese numbers are Marangoni, capillary, and modified Boussinesq, respectively. It sould be noted tat for our scaling, C = δ. Te nondimensional tangential component of interfacial stress can be written in nondimensional form as 6 ( ) ( )( ) N δ ux w + 1 δ δ + w x = σ N + Ω N u + u + w + N Ω Γ u + w 3 3 M S ( δ δ ) S Γ ( δ ) [6] In our model, using [9] for te variations of surface tension wit surfactant concentration, te Marangoni number would be defined as βα δ µ W M =. [7].. Lubrication Teory Te solution is expanded in te form of a regular perturbation expansion using δ as te perturbation coefficient as follows (, p,, ) (, p,, ) ( ) δ (, p,, ) ( 1) v Γ = v Γ + v Γ + L [8] 7

8 Te above expansion is ten substituted into te scaled equations. After dropping te superscripts, te leading order of equations may be written as Te boundary condition at te wall, x =, is u + w =, [9] x = p x, [3] = w p + 1. [31] xx u = w=, [3] and te boundary conditions on te free surface of te film, x = (, t), can be written as u+ w=, t [33] 1 1 SΩ w + SΩΓΓw wx M σ Γ, Γ = [3] N N p = η, [3] were tese equations are te leading order of kinematic, tangential stress, and normal stress boundary conditions, respectively. Here, η is te film surface curvature, wic is defined by η = = 3 N ( 1+ δ ) Using [31]-[33], te vertical component of velocity vector may be written as Here A ( t, ) (,, ) ( η 1 ) (, ) w xt 3. [36] x = + +A t x. [37] is an unknown function so far. Now defining te vertical surface velocity as ( ) ( ( ) It may now be expressed by te following equation based on [37] ϖ ) ϖ t, = w t,, t,. [38], = + 1 +A t,. [39] ( t) ( η ) ( ) Te governing equation for te surface velocity will be derived by substituting te above expression into [3] 1 1 ϖ 1 S( Vϖ) + ( η 1) σ + Γ Γ N N M =. [] 8

9 Using te definition for surface velocity, te kinematic boundary condition [33] can be written as follows t + ϖ+ ( η + 1) 1 =. [1] Tese two equations are coupled wit te surfactant transportation equation [19], wic as te following form in te leading order 1 P Γ t + N Γϖ N Γ =. [] Equations []-[], are te tree coupled differential equations wic sould be solved for tis matematical model along wit te boundary conditions defined in te subsequent section... Boundary Conditions At te top of te film, i.e. =, we assume tat te film is fixed to a film older, as used in te experiments. Also we assume tat te no flux condition olds for te top of te film. In te experiments, tis is accomplised by using a film older made of inert material suc as Teflon. Tese conditions may be summaried into ( t ) = ( t ) = Γ ( ) = ( ),,,,, t, and ϖ, t =. [3] At te bottom of te film, i.e. = L, te film sould be matced into te static meniscus wic is governed by te Young-Laplace equation, Te solution for a two-dimensional case is parameteried by ν, η = 1. [] ν δ = η + 1+ δ, [] wic is a constant in te first integral of []. Te value of tis parameter for an infinite bat, wic olds for our case, is ν =. However, aving a fixed initial condition for at te bottom of te film, we can define using [] by te equation 3 1 = 1 ( + δ ) δ 1+ δ 1. [6] 9

10 Imposing and at te end of te domain is an asymptotic patc because te solutions only agree at a single point. In standard matced asymptotics, tere is a finite region of overlap between te two solutions. Te patcing will conserve te amount of surfactant over te film region and it conserves surfactant over te entire fluid surface (union of film and bat surfaces) as may be verified by a simple integration along te fluid surface. However, te surfactant concentration may jump across te patc point from te film side to te meniscus side. We coose to use tis approac because tis coice will accentuate any effect of surfactant build up at te bottom of te film as it is wased down during drainage. Te oter boundary conditions at te film end are no flux for bot bulk pase and te surfactant, ϖ ( ) = ( ) Lt,, Γ Lt, =, [7] were L is te lengt of te film. Tis furnises all te necessary boundary conditions tat we need at te ends of te film..6. Initial Conditions Linear initial conditions are assumed for te film sape and surfactant distribution. Te following form is considered for te film surface ( ), = 1 + s, L, [8] were s is te initial slope. Te initial surfactant concentration is cosen to be of te form ( ) ˆ L, γ Γ =Γ+, [9] were ˆΓ is te average surfactant concentration and γ is te initial surfactant gradient, i.e. γ ( ) = Γtop Γbottom L. 3. NUMERICAL SCHEME In order to numerically solve te system of nonlinear PDEs of []-[], te spatial variable is discretied using a second-order accurate finite difference sceme and te temporal variable is left continuous. Te resulting system may be written in te form of a system of differential-algebraic equations (DAEs) wic is ten solved by te package DASSL (9). In order to derive te DAE system, te following discretiation in te spatial direction is employed. Te kinematic boundary condition [1] is first written in terms of flux as follows t = Φ, [] 1

