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1 Journal of Computatonal Physcs 233 (2013) Contents lsts avalable at ScVerse ScenceDrect Journal of Computatonal Physcs journal homepage: Fnte element smulaton of dynamc wettng flows as an nterface formaton process J.E. Sprttles a,, Y.D. Shkhmurzaev b a Mathematcal Insttute, Unversty of Oxford, Oxford, OX1 3LB, UK b School of Mathematcs, Unversty of Brmngham, Edgbaston, Brmngham, B15 2TT, UK artcle nfo abstract Artcle hstory: Receved 29 February 2012 Receved n revsed form 27 June 2012 Accepted 4 July 2012 Avalable onlne 20 July 2012 Keywords: Flud mechancs Dynamc wettng Interface formaton Shkhmurzaev model Computaton Capllary rse A mathematcally challengng model of dynamc wettng as a process of nterface formaton has been, for the frst tme, fully ncorporated nto a numercal code based on the fnte element method and appled, as a test case, to the problem of capllary rse. The motvaton for ths work comes from the fact that, as dscovered expermentally more than a decade ago, the key varable n dynamc wettng flows the dynamc contact angle depends not just on the velocty of the three-phase contact lne but on the entre flow feld/geometry. Hence, to descrbe ths effect, t becomes necessary to use the mathematcal model that has ths dependence as ts ntegral part. A new physcal effect, termed the hydrodynamc resst to dynamc wettng, s dscovered where the nfluence of the capllary s radus on the dynamc contact angle, and hence on the global flow, s computed. The capabltes of the numercal framework are then demonstrated by comparng the results to experments on the unsteady capllary rse, where excellent agreement s obtaned. Practcal recommendatons on the spatal resoluton requred by the numercal scheme for a gven set of non-dmensonal smlarty parameters are provded, and a comparson to asymptotc results avalable n lmtng cases confrms that the code s convergng to the correct soluton. The appendx gves a user-frendly step-by-step gude specfyng the entre mplementaton and allowng the reader to easly reproduce all presented results, ncludng the benchmark calculatons. Ó 2012 Elsever Inc. All rghts reserved. 1. Introducton Relable smulaton of flows n whch a lqud advances over a sold, known as dynamc wettng flows, s the key to the understandng of a whole host of natural phenomena and technologcal processes. In the technologcal context, the study of these flows has often been motvated by the need to optmze contnuous coatng processes that are routnely used to create thn lqud flms on a product [1], for example, n the coatng of optcal fbres [2,3]. However, more recently, dscrete coatng, n partcular nkjet prntng of mcrodrops [4], has matured nto a vable, and often preferable, alternatve to tradtonal fabrcaton processes, e.g. n the addtve manufacturng of 3D structures or the creaton of P-OLED dsplays [5,6], and t s becomng a new drvng force behnd research nto dynamc wettng phenomena. In most cases, such flows can be regarded as mcrofludc phenomena, where a large surface-to-volume rato brngs n nterfacal effects on the flow that are not observed at larger scales. Obtanng accurate nformaton about mcro and nanofludc flows expermentally s often dffcult and usually costly so that, consequently, a desred alternatve s to have a relable theory descrbng the physcs that s domnant for ths class of Correspondng author. Tel.: +44 (0) E-mal addresses: sprttles@maths.ox.ac.uk (J.E. Sprttles), yul@for.mat.bham.ac.uk (Y.D. Shkhmurzaev) /$ - see front matter Ó 2012 Elsever Inc. All rghts reserved.

