Steady rimming flows with surface tension

Transcription

1 J. Fluid Mec. (8), vol. 597, pp c 8 Cambridge University Press doi:.7/s79585 Printed in te United Kingdom 9 Steady rimming flows wit surface tension E. S. BENILOV,M.S.BENILOV AND N. KOPTEVA Department of Matematics, University of Limerick, Ireland Department of Pysics, University of Madeira, Portugal (Received 8 Marc 6 and in revised form October 7) We examine steady flows of a tin film of viscous fluid on te inside of a cylinder wit orizontal axis, rotating about tis axis. If te amount of fluid in te cylinder is sufficiently small, all of it is entrained by rotation and te film is distributed more or less evenly. For medium amounts, te fluid accumulates on te rising side of te cylinder and, for large ones, pools at te cylinder s bottom. Te paper examines rimming flows wit a pool affected by weak surface tension. Using te lubrication approximation and te metod of matced asymptotics, we find a solution describing te pool, te outer region, and two transitional regions, one of wic includes a variable (depending on te small parameter) number of asymptotic zones.. Introduction Rimming flows, i.e. flows of a viscous fluid on te inside of a rotating orizontal cylinder, ave important industrial applications and are also of great interest to teoreticians. Te case wen te fluid layer is sufficiently tin and is fully entrained by te cylinder s rotation as been examined by Moffatt (977), wo found a family of asymptotic solutions describing a steady-state distribution of liquid film on te cylinder s surface. If, owever, te net mass M of te fluid inside te cylinder exceeds a certain tresold, M, te film is no longer tin enoug for viscous entrainment to overcome gravity. As a result, fluid parcels cannot climb past te point were te tangent to te cylinder s surface is vertical and te effect of gravity is, tus, te strongest. Tese parcels accumulate on te rising side of te cylinder, were te film becomes so tick, tat te fluid starts falling back and forms a sock (Benjamin, Pritcard & Tavener 99, O Brien & Gat 998) similar to tat of a ydraulic jump or a tidal bore. It turns out, owever, tat te net mass of a sock solution may not exceed a certain tresold value, M. Pysically, tis means tat te cylinder s rising side can accommodate no more tan a certain amount of fluid, wit te excess fluid pooling at te bottom. As sown by Asmore, Hosoi & Stone (), te pool can be strongly affected by surface tension, but (as we sall see in 7) teir results are applicable only if M M. Te present paper re-examines te setting considered by Asmore et al. (), i.e. a steady rimming flow affected by weak surface tension, for M M.In, we formulate te problem and, in 4, 6, solve it asymptotically. In 5, asymptotic results are verified and complemented numerically.

2 9 E. S. Benilov, M. S. Benilov and N. Kopteva Ω R - θ R g Figure. Liquid film in a rotating orizontal cylinder.. Formulation of te problem Consider a tin liquid film on te inside of a cylinder of radius R wit orizontal axis, rotating about tis axis wit a constant angular velocity Ω (see figure ). We are concerned wit two-dimensional flows, described by polar coordinates (r,θ ), so te tickness of te film depends on te azimutal angle θ and time t (asterisks denote dimensional variables). We sall also introduce te acceleration due to gravity, g, te fluid s density ρ, kinematic viscosity ν, and surface tension σ. In wat follows, we use te following non-dimensional variables: = αr, θ = θ, t = Ωt, (.) were ( ) / νω α =. (.) gr We also introduce a non-dimensional parameter caracterising surface tension, ɛ = σ ( ) / νω. (.) ρgr gr.. Te governing equations Following most studies of rimming flows, we employ te so-called lubrication teory, assuming te film to be tin and te slope of its surface small resulting in te following evolution equation: t + θ [ ( cos θ + ɛ θ + θ )] = (.4) (Jonson 99). In equation (.4), te first term in square brackets describes viscous entrainment of te film by te rotation of te cylinder, te second term describes te effect of gravity, and te term involving ɛ, surface tension.

3 Steady rimming flows wit surface tension 9 We are concerned wit steady-state solutions, = (θ), in wic case equation (.4) yields ( ) d cos θ + ɛ dθ + d = q, (.5) dθ were te constant of integration q is, pysically, te non-dimensional flux. Equation (.5) sould be supplemented by te periodicity condition, (θ +π) =(θ). (.6) Problem (.5) (.6) describes a family of solutions wit various values of q. It is more convenient, owever, to caracterize te solutions by te non-dimensional net mass M, i.e. impose an additional constraint, π dθ = M. (.7) Te flux in tis case sould be treated as a function of te mass, q(m), to be determined from (.5) (.7). Following all previous work on tis and similar problems, we sall assume ɛ... Te leading-order results (ɛ =) To leading order, equation (.5) becomes cos θ = q. (.8) Tis cubic equation was examined by Moffatt (977), wo demonstrated tat, if q<, (.8) as a smoot unique solution (see figure a). For q =, (θ) as a corner at θ = (see figure b); note also tat te net mass of tis solution is M 4.44, as computed numerically. Recall tat M is te tresold separating continuous and sock solutions. In order to understand te nature of te sock solutions, note tat, in te first and fourt quadrants, equation (.8) as two positive roots, wit te smaller of te two corresponding to continuous solutions (see te left-and panels of figure ). For q =, te two roots touc at θ = and allow a continuous transition from te smaller root to te larger one (see te left-and panels of figure ). Observe tat, as θ π, π,te larger root becomes infinite ence, te solution must jump back to te smaller root before tat. As a result, two types of solutions exist (Benjamin et al. 99; O Brien & Gat 998), wit socks in te first and fourt quadrants see figures (b) and(a) respectively. Te former, owever, is unstable (Benjamin et al. 99; O Brien ), making te latter te only meaningful sock solution. Note tat all sock solutions correspond to te same value of flux, q =, i.e. tere exists a limiting amount of fluid per unit time wic can be transferred across te roof of te cylinder. Te excess fluid accumulates on te cylinder s rising side, between In real rimming flows, socks ave never been observed, as tey are always smooted by surface tension and/or a breakdown of lubrication teory due to large slopes of te film s surface. In te latter case, a cusp can form at te foot of te sock, similar to tat examined by Jeong & Moffatt (99).