11 were te flux is defined by Φ ϖ+ ( η + 1). [1] 1 Te conservation of flux over te film region is assured by using te above form for te film evolution equation. Equation [1] is coupled wit te surface velocity equation [] and te surfactant transport equation [] wic te later is again written in terms of surfactant flux as follows were te surfactant flux is defined by Γ + N Ψ =, [] t 1 N P Ψ Γϖ Γ [3] ϖ i Tree dependent variables wic are to be evaluated, namely, ϖ, and Γ, are ten discretied as =, t, ϖ (, t i ( ), and Γ = Γ(, t based on grid points = i, i =, K, M, were M is te number of grid points. = ) ) i i i Up to tird-order derivative of film tickness function is needed to calculate te curvature derivative term in [1], leading to te following finite difference formulation i i + i+ i+ 1 i 1 i 3 ( ), [] wic in turn produces te need for fictitious points and 1 M + 1. Based on a perturbation expansion of te boundary condition [] at te end of te film using δ as te perturbation coefficient, te following explicit formulation for 1 may be derived ( ) ( ) 8 1 = 1 + δ 1 + ( δ ) O, [] were ( ) 1 ( ) 1 = , [6] ( ) ( ) ( ) ( )( ( ) ( ) 1 ( ) 1 = [7] M + 1 M + Also and are derived based on te boundary conditions for and at te bottom of te computational domain resulting in ) 11

12 ( ) = M 1 s M M 1, [8] M+ = s + M+ 1 3 M, [9] were s 1 and s are te first and second-order derivatives at te bottom of te domain, respectively. Also, fictitious points are needed for te surfactant concentration were tey are derived based on te top [3] and bottom [7] boundary conditions. At te bottom of te domain, one-sided difference approximations are used in order obtain flux derivatives as follows 3Φ Φ +Φ Φ M M 1 M [6] and te same procedure is applied to te surfactant flux, Ψ. Having all te dependent variables discretied, te system may now be fed into te DASSL package as te following implicit form ( & ) G t, y, y = [61] were a dot denotes a temporal derivative, and y may be eiter a differential or an algebraic variable. Using a backward difference sceme, te system is discretied in temporal direction is ten solved by a modified version of Newton s metod.. RESULTS In te results to follow, a parametric study is conducted on te ydrodynamic beavior of te tin film modeled in te earlier sections. Te effect of te value of surfactant average concentration is first considered. In tis study, a low surface viscous effect and a relatively ig Marangoni effect is considered so tat te effect of Marangoni-driven flow would be dominant. Next, te effect of initial surfactant concentration gradient is discussed based on a fixed average concentration and oter parameters. Te beavior of te film is subsequently analyed in a low-viscous, low-marangoni effect regime. Ten te effect of te surface concentration, bot initial distribution and te average concentration is studied in a ig surface viscosity situation. A case of ig-viscosity and low Marangoni effect is discussed next and finally te similarity beavior of te tin film is studied based on a situation of ig Marangoni effect and various levels of surface viscosity effect. 1