2 J.E. Sprttles, Y.D. Shkhmurzaev / Journal of Computatonal Physcs 233 (2013) flows and ncorporate t nto a flexble and robust computatonal tool whch can quckly map the parameter space of nterest to allow a specfc process to be optmzed. Such computatonal software could be valdated aganst experments at scales and geometres easly accessble to accurate measurement and then used to make predctons n processes naccessble to expermental analyss. The dscovery that no soluton exsts for the movng contact-lne problem n the framework of standard flud mechancs [7,8] prompted a number of remedes to be proposed, whch are summarzed, for example, n [9, ch. 3]. Modellng approaches range from contnuum theores through to mesoscopc approaches [10] rght down to molecular dynamcs smulatons [11], see [12,13] for dscussons of the varous modellng approaches. Of these, most, based on contnuum mechancs, are what we shall refer to as conventonal or slp models, n whch the no-slp condton on the sold surface s relaxed to allow a soluton to exst, wth the Naver-slp condton [14] beng the most popular choce. As a boundary condton on the free-surface shape at the contact lne, one has to specfy the contact angle formed between the free surface and sold. In conventonal models, ths angle s prescrbed as a heurstc or emprcal functon of the contact-lne speed and materal parameters of the system, e.g. [15]. A varant of ths approach s to clam that the actual contact angle s statc, whlst the observed apparent angle s produced as a result of vscous bendng of the free surface near the contact lne [16]. Such models provde predctons that adequately descrbe experments at relatvely large scales, often around the length scale of mllmetres. It s well establshed that on such scales many of the proposed models work equally well and that fndng sgnfcant devatons between ther predctons, and hence ascertanng whch best captures the key physcal mechansms of dynamc wettng, s practcally mpossble [17,18]. A physcal phenomenon that gves an opportunty to dstngush between dfferent models came to be known under a technologcal name of the hydrodynamc assst of dynamc wettng (henceforth hydrodynamc assst or smply assst ). The essence of ths effect, frst observed n hgh accuracy experments on the curtan coatng process [19,20], s that for a gven lqud spreadng over a gven sold at a fxed contact-lne speed, the dynamc contact angle can stll be manpulated by alterng the flow feld/geometry, for example, n the case of curtan coatng, by changng the flow rate or the heght from whch the curtan falls. Ths effect has profound technologcal mplcatons as t allows the process to be optmzed by offsettng the ncrease of the contact angle wth ncreasng contact-lne speed by manpulatng the flow condtons and hence postponng ar entranment [19]. The dependence of the dynamc contact angle on the flow feld has also been reported n the mbbton of lqud nto capllares [21,22], n the spreadng of mpacted drops over sold substrates [23,24] and n the coatng of fbres [3]. However, n many of these flows t s yet unclear whether hydrodynamc assst actually occurs, or whether the free surface bends sgnfcantly beneath the spatal resoluton of the experments, whereas for curtan coatng the hope of attrbutng assst to the poor spatal resoluton of experments has been quashed by careful fnte element smulatons whch show that the degree of free-surface bendng under the reported resoluton of the measurements s too small to account for the observed effect and that conventonal models cannot n prncple descrbe ths mportant physcal effect [25]. Currently, the only model known to be able to even qualtatvely descrbe assst [26,27] s the model of dynamc wettng as an nterface formaton process, frst ntroduced n [28] and dscussed n detal n [9]. Ths model s based on the smple physcal dea that dynamc wettng, as the very name suggests, s the process n whch a fresh lqud sold nterface (a newly wetted sold surface) forms. Qualtatvely, the orgn of the hydrodynamc assst s that the global flow nfluences the dynamcs of the relaxaton of the formng lqud sold nterface towards ts equlbrum state and hence the value of ths nterface s surface tenson at the contact lne, whch, together wth the surface tenson on the free surface, determnes the value of the dynamc contact angle. When there s a separaton of scales between the nterface formaton process and the global flow, the movng contact-lne problem can be consdered locally and asymptotc analyss provdes a speed-angle relatonshp whch s seen to descrbe experments just as well as formulae proposed n other models [9]. However, n the stuaton where the scale of the global flow and that of the nterface formaton are no longer separated, the nfluence of the flow feld on the dynamc contact angle wll occur and hence no unque speed-angle relatonshp wll be able to descrbe experments. Then, the nterface formaton model becomes the only