4 94 E. S. Benilov, M. S. Benilov and N. Kopteva (a) (b) θ Figure. Rimming flows wit continuous profile: (a) q =.6; (b) q =. Te left-and panels sow te non-dimensional solution (θ): te solid/dased line corresponds to te smaller/larger root of equation (.8), te first and fourt quadrants are saded. Te rigt-and panels sow te real solution, te scale of wic implies α =.4, were α is given by (.) (te relatively large value of α as been cosen to improve visibility of te fine structure of te flow). te sock and te point θ =, wic seems to suggest tat te latter is a stagnation point. To clarify tis issue, we computed te streamlines (see Appendix A), according to wic θ = is not a stagnation point see figure. To resolve te paradox, consider te local flux Q(, θ) = cos θ, wic is, essentially, te left-and side of equation (.8). For points on te rising side of te cylinder, i.e. for θ ( π, π), tis expression as a maximum at =(cosθ) /,

5 Steady rimming flows wit surface tension 95 (a) (b) θ Figure. Rimming flows wit socks (as in figure, te spatial scale of te rigt-and panels corresponds to α =.4). (a) Te sock is located at θ = 5 π, in te fourt quadrant (stable solution); (b) te sock is located at θ = π, in te first quadrant (unstable solution). suc tat Q max = (cos θ) /. Clearly, te maximum global flux may not exceed te local flux at te most unfavourable point, i.e. were cos θ = and Q max is precisely. Te occurrence of te maximum flux at θ = expslains wy te excess fluid cannot pass troug and accumulates below it, and also wy te limiting flux is q = (tis value can only be canged by iger-order effects, suc as surface tension). Finally, as follows from equation (.8) for its larger root, (θ π) / as θ π, wic is an integrable singularity. Hence, no matter ow close te jump is to te cylinder s bottom, te net mass will not exceed a certain critical value, wic can be

6 96 E. S. Benilov, M. S. Benilov and N. Kopteva Outer zone Pool θ Figure 4. Numerical solution of equation (.5) wit M =9.5, ɛ =, and te corresponding streamlines. Te dotted line sows te boundary of te re-circulation region. Te spatial scale of te rigt-and panel corresponds to α =.4, were α is given by (.). Te lower left panel sows te asymptotic structure of te solution (te transitional regions are saded). found troug numerical integration of te limiting solution, M 6.9. Note tat te leading-order equation (.8) does not ave solutions wit M>M owever, tis does not necessarily mean tat te full problem (.5) (.7) as no suc solutions too. In oter words, a steady state wit M>M may well exist, but it must be crucially affected by surface tension.. Asymptotic analysis of problem (.5) (.7) In tis section, we sall take advantage of te smallness of ɛ and examine te problem troug matced asymptotics. Wit tis metod, it is crucial to correctly guess te solution s asymptotic structure (a task usually assisted by numerical results and pysical intuition), ten verify te guess by matcing te solutions found for different asymptotic zones. In te present problem, we sould expect a pool located about te cylinder s bottom, θ π, and strongly affected by surface tension. Because of te cylinder s rotation, fluid is being witdrawn from te pool and taken into te outer zone, θ π = O(), were surface tension is weak. Te two main zones sould be matced in te regions were te film leaves/re-enters te pool (te rigt/left-and transitional regions respectively). In order to make te general picture more compreensible to te reader, a typical solution (calculated numerically see 5) and te corresponding streamlines are sown in figure 4. Te asymptotic structure of te solution is sown in te lower left panel.

7 Steady rimming flows wit surface tension θ Figure 5. Te outer solution (.)... Te outer zone In te outer zone, capillary effects are weak ence, te full equation (.5) can be replaced wit its truncated version, (.8). Ten, if tere is a supply of fluid at te cylinder s bottom (te pool), we can assume tat te cylinder s rotation witdraws from it as muc fluid as it can, and te flux is close to its limiting value, q. (.) It is also clear tat fluid accumulates on te rising side of te cylinder ence, te larger root of (.8), (.) sould be cosen tere. Tus, in te outer zone, te smaller positive root of(.8), (.) if θ [, π), te only positive root of(.8), (.) if θ [ π, π], te larger positive root of(.8), (.) if θ ( (.) π, π]. Note tat, at θ = π, te rigt-and and left-and limits of solution (.) do not matc, ( ) / as θ π, (.) θ π as θ π (.4) (see figure 5). Tus, in te inner zones, te solution coming from te rigt will need to be brougt down and matced to tat coming from te left.

8 98 E. S. Benilov, M. S. Benilov and N. Kopteva.. Te pool zone In te inner zones (i.e. in te pool and transitional regions), it is convenient to replace θ wit x = θ π. Ten, equation (.5) becomes ( ) d sin x + ɛ dx + d = q. (.5) dx Since all inner zones are located near x =, we can assume, to leading order, sin x x. Note also tat te widts of te inner zones are small ence, te first derivative in (.5) is muc smaller tan te tird derivative and, tus, can be omitted. Ten, replacing te flux wit its limiting value (as justified in.), we rewrite (.5) in te form x + d ɛ dx =. (.6) In wat follows, inner zones will be given (positive or negative) numbers, wit denoting te pool zone. For example, te stretced variables for te pool are = H, x = x X, (.7) were H and X are te te pool s caracteristic dept and widt. We assume tat te sape of te pool is determined by te balance of gravity (te second term on te left-and side of (.6)) and surface tension (te tird term). Ten, substituting (.7) into equation (.6) and balancing its second and tird terms, we obtain X = ɛh. (.8) X Pysically, te most interesting regime is te one were te pool contains an order-one amount of fluid (i.e. its mass is comparable to tat elsewere in te flow), Equations (.8) (.9) yield H X =. (.9) H = ɛ /5, X = ɛ /5. (.) Equation (.) will be referred to as te supercritical regime as, in tis case, te net mass of te flow exceeds M by an order-one value. We sall also consider (in 6) a near-critical regime, suc tat M M and H X. Now, rewrite equation (.6) in terms of te stretced variables (.7), (.) and omit small terms: x + d =, (.) dx wic yields = const + const x +const x + 4 x4. (.) As verified aposteriori, tis polynomial can be matced to te solutions in te oter zones only if it as two pairs of double roots in wic case (.) can be rearranged as = 4( x W ), W x W, (.)