13 For all te following results te pysical properties of te fluid/surfactant system is considered as follows: µ=.1 g cm s, 3 ρ= 1g cm, d = 1 µ m, l =.8mm, β= 6.3dyne cm, α= 6., 6 s = 1 cm s D, δ=.189. Based on te above parameters, a Marangoni number of M = 1 and a Peclet number of P = 7 are used s in te following results. Two different values of te ratio of dilatational surface viscosity κ to surface sear viscos- s ity µ are used in our computations in order to cover bot low and ig viscosity. Hence, based on θ = 1 and θ= 1, two different values of S =. and S = 1 are used in te computations. Also for limiting cases of surface viscous effects and/or Marangoni effects, oter artificial values of S and M are used as described in te following sections..1. Effect of Average Concentration Te effect of te average concentration is studied based on te tree values of surface concentration ˆ.,.6,.8 Γ=, wic are considered to be relatively low, medium, and ig, respectively (see FIGURE ). In all tree cases, oter parameters are considered to be fixed at te following values: S =., M = 1, P = 7. Terefore, te flow regime may be described as low surface viscosity and ig Marangoni effect. Also no initial surfactant concentration gradient is assigned for tese cases. FIGURE 3 illustrates te film beavior aving a relatively low average surface concentration of surfactant. At very early times ( t = ), upper alf of te film as a positive surface velocity distribution terefore te surfactant is being swept down in tat region. On te oter and, te lower alf as a negative surface velocity distribution, wic is negligible compared to tat of te upper alf. According to FIGURE, te dependence of te surface tension on te surfactant concentration in te neigborood of ˆ. Γ=, is very weak. Terefore, only around te iggradient area of newly developed surface concentration distribution at t =, a surfactant-driven flow begins to counteract gravity, dragging up te surfactant at later times. However, at te very top of te film, concentration continues to decrease and so does te film tickness due to te downward velocity. A ealing effect sapes in te film free surface, moving up wit time, around te point were te concentration gradient is ig. Tis upward-traveling wave, is actually dividing te film surface into a rigid, lower part of te film, and a mobile film in te upper part of te film. 13

14 In te next case, illustrated in FIGURE, te surface tension is more sensitive to te canges in surfactant concentration distribution due to a iger value of average concentration, Γ= ˆ.6. Te surface tension function begins to curve downward in te neigborood of tis value of concentration (see FIGURE ). Due to tis effect, stronger upward flow occurs at te lower ¾ of te film tending toward redistributing te surfactant along te film eigt. Also te film tinning is more uniform and slower at te top comparing te previous case due to te negative surface velocities. A very weak wave is observed traveling up te film and gradually disappearing due to smooted concentration distribution by about t = 3. Consequently, most of te film surface may be described as rigid in tis case. At te iger average concentration in te tird case, Γ= ˆ.8, sown in FIGURE, te surface velocity is completely negative (upward) from early time. No ealing is observed on te film surface and te surfactant concentration is symmetrically redistributed after a sligt gravitational was-down effect at t = in te upper alf of te film. Te surface tension is igly sensitive to te concentration and tis is te reason for te dominance of te Marangoni effect and upward surface flow in tis case. Due to te strong Marangoni effect, te film is acting more rigid in tis case. By fixing te value of average concentration at.. Te Effect of Initial Concentration Gradient, te effect of initial concentration gradient is discussed in tis section. As te first case, te beavior of te film for a relatively low concentration gradient is illustrated infigure 6, using ˆ. Γ= γ =. L. By comparing wit te case of FIGURE 3, it is clear tat tere is no significant difference in te ydrodynamic beavior of te film except tat te negative surface velocity is sligtly larger in te lower alf of te film for te case wit an initial gradient in Γ. Tis is due to te stronger initial Marangoni effect wic puses te stagnation point on te free surface upward from around to 13. Consequently, te same wave observed in FIGURE 3 can be identified in tis case, starting sligtly closer to te top of te film at te first illustrated time step and ten continuing to travel upward keeping te same position as te previous case. Tis is due to te balanced Marangoni effect after te redistribution wic starts at about t =. Film may again be described as rigid in te lower part and mobile in te upper part, separated by te wave front. 1

