J. Fluid Mec. (28), vol. 617, pp. 28 299. c 28 Cambridge University Press doi:1.117/s22112841x Printed in te United Kingdom 28 On te drag-out problem in liquid film teory E. S. BENILOV AND V. S. ZUBKOV Department of Matematics, University of Limerick, Ireland (Received 15 February 28 and in revised form 4 September 28) We consider an infinite plate being witdrawn (at an angle α to te orizontal, wit a constant velocity U) from an infinite pool of viscous liquid. Assuming tat te effects of inertia and surface tension are weak, Derjaguin (C. R. Dokl. Acad. Sci. URSS, vol. 9, 194, p. 1.) conjectured tat te load l, i.e. te tickness of te liquid film clinging to te plate, is l =(μu/ρg sin α) 1/2,wereρ and μ are te liquid s density and viscosity, and g is te acceleration due to gravity. In te present work, te above formula is derived from te Stokes equations in te limit of small slopes of te plate (witout tis assumption, te formula is invalid). It is sown tat te problem as infinitely many steady solutions, all of wic are stable but only one of tese corresponds to Derjaguin s formula. Tis particular steady solution can only be singled out by matcing it to a self-similar solution describing te non-steady part of te film between te pool and te film s tip. Even toug te near-pool region were te steady state as been establised expands wit time, te upper, non-steady part of te film (wit its tickness decreasing towards te tip) expands faster and, tus, occupies a larger portion of te plate. As a result, te mean tickness of te film is 1.5 times smaller tan te load. 1. Introduction Many industrial coating processes can be modelled by a simple setting were an infinite sloping plate is witdrawn from an infinite pool of viscous liquid (see figure 1a). Given tat te plate s speed is constant and, eventually, a steady state is establised, one usually needs to know te so-called load, i.e. te tickness of te liquid film tat te plate carries away from te pool. Tis classical problem was originally formulated by Derjaguin in 194, wo conjectured tat, if surface tension is negligible and te plate is being witdrawn slowly (ence, te effect of inertia is weak), te load l is ( ) 1/2 μu l =, (1.1) ρg sin α were ρ and μ are te liquid density and dynamic viscosity, U is te plate speed, α is te angle between te plate and te orizontal, and g is te acceleration due to gravity. In 1945, Derjaguin supported is conjecture by a quantitative argument; formula (1.1) as also been verified experimentally (Derjaguin & Titievskaya 1945) and numerically (Jin, Acrivos & Münc 25). Note, owever, tat Derjaguin s (1945) argument supporting formula (1.1) was more intuitive tan rigorous (see a discussion in Appendix A), and it left several Email address for correspondence: eugene.benilov@ul.ie
284 E. S. Benilov and V. S. Zubkov y * Liquid film U (a) α Pool x * x (b) x = t x Figure 1. Formulation of te problem: (a) te setting, (b) te matematical model. important issues unresolved. First, one can sow tat te problem admits infinitely many steady solutions, all corresponding to different loads and it is unclear wy te observed value, (1.1), is selected among all te oters. Secondly, since formula (1.1) does not ave a rigorous matematical foundation, it is difficult to establis its pysical limitations. In tis work, we sall remedy te above sortcomings. We sall demonstrate tat te observed steady state (and, ence, te load) is, paradoxically, determined by te unsteady part of te film adjacent to its tip (no matter ow far tat as moved away). Furtermore, it will be sown tat te film s tickness is close to te load only near te pool s edge, wile most of te region between te pool and film s tip is unsteady. As a result, te load (being a steady-state caracteristic) is not representative of te mean tickness of te film drawn from te pool but differs from it by an order-one factor. Finally, formula (1.1) will be sown to old only for moderate values of te plate s slope.