9 Steady rimming flows wit surface tension x Figure 6. Te pool solution (.) (te curves are marked wit te corresponding values of W). were W is te scaled alf-widt of te pool. In te rest of our analysis, W will remain free (i.e. we expect to find a one-parameter family of solutions); W also determines te net mass of te solution, M M + ɛ /5 W ɛ /5 W ( ɛ /5 ) d(ɛ /5 x )=M + 45 W 5, were M is te outer-zone s contribution. Examples of solution (.) are sown in figure 6. Finally, we express te asymptotics of (.) at te boundaries of te pool in terms of te non-scaled variables, 6 ɛ /5 W ( x + ɛ /5 W ) 6 ɛ /5 W ( x ɛ /5 W ) for x ɛ /5 W, (.4) for x ɛ /5 W, (.5) wic will be used for matcing te pool solution to te neigbouring zones... Discussion Observe tat te pool solution (.) involves a free parameter, W, wereas te outer solution (.) is entirely fixed, i.e. te latter cannot adjust to various values of te former. Te adjustment as to occur in te transitional regions, wic sould be expected to ave fairly complicated structure. Alternatively, one could try to utilize an outer solution wit a non-limiting flux (q< ), so q could adjust to te current value of W (as done by Asmore et al. (), see 7). It turns out, owever, tat suc outer solution cannot be matced to te pool wit supercritical parameters (.). Still, since solutions wit non-limiting flux migt occur in oter parameter regimes, tey will be examined in 7.

10 E. S. Benilov, M. S. Benilov and N. Kopteva.4. Te left-and transitional region Since Landau & Levic (94), te structure of transitional regions between a viscosity-dominated zone and a surface-tension-dominated zone as been examined using te so-called Landau Levic (LL) equation. In te present problem, it can be obtained from te general equation, (.6), by neglecting its second (gravity) term, + d ɛ dx =. (.6) Matcing te solution of (.6) to te outer solution (.) implies as x. Matcing of (.6) to te pool is less obvious, as none of te tree possible asymptotics of te LL equation, as x +, (.7) const x as x +, (.8) x (9lnx) / as x +, (.9) matces te pool solution (.4). Still, a zone wit te LL equation does exist in te problem at and, and te correct boundary condition is, in fact, (.9). Tere is, owever, an auxiliary zone, separating te LL zone from te pool and transforming te linear/logaritmic beaviour of (.9) into te quadratic beaviour necessary for matcing to te pool. In te remainder of tis subsection, te above qualitative picture will be cast into te formal framework of matced asymptotics. Te auxiliary and LL zones will be denoted by and respectively. Anticipating tat te logaritm in asymptotics (.9) will give rise to a logaritmically small parameter, we introduce δ = O[(ln ɛ) ]. Ten, te stretced variables for te auxiliary zone are = ɛ /5 δ /, x = x + ɛ/5 W ɛ 4/5 δ / (.) (observe tat tis zone is located near te pool s left-and boundary, x = ɛ /5 W). Substituting (.) into equation (.6) and omitting small terms, we obtain d dx =. (.) Matcing of wit te pool requires W x 6 as x +, (.) wereas matcing of to te LL zone, as ascertained aposteriori, requires te solution to approac zero wit a certain value of its derivative, namely, d 9 / dx as. (.) Equation (.8) appears to matc te quadratic beaviour of (.4), but one can verify tat te orders of te two solutions do not matc.

11 Steady rimming flows wit surface tension 5 (a) 5 (b) x x Figure 7. Te left-and transitional region: (a) te Landau Levic zone (.6) (.8); (b) te auxiliary zone (.4) (te curves are marked wit te corresponding values of W). Te solution of te boundary-value problem (.) (.) is =9 / (x + D)+ W (x 6 + D), (.4) were te integration constant D is, essentially, te distance by wic zone sifts te left-and boundary of te pool (it sould eventually be related to W). Examples of solution (.4) are sown in figure 7b. Te stretced variables of te LL zone are =, x = x + ɛ/5 W + ɛ 4/5 δ / D ɛ / (.5) (observe tat tis zone is located near te sifted left-and boundary of te pool, x = ɛ /5 W ɛ 4/5 δ / D). As expected, substitution of (.5) into (.6) and omission of small terms yields te LL equation, + d = dx. (.6) Te boundary conditions for are as x, (.7) x (9 ln x ) / as x +, (.8) wic ave been discussed above and will be verified below (by matcing to te neigbouring zones). Te boundary-value problem (.6) (.8) as been solved numerically via sooting, and te solution is plotted in figure 7(a). It remains to ascertain tat matces. As follows from (.4), 9 / (x + D) as x D, wic, in terms of te non-scaled variables, corresponds to 9 / ɛ / δ / x for x = O ( ɛ 4/5 δ /), (.9) were x = x + ɛ /5 W + ɛ 4/5 δ / D.

12 E. S. Benilov, M. S. Benilov and N. Kopteva Te corresponding expression for zone follows from (.8) and (.5), [ ( )] / x ɛ / x 9ln for x = O(ɛ / ). (.) ɛ / Observe tat te logaritmic function in (.) corresponds to te logaritmic constant δ in (.9). Since use of van Dyke s principle is not entirely safe for solutions involving logaritms (Hinc 99, 5..6), we sall verify (.) via matcing by an intermediate variable. Estimating te terms omitted in te derivation of (.), one can sow tat solution (.4) is applicable for x ɛ 4/5 ence, te intermediate variable can be taken to be x = ɛ 4/5 δ k const, k (, ). Substituting tis expression into (.9) and (.) and equating tem, one sould neglect te logaritms of const and δ, but retain tose of ɛ, wic yields δ = 5 ln ɛ. (.) Tus, (.9) matces (.) subject to δ satisfying (.). Finally, recall tat te sift D of (x ) still remains undetermined (see (.4)); furtermore, te problem (.6) (.8) is invariant wit respect to replacing x x + const. Tus, to fix te location of te solution, we need more boundary conditions, wic, owever, can only be obtained in te next order, were te equivalents of (.) and (.6) involve te spatial variable. Following Landau & Levic (94), and oter autors wo encountered similar difficulties in similar problems, we sall not carry out tis cumbersome calculation, as its result does not affect te leading-order matcing. Still, one sould keep in mind tat te true solutions can be located at an order-one distance from tose sown in figure Te rigt-and transitional region First, observe tat, in te vicinity of te pool, te outer solution is strongly asymmetric (see figure 5), wic suggests tat te rigt-and transitional region differs from its left-and counterpart. Tis is indeed te case: te latter, for example, includes only two asymptotic zones, wereas te former will be sown to comprise a variable (depending on ɛ) number of zones. Tis is an unusual feature, aving bot matematical and pysical implications, wic will be discussed in tis subsection and 4. respectively. Note also tat tis subsection involves some lengty calcualtions, so te readers wit interests more in pysics tan matematics are advised to skip it and move to 4. First we sall consider zone, located next to te pool. It can be sown tat te only set of stretced variables tat guarantees te two zones matcing is = ɛ, x /5 = x ɛ/5 W. (.) ɛ 4/5 Ten, equation (.6) yields, to leading order, + d =, (.) dx wic is similar to te LL equation (.6). Te left-and boundary condition follows from te matcing of (.) to te pool asymptotics (.5), W x 6 as x. (.4)