15 .3. Te Effect of Surface Viscosity To study te effect of surface viscosity on te ydrodynamic beavior of te film, te modified Boussinesq number of te previous case illustrated in FIGURE 6, is increased to S = 1. Te Boussinesq number of tis case, illustrated in Figure 7, represents an increased value surface viscosity ratio θ (see equation [1]) from 1 to 1, s resulting in muc larger surface dilatational viscosity κ and consequently muc stronger surface viscous effects. Te sensitivity of surface tension to te surfactant concentration is relatively low in tis case as te curve in FIGURE sows a nearly flat beavior around Γ=.. Te significant surface viscous effect as decreased te surface velocity gradient especially around te surface stagnation point, wic in turn as made te traveling wave on te surface disappear at te early times. However, as te total Marangoni effect increases at later times te wave starts to form again wic can be clearly identified in t = 3. Still in tis case, te Marangoni and viscous effects are weak enoug at te very top of te film allowing te surface to get swept clean of te surfactant. Te magnitude of te surface velocity peak is approximately 3% lower tan te low surface-viscosity case, making te film surface relatively more sluggis... Limiting Cases for low Marangoni Effect Two limiting cases are studied in tis section wic correspond to a weak Marangoni effect, aving eiter a relatively low or a ig Boussinesq number. Te first case corresponds to a film wit S = 1 and M =, wic is illustrated in Figure 8. Te average surfactant concentration is set to te flat region of te equation of state and also a relatively mild initial gradient is considered. As it is expected for tese parameters, te film is acting completely mobile for te early time. Due to te very weak Marangoni effect, most of te surfactant is swept down and as built up near te meniscus as it is sown in te curves for t =. Te surface velocity is also completely positive wit te only mecanism acting in tis stage being te gravitationally-driven flow. Te viscous effects are also small at te top of te film wic along wit te weak Marangoni effect, causes a rapid tinning in te top of te film. However, te redistributed surfactant concentration enances te total effective Marangoni and viscous effect and causes te flow to reverse as is illustrated in t =. A wave front forms and starts traveling upward rigt in te position of te stagnation point wic is te equilibrium of te gravitationally- and viscous-driven flow. However, due to low overall Marangoni effect, te traveling wave does not get close to te top of te film for te times we computed and te 1

16 mobile portion of te film is muc larger compared to te case wit te same strengt of viscous effect and iger Marangoni number wic is illustrated in Figure 7. Next, a case wit a very strong viscous effect and te same weak Marangoni effect is sown in Figure 9. Due to te igly viscous free surface, te sweeping effect of te surface concentration in te previous case is not observed ere. Te concentration is gradually reduced in te upper alf of te film and starts to buildup in te lower alf until t = 18. At tis time, te Marangoni effect starts to turn on due to te adequate concentration gradient wic is provided by te downward gravitationally-driven flow. Te surface velocity is positive along te film eigt and no stagnation point is observed. At te last curve sown in te grap wic corresponds to t = 1, a very sligt flow reversal starts to occur and te surfactant is driven upward in a small region just over te middle point of te film... Power-Law Beavior of Drainage In an attempt to study te similarity beavior of te film drainage, te time-istory of te film tickness for tree locations along te film eigt is illustrated in FIGURE 1. Graps are illustrated for points near te top ( = L ), middle ( = L ), and bottom ( = 3L ) of te film and tere are tree cases of te surface viscosity strengt for eac location. Te curves are on a logaritmic scale ence a straigt line sows power-law beavior of te film drainage proc- 1 ess. Te bottom and middle of te film bot taken on linear forms wit te slope for; tey are essentially rigid for t 1. Tis beavior is consistent wit te dominance of te Marangoni effect causing te film to be rigid at tese locations. However, te top of film beaves differently and sows more sensitivity to te strengt of surface viscous effect. For very ig surface viscosity ( S = 1 ), te film acts essentially immobile from very early times, i.e. around. Before tis time, mobility is observed on te film surface. For a lower surface viscosity ( S = 1 ), te film surface starts by being more mobile up to about t =, and ten entering a transition time t 6. From approximately 1 t = 6, te surface becomes immobile and follows a straigt line wit slope. Terefore, it may be concluded tat in tis case, te film goes troug two transitions. Decreasing te surface viscous effect 1 furter by setting S =.1, sows a time interval t 1 wit slope > wic indicates a mobile surface. Tere follows a transition interval, and for t te top part of te film is rigid.