On te drag-out problem in liquid film teory 285 Te paper as te following structure: in 2, te problem is formulated matematically; in, we examine te case were te plate s slope is small; and, in 4, te general case is tackled. 2. Formulation of te problem Let te x -axis of te (x,y ) coordinate system (asterisks indicate tat te corresponding variables are dimensional) be directed along te plate and downwards see figure 1(a). We sall also introduce te (similarly oriented) velocities (u,v ), te pressure p, te film s tickness, and te time t. 2.1. Te limit of zero surface tension Following Derjaguin (1945), we sall assume tat inertia and surface tension are negligible, in wic case te flow is governed by te Stokes equations, ( ( 2 u 2 v p x = ρg sin α + μ x 2 + 2 u y 2 ), p y = ρg cos α + μ x 2 + 2 v y 2 ), (2.1) u + v =. (2.2) x y Te no-slip and no-flow conditions at te plate are u = U, v = at y =. (2.) Te kinematic condition at te liquid s surface can be written in a form reflecting mass conservation, + t x wereas te dynamic condition is were u dy =, (2.4) S n = at y =, (2.5) 2μ u ( u p μ + v ) [ S = ( x y x ] u μ + v ) 2μ v, n = x (2.6) p 1 y x y are te stress tensor and a (not necessarily unit) normal to te liquid s surface. Note tat our setting involves a contact point, separating te dry and wet parts of te plate wic, generally speaking, requires a certain boundary condition (see Dussan V. 1979; Sikmurzaev 26, and references terein). As a result, a boundary layer occurs near te contact point. Te widt of tis layer, owever, is extremely small, and all of te previous researcers dealing wit te drag-out problem neglected its effect. Accordingly, we sall assume tat te effect of dewetting is negligible and te contact point moves togeter wit te plate wic corresponds to te following boundary condition: = at x = Ut. (2.7)
286 E. S. Benilov and V. S. Zubkov At large distances from te edge of te pool, te pool s surface is unperturbed, i.e. (tan α)x as x +. (2.8) Te boundary-value problem (2.1) (2.8) determines te flow in te domain Ut <x < +, <y <. Next we introduce te following non-dimensional variables: x = εx, y = y l l, t = εut, = l l, (2.9) u = u U, v = v εu, p = (1 + ε2 ) 1/2 p, (2.1) ρgl were l is defined by (1.1) and ε =tanα (te use of symbol ε does not necessarily imply tat tan α 1). Substituting (2.9) (2.1) into te Stokes set (2.1) (2.8), we obtain p 2 u x =1+ε2 x + 2 u 2 y, 2 ( 2ε 2 u x p ( u ε 2 v + ε2 y x u x + v y p y = 1+ε4 2 v x 2 + ε2 2 v y 2, (2.11) =, (2.12) u = 1, v = at y =, (2.1) ) ( ) u x v + ε2 = y x at y =, (2.14) ) ( x 2ε 2 v ) y p = at y =, (2.15) t + u dy =, (2.16) x = at x = t, (2.17) x as x +. (2.18) In wat follows, equations (2.11) (2.18) will be used wen te plate s slope is orderone. If, owever, it is small, ε 1, one can take advantage of te lubrication approximation. Following te usual routine of lubrication analysis, one can treat (2.11) (2.15) as a boundary-value problem for u, v, andp and express tem as expansions in ε 2,e.g. )( u = 1+ (y y2 1 ) + O(ε 2 ). (2.19) 2 x Substituting tis equality into (2.16) and omitting te O(ε 2 ) terms, we obtain t + ( ) + x =. (2.2) x
On te drag-out problem in liquid film teory 287 Note tat te first term in te brackets corresponds to entrainment of te liquid by te plate s motion, te second term describes te effect of gravity, and te last one te ydrostatic pressure gradient due to variations of te film s tickness. Similar equations ave been derived, for similar problems, by Moriarty, Scwartz & Tuck (1991), Bertozzi & Brenner (1997), and Kalliadasis, Bielarz & Homsy (2). Finally, te asymptotic equivalents of te boundary conditions (2.17) (2.18) are = at x = t, (2.21) x as x +. (2.22) Te boundary-value problem (2.2) (2.22) is illustrated in figure 1(b). 2.2. Te effect of surface tension As mentioned before, tis paper is concerned wit te limit of negligible surface tension, but it is still instructive to briefly discuss te capillary-modified version of equation (2.2). Its derivation is very similar to tat of (2.2), resulting in t + x ( + x + γ x ) =, were σ (tan α) γ = (2.2) μu is te non-dimensional parameter caracterizing te importance of capillary effects (σ is te surface tension, recall also tat μ is te viscosity and U, te plate s velocity). To illustrate expression (2.2), let te liquid under consideration be glycerine at 2 C, for wic μ 1.41kg m 1 s 1, σ.6 N m 1. Ten, for a moderately small angle α = 1π, and a velocity U =.1m 