modellng tool, and, gven that the processes of practcal nterest are free-surface flows under the nfluence of, at least, vscosty, capllarty and nerta, t s nevtable that, to descrbe such flows, one needs computer smulaton,.e. the development of accurate CFD codes, whch, for the effect of hydrodynamc assst to be captured, have to ncorporate the nterface formaton model. Snce the nterface formaton model came nto wder crculaton, there has been consderable nterest, e.g. [29,23,24], n usng t n ts entrety as a practcal tool for descrbng dynamc wettng phenomena, especally on the mcroscale. Revew artcles have also referred to the descrpton of assst usng ths model as one of the man challenges n the feld [12,13]. Although robust computatonal codes based on the conventonal models already exst, e.g. [30,31], the extenson of these codes to a numercal tool that ncorporates the nterface formaton model s far from trval. In partcular, one has to solve numercally the Naver Stokes equatons descrbng the bulk flow subject to boundary condtons whch are themselves partal dfferental equatons along the nterfaces and n ther turn have to satsfy certan boundary condtons at contact lnes confnng the nterfaces. These condtons determne the dynamc contact angle and hence nfluence the free-surface shape, whch exerts ts nfluence back on the bulk flow. Thus, the bulk flow, the dstrbuton of the surface parameters along the nterfaces and the dynamc contact angle that these dstrbutons negotate become nterdependent, wth the dynamc contact angle beng an outcome of the soluton as opposed to conventonal models where t s an nput. Ths nterdependency s, on the one hand, the physcal essence of the expermentally observed effect of hydrodynamc assst to be descrbed but, on

3 36 J.E. Sprttles, Y.D. Shkhmurzaev / Journal of Computatonal Physcs 233 (2013) the other hand, t s ths very nterdependency that, coupled wth the nonlnearty of the bulk equatons and the flow geometry, s the reason why the model s dffcult to mplement robustly nto a numercal code. Some numercal progress has been made for the computatonally less complex steady Stokes flows [27], but what s lackng s a step-by-step gude to the mplementaton of ths model n the general case, wth unsteady effects n the bulk and nterfaces as well as nonlnearty of the bulk equatons fully mplemented. Ths would pave the way for ncorporatng the nterface formaton model nto exstng codes as well as developng new ones. Therefore, the frst objectve of the present paper s to address these ssues and provde a dgestble gude to the model s mplementaton nto CFD codes. Then, after gvng benchmark computatonal results to verfy ths mplementaton, we consder a problem of mbbton nto a capllary, compare the outcome wth experments and predct essentally new physcal effects. A startng pont n the development of the aforementoned CFD code s to frst develop an accurate computatonal approach for the smulaton of dynamc wettng flows usng the mathematcally less complex conventonal models and ths was acheved n [32]. It was shown that many of the prevous numercal results obtaned for dynamc wettng processes are unrelable as they contan uncontrolled errors caused by not resolvng all the scales n the conventonal model, most notably the dynamcs of slp and the curvature of the free surface near the contact lne. Benchmark calculatons n [32] for a range of mesh resolutons resolved prevous msunderstandngs about how to mpose the dynamc contact angle and showed that mplementng t usng the usual fnte element deology, as opposed to strong mplementaton of the contact angle, works most effcently: t allows errors to be seen, and hence controlled, as the computed contact angle vares from ts mposed value, nstead of them beng masked elsewhere n the code. Furthermore, n [33], we have shown that numercal artfacts whch occur at obtuse contact angles are present n both commercal software and n academc codes where, msleadngly, they have prevously been nterpreted as physcal effects. By comparng computatonal results to analytc near feld asymptotc ones, we have shown that the prevously obtaned spkes n the dstrbutons of pressure observed near the contact angle are completely spurous, and, to rectfy ths ssue, a specal method, based on removng the hdden egensolutons n the problem pror to computaton, has been developed [33]. In[34], we showed that our code s capable of smulatng unsteady hgh deformaton flows by comparng to benchmark calculatons publshed n the lterature and performed by varous research groups. In contrast, n [35] t has been shown that when commercal software s used to smulate relatvely smple benchmark free surface flows, the soluton, albet converged, s not the correct one. In ths seres of artcles [32 34], our approach has been to carefully develop a robust CFD algorthm for the smulaton of dynamc wettng flows. Thus far, ths has been acheved for the conventonal models, where