13 Steady rimming flows wit surface tension (a).5 (b) x x Figure 8. Te rigt-and transitional region (te curves are marked wit te corresponding values of W): (a) te numerical solution of problem (.) (.5); (b) te explicit solution (.4) (.4). As x +, equation (.) admits eiter quadratic or linear/logaritmic asymptotics, similar to (.7) and (.8) respectively. To coose te correct one, note tat, in te next zone, te former gives rise to anoter peak, wit parameters comparable to tose of te pool, wic as never been observed in numerical simulations (see 6). Tus, we sall assume x (9lnx ) / as x +. (.5) Tus, grows as x +, wic, again, indicates a peak in te next zone but te amplitude of tis peak will be muc smaller tan tat of te pool, as indeed corroborated by our numerical results. Problem (.) (.5) was solved numerically via sooting. Several examples of (x ) are sown in figure 8(a). In te next zone, te stretced variables are = ɛ / ξ /, x = x ɛ/5 W, (.6) ɛ 7/ ξ /6 were te (logaritmically small) parameter ξ is related to ɛ by ξ = 5 lnɛ. (.7) Substitution of (.6) into equation (.6) yields, to leading order, W + d =. (.8) dx Te solution to tis equations is given by a cubic polynomial wic, generally, as tree roots. However, it can be sown tat tis zone can be matced to te next one only if te two larger roots coincide. Hence, denoting te resulting double root by F, we require d =, = at x = F. (.9) dx

14 4 E. S. Benilov, M. S. Benilov and N. Kopteva Matcing of tis zone wit te previous one is similar to tat of zones and, and yields d =, =9 / at x = G, (.4) dx were G is te solution s smaller root. Finally, observe tat te previous zone, zone, is muc narrower tan eiter of its neigbours, zone and te pool compare (.) to (.6) and (.7), (.). Tus, to leading order, we sall require tat te left-and boundary of zone coincide wit te rigt-and boundary of te pool, i.e. Te solution of (.8) (.4) is were G =. (.4) = 6 Wx (x F ), x F, (.4) F = / 5/6 (.4) W / is te widt of tis zone. Examples of tis solution are sown in figure 8(b). Observe also tat (.4) (.4) imply W / / (x /6 F ) as x F, (.44) wic we sall use wen matcing tis zone to te next one. Te next zone s scaling is, in fact, fully determined by matcing to (.44), = ɛ / ξ /6, x = x ɛ/5 W, (.45) ɛ 7/ ξ /6 were W = W ɛ / ξ /6 F (te second term in W reflects te fact tat zones and are separated by te widt of zone ). Ten, te boundary-value problem for is + d dx =, (.46) W / / /6 x as x, (.47) x (9lnx ) / as x +. (.48) Comparison of tese equations wit (.) (.5) sows tat can be obtained by re-scaling. Zones and are followed by a sequence of similar asymptotic zones: te evennumbered ones are described by te explicit solution (.4) (.4); and te oddnumbered ones, by te boundary-value problem (.46 ) (.48). Te corresponding stretced variables are n = n+ = ɛ / ξ / n ɛ / ξ /6 n, x n = x ɛ/5 W n ɛ 7/ ξ /6 n, x n+ = x ɛ/5 W n ɛ 7/ ξ /6 n for n 4, (.49) for n +, (.5)

15 Steady rimming flows wit surface tension 5 were W n = W ɛ /( ξ /6 + ξ / ξ /6 ) n F is te cumulative widt of te even-numbered zones (te odd ones are muc narrower). ξ n are linked recursively to ξ n (and, eventually, to ξ )by ξ n = 6. (.5) ln ξ n To visualize te asymptotic structure of te solution, observe tat, as follows from (.49), te caracteristic eigts and widts of te peaks located in even-numbered zones are H n = ɛ / ξ / n, X n = ɛ 7/ ξ /6 n. Te corresponding parameters of te trougs (located in odd-numbered zones) can be derived from (.5), H n+ = ɛ / ξ /6 n, X n+ = ɛ 7/ ξ /6 n. Note also tat equations (.7) and (.5) imply tat ( ) ( ) ( ) ξ = O, ξ 4 = O, ξ 6 = O,..., ln ɛ ln ln ɛ ln ln ln ɛ i.e. ξ n increase wit n and, ence, te peaks become smaller and narrower, wile te trougs become sallower and wider. Sooner or later (say, at n = N), ξ N becomes comparable to unity, and te corresponding peak and troug are indistinguisable, as H N+ = H N = ɛ /, X N+ = X N = ɛ 7/. Tus, we need to introduce a limiting asymptotic zone, wit te stretced variables given by N = ɛ, x / N = x ɛ/5 W N. (.5) ɛ 7/ Substituting (.5) into (.6) and omitting small terms, we obtain N W N + d N N =. (.5) dxn Te matcing of N to te outer solution (.4) requires ( ) / N as x N +. (.54) W Before we discuss te left-and boundary condition, observe tat te previous equations (.) and (.8) are bot included in (.5) as limiting cases, as tey can be derived from it by re-scaling te variables. In fact, te solutions of te previous zones are simply an asymptotic description of N for large negative values of x N. Tus, we sould require tat, as x N, N consists of a sequence of maxima and minima described by (.4) (.4) and (.46) (.48). In practice, owever, we just need to integrate equation (.5) from left to rigt, starting from a maximum, and sow tat, eventually (at te positive infinity), we can satisfy condition (.54). Furtermore, since te spatial variable x N does not explicitly appear in (.5) (.54), we can start integration from x N =, i.e. te left-and