9 s 1, formula (2.2) yields γ.4 wic can be safely regarded as a small parameter. Furtermore, wit te exception of mercury, surface tensions of common liquids range witin.2.8 N m 1, wereas te corresponding viscosities may differ by up to two orders of magnitude. Te viscosity of industrial polymers, for example, often reaces 4 kg s 1 m 1, wile tat of liquid foods suc as ketcup, mustard, or peanut butter ranges from 5 to 25 kg s 1 m 1. Tus, in many applications, γ is very small and capillary effects are negligible.. Te small-slope limit It appears obvious (and will be verified later) tat, even toug a steady state emerges near te edge of te pool, te film s tip always remains non-steady. At large times (t 1), te two regions become well separated and can be examined separately see.1. Ten, in.2, te asymptotic results will be verified and complemented by direct simulation of te small-slope equation (2.2)..1. Asymptotic results for t 1 For a steady solution, = (x), equation (2.2) yields + d dx = q (.1)
288 E. S. Benilov and V. S. Zubkov 2 1 c b a 12 1 8 6 4 2 2 x Figure 2. Te film s tickness (x) of a steady drag-out flow (determined by te boundary-value problem (.1) (.4)), for various values of te non-dimensional load: (a) =1, (b) = 2, (c) =1.7. Solutions (a) and(b) correspond to te endpoints of range (.5). were te constant of integration q is, pysically, te non-dimensional flux of liquid carried by te plate. We sall also require as x, (.2) were is te non-dimensional load, i.e. te tickness of te film carried by te plate infinitely far from te pool. Taking te limit x in (.1) and taking into account (.2), we obtain q = + 1. (.) Finally, condition (2.22) (wic matces te film to te pool) yields x as x +. (.4) Equation (.1) is separable and, tus, boundary-value problem (.1) (.4) can be readily solved (te actual solution is not presented ere, as it is bulky but several examples are sown in figure 2). Most importantly, solutions exist for all 1, i.e. te steady-state problem leaves te load undetermined. Te allowable range of, owever, can be reduced by analysing expression (2.19) for te liquid s velocity. Estimating its sign at y =, one can see tat, for > 2, te film s top layers slide back towards te pool wic corresponds to a source of liquid located at minus-infinity and makes te solution irrelevant pysically. Still, all values in te range 1 2 (.5) appear to give rise to pysically meaningful steady solutions. Only one of tese, owever, corresponds to te correct (experimentally verified) formula (1.1) for te dimensional load. We note tat paradoxes resulting from multiple solutions can often be resolved by a stability analysis (wic would usually sow tat all solutions, except one, are unstable). In te present case, owever, all solutions corresponding to range (.5) are stable (see Appendix B). As it turns out, te correct value of te load can only be identified by studying te non-steady part of te film. Fortunately, tat appens to be described by a
On te drag-out problem in liquid film teory 289 relatively simple self-similar solution wic, owever, does not necessarily matc te steady solution near te pool. Te only value of for wic te two solutions do matc is te correct one. To find te solution describing te non-steady part of te film, let (x,t)=t 1 f (ξ), (.6) were ξ =(x + t)t (.7) (tis substitution corresponds to a self-similar beaviour in te reference frame associated wit te film s tip). Substitution of (.6) (.7) into equation (2.2) yields [ ] f f + f ξ (f 1) =. (.8) Te condition at te tip of te film, (2.21), implies f () =. We ave found tree distinct asymptotics of equation (.8) compatible wit te above boundary condition (and are reasonably sure tat none as been missed): f aξ 1/4 + 4ξ + O( ξ 5/4) as ξ + (.9) 7 or f = ξ + a exp[ ξ 1 2lnξ + O(ξ)] as ξ + (.1) (in bot cases, a is arbitrary) or f = aξ + a (a 1)ξ 2 + O(ξ ) as ξ + (.11) (a 1). To distinguis between te tree possible asymptotics, note tat te evolutionary equation (2.2) preserves te slope at te film s tip (see Appendix C). Ten, assuming tat our solution originates from te initial condition were te pool s surface is unperturbed (ence, / x = 1), we conclude tat te desired asymptotic is (.1). Asymptotics (.9) implies an infinite slope at ξ =, wereas (.11) implies a finite one, but it still does not matc te derivative of te initial condition. At large ξ, in turn, equation (.8) yields f = ξ 1/2 + 1 ln ξ + b + O( ξ 1/2 ln 2 ξ ) as ξ + (.12) 12 or f = ξ + b(1 + ξ 1 +9ξ 2 ln ξ)+cξ 2 + O(ξ ln ξ) as ξ +, were b and c are arbitrary. It can be verified, owever, tat te latter asymptotic cannot be matced to te steady solution (x) near te pool and, tus, is irrelevant to te problem at and. To matc te remaining asymptotic, (.12), to (x), assume tat x is far from te pool, but not too close to te film s tip, 1 x t (.1) (wic, obviously, implies tat t 1). Ten, in solution (.6) (.7), we can replace f wit asymptotics (.12) and obtain t 1 (x + t)t 1. (.14)