we valdated our results by performng mesh ndependence tests, comparng wth asymptotc solutons n lmtng cases and, where no analytc progress was possble, to prevously obtaned benchmark solutons publshed n the lterature. In ths artcle, we contnue ths seres of papers and, for the frst tme, ncorporate the nterface formaton model nto our code n full. Notably, t wll be shown that a code mplementng the mathematcally complex nterface formaton equatons can easly recover the much smpler conventonal models by settng certan parameters to zero. Ths allows the same code to be used to compare the predctons of dynamc wettng models for a range of flows and hence to determne, by a comparson wth experment, where the physcs of nterface formaton manfests tself n a sgnfcant way and where mathematcally smpler conventonal models are adequate. Here, we focus on dynamc wettng; the extenson to other flows of nterest where nterface formaton or dsappearance also occurs s a straghtforward procedure computatonally, and t wll be dealt wth n forthcomng artcles. The layout of the present artcle s as follows. Frst, n Secton 2, wthout lmtng ourself to a partcular flow confguraton, the equatons descrbng the dynamc wettng process are brefly recaptulated. Then, n Secton 3, the fnte element equatons are derved for the dynamc wettng flow as an nterface formaton process. Local element matrces, and addtonal detals about the fnte element procedure are provded n the Appendx, whch complements a user-frendly step-by step gude to fnte element smulaton gven for ths class of flows n [32] and allows one to reproduce the benchmark smulatons n Secton 4. These smulatons are performed for the dynamc wettng flow through a capllary both n the case where the menscus moton s steady and for the unsteady mbbton of a lqud nto an ntally dry capllary. Computatons are checked for convergence both as the mesh s refned and towards asymptotc results n lmtng cases. Havng establshed the accuracy of our approach, n Secton 5 new physcal effects are dscovered by consderng the nfluence of capllary geometry on the dynamc wettng process. Fnally, the computatonal tool s ablty to easly descrbe expermental data s shown n Secton 6 and a number of advantages over current approaches, partcularly n the ntal stages of a menscus moton nto a capllary tube, are hghlghted. Concludng remarks and areas for future research are dscussed n Secton Modellng dynamc wettng flows as an nterface formaton process Consder the spreadng of a Newtonan lqud, wth constant densty q and vscosty l, over a chemcally homogeneous smooth sold surface. The lqud s surrounded by a gas whch, for smplcty, s assumed to be nvscd and dynamcally passve, of a constant pressure p g. Let the flow be characterzed by scales for length L, velocty U, tme L=U, pressure lu=l and external body force F 0. In the dmensonless form, the contnuty and momentum balance equatons are then gven by r u ¼ þ u ru ¼ r P þ St F; ð1þ where

4 h P ¼ piþ ru þ ðru Þ T ; ð2þ s the stress tensor, t s tme, u and p are the lqud s velocty and pressure, F s the external force densty and I s the metrc tensor of the coordnate system. The non-dmensonal parameters are the Reynolds number Re ¼ qul=l and the Stokes number St ¼ qf 0 L 2 =ðluþ. Boundary condtons to the bulk equatons are requred at the lqud gas free surface x ¼ x 1 ðs 1 ; s 2 ; tþ, whose poston s to be found as part of the soluton, and at the lqud sold nterface x ¼ x 2 ðs 1 ; s 2 ; tþ, whose poston s known, and at other boundng surfaces whch wll be specfed by the problem of nterest; here, ðs 1 ; s 2 Þ are the coordnates that parameterze the surfaces. Boundary condtons along the free surface, the lqud sold nterface and the contact lne at whch they ntersect are provded by the nterface formaton model [9], as follows The nterface formaton model To represent the boundary condtons on an nterface wth a normal n n an nvarant form, t s convenent to ntroduce a (symmetrc) tensor I nn, whch s essentally a metrc tensor on the surface. If an arbtrary vector a s decomposed nto a scalar normal component a n ¼ a n and a vector tangental part a k, so that a ¼ a k þ a n n, we can see that, because n ði nnþ ¼ 0, the tensor ði nnþ extracts the component of a vector whch s tangental to the surface,.e. a ði nnþ ¼ a k. Hereafter, n s the unt normal to a surface pontng nto the lqud, and subscrpts 1 and 2 refer to the free surface and sold surface, respectvely. The equatons of nterface formaton on a lqud gas free surface, whch act as boundary condtons for the bulk Eq. (1), are gven vs 1 n 1 ¼ 0; h Ca n 1 ru þ ðru Þ T J.E. Sprttles, Y.D. Shkhmurzaev / Journal of Computatonal Physcs 233 (2013) ði n 1 n 1 Þþrr 1 ¼ 0; ð4þ ð3þ n h Ca p g p þ n 1 ru þ ðru Þ T o n 1 ¼ r 1 r n 1 ; ð5þ v s 1k u k ¼ 1 þ 4 a b 4b rr 1 ; u v s 1 n1 ¼ Q q s 1 qs 1e ; þ r qs 1v s 1 ¼ q s 1 qs 1e ; ð8þ ð6þ r 1 ¼ kð1 q s 1 Þ; ð9þ whlst at lqud sold nterfaces formed on a sold whch moves wth velocty U, the equatons of nterface formaton have the form v s 2 U n2 ¼ 0; ð10þ Ca n 2 P ði n 2 n 2 Þþ 1 2 rr 2 ¼ b u k U k ; ð11þ v s 2k 1 2 u k þ U k ¼ arr 2 ; ð12þ u v s 2 n2 ¼ Q q s 2 qs 2e ; þ r qs 2v s 2 r 2 ¼ kð1 q s 2 Þ: ¼ q s 2 qs 2e ; ð14þ The dfferental term r 1 r n 1, where r 1 s the (dmensonless) surface tenson on the free surface, descrbng the capllary pressure n the normal-stress Eq. (5) ndcates that ths equaton requres ts own boundary condton where the free surface termnates,.e. at the contact lne. There n 2 s known and n 1 must be specfed by settng the dynamc contact angle h d at whch the free surface meets the sold surface: ð15þ