16 6 E. S. Benilov, M. S. Benilov and N. Kopteva 4 N x N Figure 9. Te numerical solution of te boundary-value problem (.5) (.55) for W =, A =,,,. boundary condition amounts to d N N = A, = at x N =, (.55) dx N were A is a large positive constant. It turned out tat te boundary-value problem (.5) (.55) cannot be solved numerically troug sooting, owing to te exponential dependence of te solution on te initial condition and oter complicating factors. As a result, we used te same iterative procedure as used for te exact boundary-value problem (.5) (.6) (see 5.). Te results are sown in figure 9: one can see tat, no matter ow large te amplitude of te initial maximum is, te solution invariably reaces te correct limiting value as x n +. It also as te correct (oscillating) structure, wit a narrow minimum following a wide maximum. Effectively, te boundary-value problem (.5) (.55) describes all of te rigt-and transitional region starting from zone. 4. Discussion 4.. Te rigt-and transitional region from a pysical viewpoint Matematically, te most interesting aspect of te above solution is te variable number of asymptotic zones in te rigt-and transitional region (recall tat N depends on ow small ɛ is). Tis unusual feature as been found earlier for a similar setting namely, a liquid film sliding down a vertical wall into a pool (Wilson & Jones 98). In tat case, te asymptotic structure of te solution also involved indefinitely many zones, but, in te exact solution, only one or two peaks (capillary ripples) could be observed. Te discrepancy was due to te fact tat te sequence of formally small parameters wic determined te zones rapidly increased, so in some cases even te second one was comparable to unity.

17 Steady rimming flows wit surface tension 7 Figure. Te azimutal velocity profile (4.) of a rimming flow. In te present case, te sequence of small parameters ξ n, on wic te separation of scales of te asymptotic zones is based, increase exponentially, i.e. even faster tan tose in te work of Wilson & Jones (98). For ɛ =, for example, equation (.7) yields ξ.6 wic is indeed small, but ten (.5) wit n = yields ξ 4. = O(). Tus, few ripples sould be expected in te numerical solution of te present problem (see te next section). Note also tat Wilson & Jones (98) observed ripples on a film entering apool, wereas a film being witdrawn from a pool turned out to be smoot (Wilson 98). Tis is at odds wit te present case, were te variable number of zones (pysically, corresponding to ripples) were observed in te rigt-and transitional region, were te film is being witdrawn from te pool. To resolve te contradiction, we need te expression for te non-dimensional azimutal velocity v, i.e. te component tangential to te cylinder s wall, as a function of te film s non-dimensional dept z (see Appendix A, formula (A )). Ten, since satisfies te steady-state equation (.8), expression (A ) can be rearranged into ( ) z ( q) v =+ z. (4.) Now, recall tat, in bot transitional regions,,q, and ence (4.) becomes ( ) z v + z. (4.) Te dependence v vs. z is sown in figure. One can see tat, even toug te net flux q is positive (directed to te rigt), te near-surface velocity is directed to te left. Given tat capillary ripples are most sensitive to wat appens near te fluid s surface, it comes as no surprise tat tey occur in te rigt-and transitional region, were te near-surface layer flows towards te pool just as it does in te setting considered by Wilson & Jones (98).

18 8 E. S. Benilov, M. S. Benilov and N. Kopteva 4.. Te applicability of te solution obtained Te lubrication teory, on wic all results in tis paper are based, requires te film to be tin and its slope small. Tus, we sould make sure tat our asymptotic solution complies wit tese requirements. First, estimating te dept of te pool (were te film is tickest) from (.) and also recalling ow was non-dimensionalized (see (.)), we obtain αɛ /5. (4.) Secondly, it can be verified tat te steepest slope occurs in zone of te left-and transitional region. Extracting te scales for and x from (.) and (.), ( H = ɛ /5 5 ) / (, X = ɛ 4/5 5 ) /, ln ɛ ln ɛ and estimating te slope as αh/x, weobtain ( αɛ / 5 ) /. (4.4) ln ɛ Note also tat, apart from surface tension, rimming flows are affected by ydrostatic pressure. Te latter can be described using te lubrication teory (Benjamin et al. 99), resulting in ( ) ( d cos θ + ɛ dθ + d + α dθ ) d dθ sin θ + 4 cos θ = q, (4.5) were α is given by (.). By comparison wit (4.5), our equation (.5) misses te expression involving α; ence, to validate our results, we need tat to be muc smaller tan te largest of our terms. In te outer zone, were te solution is determined by te first two terms of (4.5), tis implies α, wic condition is weaker tan (4.) (4.4) and, tus, can be discarded. In te inner zones, in turn, pressure sould be compared to surface tension (wic is always a leading-order effect); one sould also keep in mind tat te spatial scale X of te solution is small. Comparing te largest of te surface-tension terms (te second one) to te largest of te pressure terms (te first one), we obtain ɛ α. (4.6) X Condition (4.6) is most stringent in te inner zone wit te largest X, wic is zone N of te rigt-and transitional region. Substituting X = X N = ɛ 7/ into (4.6), we obtain ɛ 8/5 α. (4.7) Tis condition is stronger tan (4.) (4.4) and, tus, can be regarded as te sole applicability criterion of our results. Note, owever, tat all of te above conditions are local : if tey do not old, tis affects only some of te asymptotic zones, wile te solution in oter zones remains intact.