29 E. S. Benilov and V. S. Zubkov f 2 1 1 2 4 5 6 7 8 9 1 ξ Figure. Te sape of te self-similar part of te film (i.e. te numerical solution of te boundary-value problem (.8), (.1), (.12)). Te dotted lines correspond to te asymptotics given by te first term of (.1) and te first two terms of (.12). Next, recall tat, in region (.1), te steady solution (x) is close to its limiting value ence, it can be matced to te limiting value (.14) of te self-similar solution only if =1. (.15) Recalling ow (and, ence, ) were non-dimensionalized (see (2.9) (2.1)), one can see tat (.15) agrees wit formula (1.1) for te dimensional load l. To complete our asymptotic results, we present te (implicit) solution of te steady problem (.1) (.4) corresponding to te correct value =1, ( ) 1 9 +8ln =9x. (.16) +2 1 We sall also need te function f (ξ) wic describes te sape of te non-steady part of te film. To find it, we solved te boundary-value problem (.8), (.1), (.12) numerically. Instead of zero and plus-infinity, te boundary conditions were set at a small and large values of ξ, wit te corresponding values of f determined by te first term of asymptotic (.1) and te first two terms of asymptotic (.12), respectively. Te resulting problem was solved troug an iterative procedure based on Newton s metod, and te solution is sown in figure. Summarizing tis subsection, we present te composite asymptotic solution, (x,t) (x)+t 1 f [(x + t)t] 1, (.17) were (x) is te steady solution emerging near te pool (given by (.16)); f (ξ) describes te sape of te film s non-steady upper part (determined by (.8), (.1), (.12)); and te unity represents te common part of te two solutions. In te limit t, (.17) olds for all values of x..2. Simulation of te evolutionary equation (2.2) Since our asymptotic solution (.17) is applicable only for large t, it cannot be traced back to t =. As a result, we cannot verify tat it corresponds to te initial condition tat we are interested in te one corresponding to te pool s surface being unperturbed, i.e. = x at t =. (.18) Tis issue can only be resolved by direct simulation of te small-slope equation (2.2). Te initial/boundary-value problem (2.2) (2.22), (.18) was integrated using te finite-element solver available in te COMSOL Multipysics package. Te solution
On te drag-out problem in liquid film teory 291 (a) 2 1 1 8 6 4 2 (b) 2 1 1 1 8 6 4 2 2 x Figure 4. Te evolution of te film for te small-slope case (determined by te initial/ boundary-value problem (2.2) (2.22), (.18)). Te curves are labelled wit te corresponding values of te time t. Te solid curves sow te numerical solution, te dotted curves sow: (a) te composite asymptotic solution (.17), (b) te steady solution (.16) and te self-similar solution (.6) (.7). computed, as well as te composite asymptotic solution (.17), are sown in figures 4 and 5. One can see tat, at t = 1, te two solutions agree reasonably well; at t = 2, tey are difficult to distinguis; and, for t, virtually indistinguisable. In addition to te composite asymptotic solution, we sow its components (te steady and self-similar solutions) in figure 4(b). Observe tat, even for te long-term evolution sown in figure 6(a), tere seems to be no visible sectoin were te solution is orizontal, i.e. equal to te load. Moreover, te emerging steady state becomes visible only if a small area near te pool s edge is enlarged (as in figure 6(b)). Te reason for tis is tat te steady-state region (wit approximately equal to te load) expands muc slower tan te non-steady region (wit variable ) ence, te portion of te plate occupied by te former is muc smaller tan tat occupied by te latter. Tis circumstance as crucial implications for te mean tickness of te film (wic is te most important caracteristic of te drag-out process). If te solution outside te pool were steady, te mean tickness would be equal to te load l, but, since te film is mainly unsteady and its tickness varies from l to, te mean is less tan l.