5 38 J.E. Sprttles, Y.D. Shkhmurzaev / Journal of Computatonal Physcs 233 (2013) Fg. 1. Illustraton showng the vectors assocated wth the lqud gas free surface A 1 and the lqud sold surface A 2 n the vcnty of the contact lne C cl. n 1 n 2 ¼ cos h d : In the framework of contnuum mechancs, the mass of the contact lne (.e., physcally, the three-phase-nteracton regon), as well as the mass of the lqud gas nterface, s neglected, and hence one has that the forces actng on the contact lne are n balance rrespectve of the contact lne s acceleraton. The tangental to the sold surface projecton of the balance of forces actng on the contact lne s gven by Young s equaton [36]: r 2 þ r 1 cos h d ¼ r S ; whch ntroduces the contact angle va a physcal law and where r 2 and r S are the surface tensons of the lqud sold nterface and sold-gas nterface, respectvely, and the latter s henceforth assumed to be neglgble r S 0. Eqs. (8) and (14) also requre a boundary condton at the contact lne where the two nterfaces meet, and here we have the contnuty of surface mass flux q s 1 v s 1k U c m 1 þ q s 2 v s 2k U c m 2 ¼ 0 ð18þ where U c s the (dmensonless) velocty of the advancng contact lne and m s the nward unt vector tangental to surface and normal to the contact lne (see Fg. 1). The nterface formaton model s descrbed n detal n the monograph [9] and a seres of precedng papers [e.g. 28,37] so that here only the man deas are brefly recaptulated. The surface varables are n the surface phase,.e. physcally n a mcroscopc layer of lqud adjacent to the surface whch s subject to ntermolecular forces from two bulk phases. In the contnuum lmt, ths mcroscopc layer becomes a mathematcal surface of zero thckness wth q s denotng ts surface densty (mass per unt area) and v s the velocty wth whch t s transported. The followng non-dmensonal parameters have been ntroduced a ¼ ar=ðulþ; b ¼ðbULÞ=r; ¼ Us=L; k ¼ cq s =r ð0þ and Q ¼ qs ð0þ =ðqusþ whch are based on phenomenologcal materal constants a, b; c; s and q s ð0þ ; n the smplest varant of the theory, the latter are assumed to be constant and take the same value on all nterfaces; r s the characterstc surface tenson for whch t s convenent to take the equlbrum surface tenson on the lqud gas nterface. Estmates for the materal constants have been obtaned by comparng the theory to experments n dynamc wettng, e.g. n [38], but could just as easly have been taken from an entrely dfferent process n whch nterface formaton s key to descrbng the dynamcs of the flow [39 42]. In other words, once obtaned for a partcular lqud n one set of experments, the materal constants determned can be used to descrbe all phenomena nvolvng the same flud n whch nterface formaton dynamcs kcks n. The surface tenson r s consdered as a dynamc quantty related to the surface densty q s va the equatons of state n the surface phase (9) and (15), whch are taken here n ther smplest lnear form. Gradents n surface tenson nfluence the flow, frstly, va the stress boundary condtons (4), (5) and (11),.e. va the Marangon effect, and, secondly, n the Darcy-type equatons 1 (6) and (12) by forcng the surface velocty to devate from that generated n the surface phase by the outer flow. Mass exchange between the bulk and surface phases, caused by the possble devaton of the surface densty from ts equlb-, s accounted for n the boundary condtons for the normal component of bulk velocty (7) and (13), and n the rum value q s e correspondng surface mass balance Eqs. (8) and (14). One would expect a generalzed set of boundary condtons to have the classcal condtons as ther lmtng case. For the nterface formaton model ths lmtng case follows from the double lmt! 0; b=ca!1. When the lmt! 0 s appled to (3) (9), the lqud gas nterface equatons are reduced to ther classcal form. Notably, f we apply! 0 to the lqud sold equatons (10) (15), the conventonal slp model s obtaned, that s the Naver-slp condton combned wth mpermeablty. ð16þ ð17þ 1 The analogy wth the Darcy equaton s that the tangental surface velocty v s k s the average velocty of the nterfacal layer and ts devaton from that generated by the outer flow s proportonal to the gradent of surface tenson, whch s the negatve gradent of surface pressure as p s ¼ r.