17 . SUMMARY AND CONCLUSION A matematical model is developed for te ydrodynamic beavior of a tin-liquid film draining over a vertical rigid wall in te regime of bot gravitationally- and surfactant-driven flow based on te Boussinesq-Scriven model. Bot surface tension and surface viscosities are nonlinearly dependent on te surfactant concentration on te free surface in tis model: tis follows from a property correlation for a pysical insoluble monolayer on water. Numerical experiments are performed on governing equations to cover te different regimes of te film freesurface and drainage beavior. It is found tat for low average surfactant concentration, te top part of te film can be almost swept clean of surfactant. In te same regime, a traveling wave on te film's free surface is observed wic acts as a boundary between mobile and rigid beavior of te film. Tis traveling wave was observed bot in low and ig surface viscosity cases, but wit different pase speed and coverage area. As in te free film case (9), tis natural flow can essentially clean te interface wen nonlinear properties are present in te surface. Similar beavior as been seen in driven flows as observed for a drop in an extensional flow (3,31) and for flow in a rotating cannel viscometer (3,33). In te latter case, te surface becomes immobile in te radial direction but not in te aimutal direction. In our two-dimensional film, te surface becomes essentially immobile wen te surfactant concentration becomes ig enoug. Wile te surface of a drop in an extensional flow witout nonlinear surface properties can be essentially swept clean (3), in our naturally-driven flows te nonlinear properties appear to be a requirement for tis to appen. It is sown tat a surfactant-driven upward flow is caused by aving eiter a ig initial gradient of concentration distribution, or a localied gradient after te early-time redistribution due to gravitationally-driven flow. Te surfactant concentration gradient, for an initially linear profile along te film surface, is found to ave virtually no effect on te late time film drainage beavior wen te average concentration is low enoug. Te beavior of te film was also studied for a weak Marangoni effect; it was found tat even for a weak Marangoni effect te film may act partially rigid and starts to sow a ealing effect for low surface viscosity. However, by increasing te surface viscosity, te film becomes essentially immobile except for a very localied flow reversal in te middle region of te film at very late times. Finally, it was also observed tat te drainage process sows a power-law beavior in different time periods based on te strengt of surface viscous effects. Wen te Marangoni effect is strong and fixed. However, it was 17

18 generally concluded tat all of te film surface tins wit power-law beavior at very late times regardless of surface viscous effects. An experiment is currently being run for comparison wit tis teory (3). Te results will be publised in a fortcoming paper. ACKNOWLEDGMENTS R.J. Braun gratefully acknowledges support from Dow Corning Corporation, and te National Science Foundation troug grant DMS A.H. Heidari is grateful for support from te Department of Matematical Science at University of Delaware during is extended visit. REFERENCES 1. Oron, A., Davis, S. H., and Bankoff, S. G., Reviews of Modern Pysics. 69, 931 (1997).. Cang, H. -C., Annu. Rev. Fluid Mec. 6, 13 (199). 3. Joo, S. W., Davis, S. H., and Bankoff, S.G., J. Fluid Mec. 3, 117 (1991).. Joo, S. W., Davis, S. H., Cemical Eng. Comm. 118, 111 (1991).. Cang, H. -C., Demekin, E. A., and Kopelevic, D. I., J. Fluid Mec., 3 (1993). 6. Braun, R. J., Snow, S. A., and Pernis, U. C., J. Colloid Interface Sci. 19, (1999). 7. Naire, S., Braun, R. J., and Snow, S. A., J. Colloid Interface Sci. 3, 91 (). 8. Naire, S., Braun, R. J., and Snow, S. A., SIAM J. Appl. Mat. 61, 889 (). 9. Naire, S., Braun, R. J., and Snow, S. A., Pys. Fluids, to appear. (1) 1. Stein, H. N., Adv. in Colloid and Interface Sci, 3, 17 (1991). 11. Nierstras, V. A., and Frens, G., J. Colloid Interface Sci, 7, 9 (1998). 1. Nierstras, V.A., and Frens, G., J. Colloid Interface Sci, 1, 8 (1999). 13. Baets, P.J., and Stein, H.N., Langmui, 8, 399 (199). 1. Kesgi, H. S., Kistler, S. F., and Scriven, L.E., Cem. Eng. Science 7, 683 (199). 1. Smit, M. K., Asymptotic metods for te matematical analysis of coating flows. in Liquid Film Coating, edited by Kistler, S.F. and Scweier, P.M. London: Capman and Hall., Park, C. W., J. Colloid Interface Sci., 38 (1991). 17. Hosoi, A. E. and Bus, J. W. M., J. Fluid Mec. (), in press. 18. Lope, J. M., and Hirsa, A., J. Colloid Interface Sci. 9, 7 (). 19. Boussinesq, M. J., Ann. Cim. Pys. 9, 39 (1913).. Scriven, L. E., Cem. Eng. Science 1, 98 (196). 1. Edwards, D. A., Brenner, H. & Wasan, D. T., Interfacial Transport Processes and Reology. Butterwort- Heinemann, Boston,