292 E. S. Benilov and V. S. Zubkov 2 1 4 2 1 4 2 1 x Figure 5. Te evolution of te film for te small-slope case (determined by (2.2) (2.22), (.18)). Te curves are labelled wit te corresponding values of te time t. Te solid curves sow te numerical solution, te dotted curves sow te composite asymptotic solution (.17) (for t, te numerical and asymptotic curves are virtually indistinguisable). (a) 2 1 1 5 1 9 8 7 6 5 4 2 1 x 1.4 1.2 (b) 1..8.6.4.2 12 11 1 9 8 7 6 5 4 2 1 x Figure 6. Te evolution of te film for te small-slope case (determined by (2.2) (2.22), (.18)). Te curves are labelled wit te corresponding values of te time t. Te solid curves sow te numerical solution, te dotted curve sows te steady solution (.16). Te area saded in panel (a) corresponds (b) (te latter illustrating te emergence of te steady state near te edge of te pool).
On te drag-out problem in liquid film teory 29 2 1 4 2 1 4 2 1 x Figure 7. Te same as in figure 5, but for te initial condition (.19) (.2) instead of (.18). To calculate te mean tickness for t 1, one can approximate te profile of te self-similar solution, f (ξ), by its large-distance asymptotics (.12) (wic is not valid only in a relatively small region adjacent to te film s tip). Tus, taking into account tat te lengt of te film grows linearly wit time, we obtain 1 mean non-dimensional tickness = lim t 1 (x + t) t dx = 2 t t. t Dimensionally, tis implies tat te film s mean tickness is 1.5 times smaller tan te load given by formula (1.1). Finally, we sall discuss wat appens if te pool s surface is initially perturbed. In particular, let te initial condition ave non-unit derivative at te pools s boundary, i.e. ( ) 1 at t =. x x= Ten, since te slope at te film s tip is conserved (see Appendix C), suc initial conditions seem to be incompatible wit our self-similar solution (.6) (.7), (.1) for wic / x at te film s tip equals unity. Te simplest option in tis case appears to be replacing te boundary condition (.1) wit (.11) wic allows non-unit derivative at ξ =, i.e. attefilm stip. It turns out, owever, tat te resulting boundary-value problem, (.8), (.11) (.12), as no solution. (Tis is not a matematical teorem but rater a conclusion based on numerical evidence.) To resolve tis issue, te small-slope equation (2.2) was simulated for various initial conditions wit non-unit derivative at te film s tip. Typical results, computed for = x + kxe x at t =, (.19) wit k = 1, (.2) are sown in figure 7. One can see tat te asymptotic solution (.17) agrees wit te numerical one everywere except te vicinity of te film s tip. Most importantly, te disagreement region rapidly contracts wit time and as no influence watsoever on te load or oter global caracteristics.