6 J.E. Sprttles, Y.D. Shkhmurzaev / Journal of Computatonal Physcs 233 (2013) In ths lmt, the surfaces are n equlbrum so Young s equaton (17) gves that the dynamc contact angle s fxed as a constant at ts equlbrum value h d ¼ h e. If we wsh to go further wth the conventonal model approach and descrbe the dynamc contact angle as some functon of the contact-lne speed, then Young s equaton (17), whch s essentally Newton s second law projected on the tangent to the sold surface, has to be dscarded n favour of ths functon, as the conventonal models have no mechansm that would allow the surface tensons to be dynamc and hence make the contact angle, as observed n experments, devate from ts equlbrum value whle keepng the Young equaton. Therefore, mplementng the nterface formaton model nto a numercal code allows one to test all conventonal models of wettng proposed n the framework of contnuum mechancs. By applyng the lmt b=cað¼ bl=lþ!1we recover the classcal equatons of no-slp and mpermeablty on the sold surface for whch the movng contact-lne problem s known to have no soluton [7,8]. Despte the model s complexty, n lmtng cases, analytc progress on t can be made to obtan explct relatonshps for the surface varables and, wth some further assumptons, even a formula relatng the contact-lne speed to the dynamc contact angle. Such formulae are a useful test of our numercal solutons, and we wll brefly recaptulate ther outcome Asymptotc formulae n a lmtng case When the contact-lne moton can be analyzed as a local problem, as opposed to cases where the nterface formaton and the bulk flow scales are not separated so that manpulatng the global flow nfluences the relaxaton process along the nterfaces, asymptotc progress s possble. A full dervaton of the results we use may be found n [9] and references theren; here we shall just outlne the man assumptons and results. If n the steady propagaton of a lqud gas free surface over a sold substrate n the Stokes regme (Re 1), the characterstc length scale of the nterface formaton process l ¼ Us s small compared to the bulk length scale L, we have that our non-dmensonal parameter 1. If n the lmt! 0 we also assume that the capllary number Ca! 0, then, to leadng order n Ca, the normal-stress boundary condton (5) gves that the free surface near the contact lne s planar, so that the problem may be consdered locally n a wedge-shaped doman. Then, we can dentfy the followng three asymptotc regons: (a) The outer regon, where, n a reference frame movng wth the contact lne, one has a flow n a wedge n the classcal formulaton, wth a zero tangental-stress and a no-slp boundary, descrbed n [43]; (b) The ntermedate regon wth the characterstc (dmensonless) length scale, where the surface-tenson-relaxaton process takes place and where, due to smallness of Ca, to leadng order the nfluence of the bulk flow on ths process can be neglected; (c) The nner regon, wth the characterstc length scale Ca, through whch the surface denstes and veloctes, to leadng order, stay constant. On the free surface, at leadng order n the ntermedate and nner regons, one has q s 1 ¼ qs 1e ; v s 1k ¼ u f ðh d Þ; ð19þ where u f ðh d Þ s the (dmensonless) radal velocty of the bulk flow n the far feld on the lqud gas nterface gven by [43]: u f ðh d Þ¼ sn h d h d cos h d sn h d cos h d h d ; ð20þ so that the surface mass flux nto the contact lne s q s 1e u f ðh d Þ. Then, snce the surface varables are constant through the nner regon, the boundary condtons at the contact lne (17), (18) can be appled to the dstrbutons of the surface varables n the ntermedate regon. As a result, we have two frst-order ODEs to solve for the dstrbutons of q s 2 and vs 2k along the lqud