19 . Slattery, J. C., Interfacial Transport Penomena. Springer-Verlag, New York, Stone, H. A., Pys. Fluids A, 111 (199).. Wong, H., Rumscitki, D., and Maldarelli, C., Pys. Fluids 8 (11), 33 (1996).. Avramidis, K.S., and Jiang, T.S., J. Colloid Interface Sci. 17, 6 (1991). 6. Li, D., Slattery, J. C., AICE Journal 3, 86 (1988). 7. Braun, R. J., Snow, S. A. and Pernis, U. C., J. Colloid Interface Sci., 19, (1999). 8. Wilson, S. D. R., J. Eng. Mat., 9 (198). 9. Brenan, K. E., Campbell, S. L., and Petold, L. E., SIAM, Piladelpia (1996) 3. Pawar, Y. P., and Stebe, K. J., Pys. Fluids 8, 1738 (1996). 31. Eggleton, C. D., Pawar, Y. P., and Stebe, K. J., J. Fluid Mec. 38, 79 (1999). 3. Lope, J. M., and Hirsa, A., J. Colloid and Interface Sci. 6, 31 (1998). 33. Lope, J. M., and Hirsa, A., Proceedings of te nd International Conference on CFD in Minerals and Process Industries, CSIRO, Melbourne, Australia, Stone, H. A., and Leal, L. G., J. Fluid Mec. 3, 1 (199). 3. Howell, D. J., Hirsa, A. H., Heidari, A. H., and Braun, R. J., to be submitted (1). 19

20 Figure 1. Scematic representation of te model

21 σ(γ) Γ Figure. Dimensionless surface tension σ as a function of surfactant surface concentration Γ based on [9] and [1] wit te parameters: β= 6.3 dyne / cm, α = 6., σ = 66 dyne / cm (18). m 1

22 ϖ Γ Figure 3. Free surface sape, surface velocity ϖ, and surface concentration of surfactant Γ for various nondimensional time values for te following parameters S =., M = 1, P = 7, ˆΓ =., γ =.

23 ϖ Γ Figure. Free surface sape, surface velocity ϖ, and surface concentration of surfactant Γ for various nondimensional time values for te following parameters S =., M = 1, P = 7, ˆΓ =.6, γ =. 3

24 ϖ Γ Figure. Free surface sape, surface velocityω, and surface concentration of surfactant Γ for various nondimensional time values for te following parameters S =., M = 1, P = 7, ˆΓ =.8, γ =.

25 ϖ Γ Figure 6. Free surface sape, surface velocity ϖ, and surface concentration of surfactant Γ for various nondimensional time values for te following parameters S =., M = 1, P = 7, ˆΓ =., γ =.8 / L

26 ϖ Γ Figure 7. Free surface sape, surface velocity ϖ, and surface concentration of surfactant Γ for various nondimensional time values for te following parameters S = 1, M = 1, P = 7, ˆΓ =., γ =.8 / L 6

27 ϖ Γ Figure 8. Free surface sape, surface velocity ϖ, and surface concentration of surfactant Γ for various nondimensional time values for te following parameters S = 1, M =, P = 7, Γ=., ˆ γ =. L 7

28 ϖ Γ Figure 9. Free surface sape, surface velocity ϖ, and surface concentration of surfactant Γ for various nondimensional time values for te following parameters S =, M =, P = 7, Γ=., ˆ γ =. / L 8

29 6 3 middle bottom 1 top S=. S=1 S= t Figure 1. Free surface sape at te top ( = L ), middle ( = L ), and bottom ( = 3L ) of te film as a function of nondimensional time for various values of modified Boussinesq number and te following parameters: M = 1, P = 7, ˆΓ =., γ =. / L 9