294 E. S. Benilov and V. S. Zubkov 1..8.6.4.2 1 2 4 5 6 7 8 9 α Figure 8. Te non-dimensional load (as determined by te Stokes equations (2.11) (2.18)) vs. te angle α of te plate s slope. Te dotted line corresponds to te small-α asymptotic result, (.15). Te same pattern was observed for oter values of k, as well as oter initial conditions. 4. Te case of finite slope If te slope of te film s surface is not small, te Stokes equations (2.11) (2.18) do not involve small parameters and can only be solved numerically. Tis task as been carried out originally by Jin et al. (25) for a more general model including surface tension. In te present work, in turn, surface tension is neglected, as we need to verify our asymptotic results presented above. Te Stokes equations (2.11) (2.18) were integrated using te Moving Mes (ALE) and Incompressible Navier Stokes modules of te COMSOL Multipysics. Te parameters were cosen in suc a way tat, in all runs, te Reynolds number was 1 4, so inertia was negligible. Starting from an appropriate initial condition, te problem was simulated until a steady state ad been establised everywere in te computational region, after wic te load could be measured. Te results are sown in figure 8. One can see tat te small-slope approximation works for fairly large angles, α 5. Furtermore, its error in tis range is 1, wic is a great deal better tan expected. Indeed, te small-slope equation (2.2) was derived by omitting O[(tan α) 2 ] and smaller terms from te Stokes set ence, for α 5, te truncation error sould be about (tan 5 ) 2.5. Tis discrepancy suggests tat some (probably more tan one) of te next-order corrections to te asymptotic load cancel out, improving te effective accuracy of te leading-order formula. Observe also te abrupt turn wit wic te numerical curve splits from its asymptotic counterpart at α 5. It appears tat a pitcfork bifurcation occurs at tis point: te old solution loses stability and te system switces to a lower branc. Suc an explanation implies tat tere exists anoter stable solution, corresponding to te upper branc (i.e. wit a load iger tan te observed one) owever, it as never appeared in our simulations. To resolve te discrepancy, recall tat te evolution begins from te initial condition wit a zero load; as a result, te system cannot reac te ig-load solution witout
On te drag-out problem in liquid film teory 295 passing troug te low-load one. Te latter, owever, is a steady state, and, once it is reaced, te system stops evolving and remains tere indefinitely. In oter words, te ig-load solution, if it exists, cannot be computed if te liquid is initially at rest and te pool s surface is unperturbed nor can it be observed (for te same reason) in te real world. 5. Summary and concluding remarks We ave examined te flow of a viscous liquid induced by an infinite sloping plate being witdrawn from an infinite pool. It was conjectured (Derjaguin 194) tat te load l, i.e. te tickness of te liquid film clinging to te plate, is given by formula (1.1). Tis conjecture, owever, was not supported by a consistent calculation, and none as followed since 194. In te present paper, formula (1.1) for l is derived from te Stokes set under an additional assumption tat te plate s slope is small, in wic case a relatively simple asymptotic equation, (2.2), can be derived. It is sown tat (2.2) as infinitely many steady solutions, all of wic are stable, but only one of tese corresponds to formula (1.1). Tis particular steady solution, (.16), is ten identified by matcing it to a self-similar solution for te film s non-steady upper part (described by (.6) (.7), (.8), (.1), (.12)). Te two asymptotic solutions are eventually combined as a composite solution, (.17), wic is applicable everywere. It is demonstrated (by direct simulation of equation (2.2), see.2) tat te composite solution is an attractor and, tus, provides large-time asymptotics for a wide class of initial conditions. It is also sown tat, even toug te region were te steady state as been establised expands wit time, te upper part of te film (wit its tickness decreasing towards te tip) expands faster. As a result, te