19 Steady rimming flows wit surface tension 9 5. Numerical results 5.. Te metod Te simplest way to compute te solution of te steady-state equation (.5) is to simulate te evolutionary equation (.4) wit a suitable initial condition, say, = const at t =. (5.) In te limit t, te solution becomes steady and, by varying te constant in (5.), one can compute steady states for various values of te net mass M. Tisapproac was used by Asmore et al. (), and we also attempted to use a somewat enanced version of it (wit te implicit Euler metod for te time derivative, fivepoint second-order upwind discretization of spatial derivatives, and a sopisticated mes refinement algoritm). However, even toug simulating te evolution equation (.4) yields an accurate solution for, it does not for te flux q. Te problem is tat, to calculate q, one needs to compute / θ, te accuracy of wic is muc lower tan tat of. Moreover, since all interesting particular cases in tis problem are concentrated near q,a ig-accuracy computation of tis quantity is essential. As a result, we solved te steady-state problem (.5) (.7) for and q (for a given mass M) via an iterative procedure based on Newton linearization. Equation (.5) was divided by, ten replaced wit a linear ODE, n 6 ( n+ n ) n cos θ + ɛ = q n n ( dn+ dθ + d n+ dθ 9q n ( n+ n ) 4 n ) + (q n+ q n ), (5.) n were ( n,q n ) represent te current iteration and ( n+,q n+ ), te next one. Equation (5.) was discretized using te five-point (fourt-order) symmetric approximations of te derivatives and solved togeter wit te discrete equivalents of n+ (θ +π) = n+ (θ), (5.) π n+ (θ)dθ = M (5.4) troug Crout s UL-decomposition algoritm (see Press et al. 99). Tis approac turned out to be faster and more accurate tan simulating te evolution equation (.4). In some cases, were a small cange in M gives rise to a large cange in q, itis more convenient to solve equation (.5) for a given q, witout imposing condition (.7). In tis case, te last term in equation (5.) sould be omitted and condition (5.4) discarded. 5.. Te results A typical pool solution and te corresponding streamlines are sown in figure 4. One can see tat most of te pool is occupied by a large re-circulation area. Using te computed solution, we ave also verified te accuracy of te asymptotic outer solution (.); it turned out tat, for ɛ = 4, te asymptotic and numerical solutions are indistinguisable everywere except for te pool zone (see figure a) Te agreement between te numerical and asymptotic solutions for te pool (see figure b) is worse tan tat for te outer solution. To understand wy tis is so,

20 E. S. Benilov, M. S. Benilov and N. Kopteva (a) (b) θ θ Figure. Numerical solution of te exact equation (.4) wit M =9.5 and(a) ɛ = 4,(b) ɛ =. Te dotted line sows te outer asymptotic solution (.) (te left-and panels) and te pool solution (.) (te rigt-and panels). note tat te accuracy of te former is ɛ /5 (wic can be verified by substituting te stretced variables (.7), (.) into equation (.6) and estimating te magnitude of te next-to-leading-order term). Te accuracy of te outer solution, in turn, is ɛ, wic is muc iger. However, te relatively low accuracy of te asymptotic solution for te pool manifests itself, mainly, in te pool s incorrect position (see figure b), wereas its sape is predicted quite well. To illustrate tis, we introduce te widt w.5 of te pool at alf of te maximum dept max, and similar widts, w.7 and w.9,at.7 max and.9 max (see figure a). Tese parameters are plotted as functions of max alongside teir asymptotic counterparts (based on solution (.)) in figure (b). Clearly, te numerical and asymptotic results agree reasonably well even for ɛ = (note tat te neigbouring regions affect w.5 more tan te oter two widts; ence, te accuracy of its asymptotic value is te lowest). Te disagreement between te numerical and asymptotic positions of te pool will be qualitatively explained in te next section. Note also tat no more tan one ripple as been observed in te

21 Steady rimming flows wit surface tension max.9 max w.9 (a) (b).7 max.5 max w.7 w.5 w.9, w.7, w θ max Figure. Te sape oftepool.(a) Te definitions of w.9, w.7,andw.5 ;(b) w.9, w.7, and w.5 (te triangles, squares, and circles respectively) vs. max, for te numerical solution of te exact equation (.5) for ɛ =. Te dotted lines sow te corresponding asymptotic results based on te pool solution (.). Te vertical dased line separates solutions wit a sock from tose wit a pool. rigt-and transitional region, and sometimes (for larger ɛ and smaller M) wedid not observe any ripples at all. 6. Te near-critical regime Note tat te maxima of subcritical sock solutions (for wic M < M )are positioned inside te fourt quadrant see figure (a). On te oter and, te maxima of te supercritical solutions constructed in (for wic M>M ) are located at exactly θ = π. Tus, to trace ow te former solutions transform into te latter, one sould consider a near-critical regime, for wic M M. It sould be te most general regime, suc tat te neigbouring ones can be obtained as its limiting cases. It can be verified aposteriori tat te most general regime corresponds to te following scaling of te pool: = ɛ /9 γ 4/9, x = were te (logaritmically) small parameter γ is x, (6.) ɛ /9 γ /9 γ = 9 ln ɛ. (6.) Substitution of (6.) into (.6) yields te same equation and solution as in te supercritical regime, i.e. (.) and (.) respectively. Even te rigt-and boundary condition is te same, i.e. we sould require tat te solution approac zero wit zero derivative. At te left-and boundary, owever, te derivative sould be equal to 9 / (wic can be sown to be te only coice tat ensures matcing to te left-and transitional region).

22 E. S. Benilov, M. S. Benilov and N. Kopteva x Figure. Te pool solution (6.5), (6.7) for te near-critical regime (te curves are marked wit te corresponding values of W ). Tus, denoting te left-and and rigt-and boundaries of te pool by A and B, we require d =9 /, dx = at x = A, (6.) d =, dx = at x = B. (6.4) Under conditions (6.) (6.4), solution (.) becomes = 4 A)(x + A +B)(x B), A x B, (6.5) were A and B satisfy B) =9 /. (6.6) It is convenient to introduce ere W = (B A), werew is te scaled alf-widt of te pool (te equivalent of W introduced for supercritical solutions). Ten, (6.6) can be satisfied by putting A = 5/ W W, B = 5/ W + W. (6.7) As W, solution (6.5), (6.7) moves to te rigt, wile its amplitude decays (see figure ) tis limit describes solutions wit socks located near te bottom of te cylinder. In te opposite limit, W, solution (6.5), (6.7) becomes more symmetric and tends to te supercritical pool solution (.). Te structure of te transitional regions for te near-critical regime is similar to tat of te supercritical one wit one exception: te left-and transitional region of te former consists of a single zone. Tis zone is described by te Landau Levic equation, te linear/logaritmic solution (.8) of wic matces te pool exactly, so tere is no need for an auxiliary zone.