mean value of te film s tickness is 1.5 times smaller tan te load. We ave also carried out direct simulations of te Stokes set and, tus, sowed tat te small-slope approximation is valid wen te angle of te plate s slope is less tan, approximately, 5. Remarkably, its accuracy witin tis range is about 1, wic is exceptionally ig. Finally, it would be interesting to extend te present approac to capillary effects and compare te results wit te extensive body of literature on steady drag-out flows wit surface tension (Landau & Levic 1942; Wilson 1982, etc.). Tis work was supported by te Science Foundation Ireland troug te Matematics Applications Consortium for Science and Industry (MACSI). Appendix A. Te paper by Derjaguin (1945) Unlike our work, Derjaguin (1945) (wose paper will be referred to as D45) assumes te plate to be motionless and te level of liquid in te pool to be dropping at a constant speed. Tis setting can be obtained from ours by canging te latter to te reference frame co-moving wit te plate. Te two approaces differ also in a more substantial way: unlike our work, D45 neglects te pressure gradient associated wit variations of te film s tickness. In terms of our non-dimensional variables, tis amounts to x 1. (A 1)
296 E. S. Benilov and V. S. Zubkov Accordingly, te last term in te small-slope equation (2.2) can be omitted, wic yields t + ( ) + = (A2) x (observe tat te cange of variables x co moving = x + t transforms (A 2) into D45 s equation (8)). Te general solution of (A 2) can be readily found (in an implicit form) by te metod of caracteristics, x = F ()+( 1+ 2 )t. (A ) Te function F () is determined by te initial condition, wic, somewat unexpectedly, D45 assumes to be = for t =, x > (A4) (see D45 s equation (1)). Eventually, D45 concludes tat F () = and (A ) yields x = +1. (A 5) t Note tat D45 s condition (A 4) implies tat te angle between te plate and te pool s surface is 9, wereas a more sensible assumption would be tat it is α. In terms of our non-dimensional variables (see (2.9) (2.1)), te latter amounts to = x for t =, x >. Accordingly, te undetermined function in (A ) becomes F ()=, and (A ) yields x = t +1+ 1 4t 1. (A 6) 2 2t Tis solution rectifies te inconsistency resulting from te incorrect initial condition (A 4). Note also tat, as t, te correct and incorrect solutions, (A 6) and (A 5), forget teir respective initial conditions and merge. Unfortunately, D45 contains anoter inconsistency, one tat cannot be easily rectified. It as several manifestations: first, it follows from (A 6) tat x t as x +, wic is inconsistent wit te boundary condition (2.22) and, tus, does not matc te pool. Secondly, even if it did (i.e. if x for large x), it would imply / x 1, wic violates assumption (A 1) on wic te original equation (A 2) is based. Tirdly, assumption (A 1) is also violated near te film s tip (x t) and te liquid level (x ). In oter words, D45 sould ave used te full equation (2.2) instead of te truncated one, (A 2). (Note tat (A 2) is still valid were te film s slope is small, e.g. far from bot te film s tip and te pool (provided tese regions are far apart, i.e. for large t). In tis case, te corresponding limit of solution (A 6) coincides wit te correct solution (.6) (.7) were f (ξ) is replaced wit its large-ξ asymptotics, i.e. te first term of (.12).).
On te drag-out problem in liquid film teory 297 Appendix B. Stability of steady solutions of equation (2.2) Let = (x)+ (x,t), (B 1) were is a steady solution of te small-slope equation (2.2) and is a disturbance. Substituting (B 1) into (2.2) and linearizing, we obtain t + ( + 2 d 2 x dx ) =. (B 2) x We sall confine ourselves to solutions wit exponential dependence on t, (x,t)=φ(x) e λt, for wic (B 2) yields λφ + d dx ( φ + 2 φ 2 φ d dx ) dφ =. (B ) dx We sall assume tat te disturbance is bounded at infinity, i.e. φ < as x ±. (B 4) (B ) (B 4) form an eigenvalue problem, were φ is te eigenfunction and λ is te eigenvalue. If tere exists an eigenvalue suc tat Reλ >, te steady state (x) is unstable. Observe tat, since as x, (B ) yields ( ) ( ) λ λ φ C 1 exp x + C 2 exp x as x, (B 5) were C 1,2 are constants of integration. Ten, it follows tat φ does not tend to zero (but oscillates) only if Reλ <, Imλ =; suc solutions are stable and, tus, do not need furter examination. In all oter cases, condition (B 4) requires one of te constants C 1,2 be zero, and (B 5) results in φ as x. (B 6) To analyse te beaviour of φ as x +, we sall need te corresponding asymptotics of (x) (wic can be extracted from (.1) (.4)): = x +x 1 + O(x 2 ) as x +. Ten, as follows from (B ), φ = C 1 [1 + O(x 1 )] + C 2 [x 2 + O(x )] as x +. Note tat solutions wit C 1 involve canges of te liquid s level in te pool far away from its edge and, tus, are irrelevant to te stability of te drag-out flow (moreover, it can be sown tat tey imply λ =, i.e. are stable anyway). As a result, we can assume φ as x +. (B 7) In wat follows, it is convenient to move te boundary conditions (B 6) (B 7) from infinity to finite points, φ = at x = x ±, (B 8)
298 E. S. Benilov and V. S. Zubkov ten take te limit x, x + + (te advantages of tis approac will be explained later). Next, we introduce a new variable ψ suc tat φ = ψ ) ( 2 exp 1 dx. In terms of ψ, te eigenvalue problem (B ), (B 8) is λ ψ ) [ ( ( 2 )] exp 1 dx = 1 d dψ 2 dx dx exp 1 dx, (B 9) ψ = at x = x ±. (B 1) Now, multiply (B 9) by ψ (were te asterisk denotes complex conjugate), integrate from x to x +, integrate te rigt-and side by parts, and take into account boundary conditions (B 1), wic yields x+ ( ψ 2 ) 2 1 x+ 2 ( ) λ exp dx dx = x 1 dψ 2 1 x dx exp dx dx. (B 11) Tis equality proves tat all eigenvalues λ are real and negative ence, te steady state under consideration is asymptotically stable. Ten, in te limit x ± ±,all λ are real and non-positive, i.e. te steady state is eiter asymptotically or neutrally stable. (Note tat wen te region were an eigenvalue problem is defined tends to infinity, some of te eigenvalues may accumulate near zero.) Observe tat, if te boundary conditions were set at infinity from te start, te integrals in (B 11) would ave infinite limits and we would need to deal wit teir convergence (wic is not difficult in principle, but involves cumbersome algebra involving large-distance asymptotics of ψ and te coefficients of equation (B 9)). Appendix C. Conservation of te film s slope at its tip Denote te film s slope at its tip by s, i.e. ( ) s = x, x= t and consider ( ) ds 2 dt = t x 2. (C 1) x 2 x= t Recalling tat (x,t) is governed by te small-slope equation (2.2), one can replace te t-derivative in (C 1) troug te x-derivatives and obtain { ( ds dt = x + 2 2 x x + ) [( ) 2 ( ) 2 ]} 2 x x 2 +2. 2 x 2 x x x= t Since, by definition, te film s tickness vanises at te tip (i.e. () x = t = ), te rigt-and side of te above equality is zero. REFERENCES Bertozzi, A. L. & Brenner, M. P. 1997 Linear stability and transient growt in driven contact lines. Pys. Fluids 9, 5 59.
On te drag-out problem in liquid film teory 299 Derjaguin, B. 194 Tickness of liquid layer adering to walls of vessels on teir emptying and te teory of poto- and motion-picture film coating. C. R. (Dokl.) Acad. Sci. URSS 9, 1 16. Derjaguin, B. 1945 On te tickness of te liquid film adering to te walls of a vessel after emptying. Acta Pysicocim. URSS 2, 49 52. Derjaguin, B. V. & Titievskaya, A. S. 1945 An experimental study of te tickness of a liquid layer left on a rigid wall by a retreating meniscus. Dokl. Acad. Nauk SSSR 5, 4. Dussan V., E. B. 1979 On te spreading of liquids on solid surfaces: static and dynamic contact lines. Annu. Rev. Fluid Mec. 11, 71 4. Jin, B., Acrivos, A. & Münc, A. 25 Te drag-out problem in film coating. Pys. Fluids 17, 16. Kalliadasis, S., Bielarz, C. & Homsy, G. M. 2 Steady free-surface tin film flows over topograpy. Pys. Fluids 12, 1889 1898. Landau, L. & Levic, B. 1942 Dragging of liquid by a plate. Acta Pysiocim. USSR 17, 42 54. Moriarty, J. A., Scwartz, L. W. & Tuck E. O. 1991 Unsteady spreading of tin liquid films wit small surface tension. Pys. Fluids A, 7 742. Sikmurzaev,Y.D.26 Singularities at te moving contact line. Matematical, pysical and computational aspects. Pysica D 217, 121 1. Wilson,S.D.R.1982 Te drag-out problem in film coating teory. J. Engng Mats 16, 29 221.