23 Steady rimming flows wit surface tension It sould be noted, owever, tat te near-critical solution as a muc lower accuracy tan te supercritical one, as te parameter γ on wic te former is based is rarely small (if, for example, ɛ = 6, ten (6.) yields γ.65, wic can ardly be regarded as small). Tus, solution (6.5), (6.7) sould, rater, be viewed as a qualitative aid for explaining subtle features of rimming flows, suc as te above-mentioned sift of te pool zone, etc. 7. Comparison wit Asmore et al. () Before discussing te paper by Asmore et al. (, encefort referred to as AHS), we sall rewrite te pool solution (.) in terms of te original non-scaled variables (see (.7) (.8)) and te maximum dept max = H 4 W 4 of te pool, [ ] x = max (4ɛ max ), (4ɛ max) /4 x (4ɛ max) /4. (7.) / Ten, at te boundaries of te pool, we ave ( max ɛ ) / [ x +(4ɛmax) /4] as x (4ɛ max ) /4, (7.) ( max ) / [ x (4ɛmax) /4] as x (4ɛ max ) /4, (7.) ɛ wic are equivalent to (.4) (.5) (wit max replacing W as te free parameter of our family of solutions). Te pool solution (7.) agrees wit te corresponding solution of AHS, but teir outer solution does not agree wit our (.). Tis is a result of AHS s assumption tat te flux is small, q, (7.4) as opposed to our ypotesis q (see.). Under assumption (7.4), te solution of equation (.5) can be sougt in te form of a series in powers of q, = q + O(q ), (7.5) i.e. te film s tickness in te outer zone is almost uniform. AHS ten, effectively, used te assumption q = ɛ /6 max / ˆq, (7.6) were ˆq is an order-one constant to be determined later. Keeping (7.6) in mind, consider te rigt-and transitional region (unlike our case, it includes only one zone). It can be verified aposterioritat te stretced variables tere are =, x ɛ /6 / = x ( ) /4 4ɛ max. (7.7) max ɛ /6 / max Te matcing of (7.7) to te outer and pool solutions, (7.5) (7.6) and (7.), requires ˆq as x +, (7.8) ( /x ) as x. (7.9) Next, substituting (7.7) into (.6), we obtain + d = ˆq. (7.) dx

24 4 E. S. Benilov, M. S. Benilov and N. Kopteva Type of solution M max q (a) continuous M M max = O() q (b) sock M (M,M ) max = O() q = (c) near-critical M M max = O(ɛ /9 γ 4/9 ) q (d) supercritical M = M + O() max = O(ɛ /5 ) q (e) strongly supercritical M ɛ /6 max O(ɛ / ) q Table. Te maximum tickness, flux, and net mass for various types of rimming flows. Te caracteristics of solutions (a) (b) ave been calculated wit surface tension neglected, ɛ =; tose of solutions (c) (e) calculated for weak surface tension, <ɛ. For te near-critical solution, γ = 9/ ln ɛ. It can be sown tat te boundary-value problem (7.8) (7.) as a solution only for a certain value of ˆq, wic can be found numerically, ˆq.8 (see Appendix B). Ten, (7.6) yields te following expression for te flux: q.8 ɛ /6 / max. (7.) Tis result is compared to te numerical solution of te exact problem in figure 4(b), below. Observe also tat (7.) is consistent wit te small-q assumption (7.4) only if max ɛ /. In terms of te net mass (wic is mainly determined by te pool solution (7.)), tis restriction as te form M ɛ /6. Naturally, since we assumed q =, our results are applicable in te opposite limit, max ɛ /, M ɛ /6. One can see tat te present work is complementary to AHS s analysis. 8. Concluding remarks: te complete classification of rimming flows wit surface tension Steady rimming flows wit surface tension are governed by equation (.5) and are caracterized by two parameters: te non-dimensional capillary coefficient ɛ defined by (.) and te non-dimensional mass M defined by (.7). A classification of rimming flows depending on M and ɛ is presented in table and briefly summarized below (te numbering corresponds to tat of table ). (a) Continuous rimming flows (see figure ) were examined by Moffatt (977) using te leading-order equation (.8). As M M 4.44, ten q and te solution develops a corner at θ =. (b) Flows wit socks (see figure a) were examined by Benjamin et al. (99) and O Brien & Gat (998), using te leading-order teory (wic allows te solution to ave discontinuities). As M M 6.9, te sock approaces te bottom of te cylinder (θ = π), wile its amplitude grows. (c) Near-critical rimming flows are examined in 6 of tis paper. Tey can be interpreted as sock solutions wit te sock being close to te bottom of te cylinder, modified by surface tension. As te amplitude of te solution grows, its peak

25 Steady rimming flows wit surface tension (a) (b).5.4 q... Continuous solutions Solutions wit a sock Solutions wit a pool M 4 5 M Figure 4. Te flux q vs. te net mass M, forɛ =. Te numerical solution is sown as a solid line, te asymptotic solution (7.) is sown as dotted line in (b). A blow-up of te region saded in (a) is sown above it. (te pool ) canges sape: initially, it is skewed to te left (wic sows its relation to socks); but, for larger amplitudes, te pool becomes symmetric (see figure ). From a matematical viewpoint, tis regime, as well as te next one, are unusual, as tey involve a variable number of asymptotic zones, depending on te small parameter ɛ. (d) Supercritical rimming flows are examined in of tis paper. Tis regime can be treated as a limiting case of near-critical flows (in particular, it as a symmetric pool). It can be demonstrated tat, even toug tis regime was studied using te scaling max = O(ɛ /5 ), te results obtained are applicable to all cases wit max ɛ /. (e) Strongly supercritical flows, i.e. suc tat max ɛ / were considered by Asmore et al. () and in 7 of tis paper. Te main caracteristic feature of tis regime is te small tickness out of te film outside te pool and small flux q (for all oter regimes wit socks and pools, out = O(), q ). Matematically, tis case is muc simpler tan its two predecessors (c) and(d), as it involves only four asymptotic zones. Observe tat our classification as a gap, namely te regime wit M = O(ɛ /6 ), max = O(ɛ / ), (8.) wic describes te transition from te tick-outer-film regime (d) to te tin-outerfilm regime (e). Te regimes introduced above are best illustrated by plotting te flux q as a function of te net mass M see figure 4. One can see tat te intermediate regime (8.) is an interesting one, as it allows for up to tree solutions wit te same net mass (see te tree brances wic exist in te interval M.5 in figure 4a). Te lower branc is an extension of te small-q solution considered by Asmore et al. () and, tus, corresponds to a non-limiting outer solution. Te middle and upper

26 6 E. S. Benilov, M. S. Benilov and N. Kopteva 9 8 (a). (b) 7. M U.8 L θ θ Figure 5. Examples of steady rimming flows (numerical solution) wit te same net mass (M =.5) but different fluxes q, corresponding to te tree brances sown in figure 4 (in all tree cases, ɛ = ). (b) A blow-up of te region saded in (a). Curve L corresponds to q.6598 (lower branc); curve M, to q.665 (middle branc); and curve U, to q.6686 (upper branc). brances bot involve te limiting outer solution and differ only in te rigt-and transitional region (see figure 5). Note tat figure 4 also validates te assumptions about te flux q on wic te present paper and Asmore et al. () are based: tat q for te near- and supercritical regimes and q for te strongly supercritical one. Finally, simulations of te evolution equation (.4) indicate tat, for all regimes up to and including te supercritical regime, te solution always converges to te steady state wit te corresponding value of M, wic implies stability. Wat appens in furter regimes is unclear, as te straigtforward numerical metods are eiter too slow or too inaccurate, and tis question requires furter investigation. Appendix A. Te streamlines of a rimming flow Let us introduce te non-dimensional radial and azimutal velocities, u = u αrω, v = v RΩ, and, also, te non-dimensional dept of te film z = R r αr, were α is given by (.) and asterisks, as before, mark dimensional variables. Witin te framework of te lubrication approximation, u and v are given by (e.g. Benilov & O Brien 5) ( z u = 6 z )[ ( d sin θ + ɛ ( z v =+ z )] dθ + d4 θz dθ 4 )[ cos θ ɛ [ ( d cos θ ɛ ( d dθ + d dθ dθ + d dθ )],(A ) )], (A )

27 Steady rimming flows wit surface tension 7 It can be verified by inspection tat (A ) (A ) correspond to te following streamfunction: ( )[ ( )] z ψ = z + 6 z d cos θ ɛ dθ + d. (A ) dθ Now, te streamlines can be obtained by equating ψ to a constant, ψ(z, θ) =const. (A 4) For a given (θ), (A ) (A 4) constitute a cubic equation for z as a function of θ (obviously, only tose roots sould be taken for wic Im z =, Re z ). Appendix B. Solution of problem (7.8) (7.) Let us rewrite (7.8) (7.) in terms of wic yields ξ = x ˆq, η = ˆq, η as ξ +, (B ) η ˆq ( ) / ξ as ξ. (B ) η + d η η =. (B ) dξ To find asymptotics of η as x +, seek a solution in te form η =+ η(x), (B 4) substitute (B 4) in equation (B ), linearize it, η + d η dξ =, and solve: η = exp ( [ ( ) ( )] / ξ) ξ / ξ c sin + c cos + c exp ( ξ), (B 5) were c, c, c are constants of integration. Observe tat (B 4) (B 5) satisfy te boundary condition (B ) only if c = c = ; note also tat (B ) (B ) are invariant wit respect to te transformation ξ ξ + real constant, and we can assume real constant =ln c wic yields eiter η = + exp( ξ) as ξ + (B 6) or η = exp( ξ) as ξ +. (B 7) Using (B 6) or (B 7), we can soot te solution (numerically) from a large positive value of ξ towards. Once te solution s second derivative becomes close to a constant (ence te boundary condition (B ) is almost satisfied), we can collect te value of ˆq from ( ˆq = ) / d η lim ξ dξ.

28 8 E. S. Benilov, M. S. Benilov and N. Kopteva It turns out tat te boundary condition (B 7) does not yield a solution wit te required (parabolic) asymptotics for ξ, as η(ξ) passes troug zero at a finite value of ξ and becomes negative. Condition (B 6), in turn, yields te required asymptotics, wit ˆq.8. REFERENCES Asmore, J., Hosoi, A. E. & Stone, H. A. Te effect of surface tension on rimming flows in a partially filled rotating cylinder. J. Fluid Mec. 479, Benilov, E. S. & O Brien, S. B. G. 5 Inertial instability of a liquid film inside a rotating orizontal cylinder. Pys. Fluids 7, 56. Benjamin, T. B., Pritcard, W. G. & Tavener, S. J. 99 Steady and unsteady flows of a igly viscous liquid inside a rotating orizontal cylinder. Unpublised manuscript. Hinc, E. J. 99 Perturbation Metods. Cambridge University Press. Jeong, J.-T. & Moffatt, H. K. 99 Free-surface cusps associated wit a flow at low Reynolds number. J. Fluid Mec. 4,. Jonson, R. E. 99 Coating flow stability in rotating molding. In Engineering Science, Fluid Dynamics: A Symposium to Honor T.Y. Wu (ed. G. T. Yates), pp World Scientific. Landau, L. & Levic, B. 94 Dragging of liquid by a plate. Acta Pysiocim. USSR 7, Moffatt, H. K. 977 Beaviour of a viscous film on te outer surface of a rotating cylinder. J. Fluid Mec. 6, O Brien, S. B. G. Linear stability of rimming flows. Q. Appl. Mats. 6,. O Brien, S. B. G. & Gat, E. G. 998 Te location of a sock in rimming flow. Pys. Fluids, 4 4. Press, W. H., Teulkolsky, S. A., Vetterling, W. T. & Flannery, B. P. 99 Numerical Recipes, nd Edn. Cambridge University Press. Wilson,S.D.R.98 Te drag-out problem in film coating teory. J. Engng Mats 6, 9. Wilson,S.D.R.&Jones,A.F.98 Te entry of a falling film into a pool and te air-entrainment problem. J. Fluid Mec. 8, 9.