sold nterface dq s 2 ds ¼ 4V 2 ð1 v s 2k Þ; dðq s 2v s 2k Þ ds ¼ ðq s 2 qs 2eÞ ðs > 0Þ ð21þ where s ¼ s= s the ntermedate regon s varable and V 2 ¼ b=ðð1 þ 4a bþkþ s the non-dmensonal contact-lne speed, subject to (a) boundary condton (18), now takng the form q s 2v s 2 ¼ qs 1e u f ðh d Þ at s ¼ 0, (b) the matchng condton q s 2! qs 2e as s!1, and, as boundary condton (a) mplctly depends on the parameter h d, (c) Young s equaton (17), now takng the form cos h d ¼ kðq s 2 1Þ. The equlbrum contact angle h e s obvously related to q s 2e by cos h e ¼ kðq s 2e 1Þ, whch can be used to replace q s 2e wth h e. The above problem s easly solved numercally; however, by takng an addtonal assumpton k 1, one can obtan an analytc relaton between the contact angle and the non-dmensonal speed of the contact lne V: h 2V cos h e þ 1 q s 1 1e 1 þ q s 1e u f ðh d Þ cos h e cos h d ¼ h V þ V 2 þ 1 þ cos h e 1 q ; ð22þ 1=2 s 1e where

7 40 J.E. Sprttles, Y.D. Shkhmurzaev / Journal of Computatonal Physcs 233 (2013) =2 k ¼ 2Vðq s 2e Þ 1 V 2 þ q s 2e V ; C ¼ 2V qs 2e þ qs 1e u f ðh d Þ 1=2 : V 2 þ q s 2e þ V The presence of u f ðh d Þ n (22) hghlghts a connecton between the flow n the outer asymptotc regon and the value of the dynamc contact angle. When the contact lne s nsulated from the global flow by the low-reynolds-number regon, the flow near the contact lne s completely determned by the contact-lne speed, and hence the mass flux nto the contact lne that feeds the lqud sold nterface can be found from the local soluton. Ths s the case consdered n [28,44], where the theory shows excellent agreement of (22) wth experments. If the outer flow s to nfluence the mass flux nto the contact lne [27], ths wll affect the value of the contact angle. The most mportant length scale L f assocated wth the nterface formaton process s the characterstc dstance over whch the sold surface returns to equlbrum and the asymptotc result ndcates that ths s gven, n non-dmensonal qffffffffffffffffffffffffffffffffffff unts, by kl f = ¼ 1n(19) so that L f ¼ q s 2e = 2Vð V 2 þ q s 2e VÞ. The expressons gven above wll be used n Secton 4 below to compare our computatons to n the stuatons where the underlyng assumptons of the asymptotcs are satsfed. Now, we shall consder the development of ths computatonal algorthm for the general case wthout makng any smplfyng assumptons. 3. Fnte element procedure A fnte element framework for the smulaton of dynamc wettng flows usng the conventonal models of dynamc wettng was descrbed n [32]. To handle the evoluton of the free surface ths framework uses an arbtrary Lagrangan Euleran (ALE) scheme based on the method of spnes, a computatonal approach whch has been successfully appled to a range of coatng flows over the past thrty years, e.g. n [45,46,25]. In[34], the framework was extended for the smulaton of tmedependent free surface flows, wth the code provdng accurate solutons for the benchmark test case [47,48] of a freely oscllatng lqud drop. Ths confrmed that the mplementaton of the vscous, nertal and capllarty effects s accurate, even when the mesh undergoes Oð1Þ deformatons. Although the nterface formaton model ntroduces addtonal varables nto the problem, these are confned to the surfaces of the doman so that ths does not notably ncrease the computatonal cost of the scheme. Furthermore, even n the conventonal models a small length scale assocated wth slp has to be resolved to obtan mesh-ndependent results (see [32]), and, as we shall see, to also capture the dynamcs of nterface formaton does not sgnfcantly ncrease the computatonal burden. In ths work, our am was to mplement the nterface formaton model nto a computatonal scheme for the frst tme, so that we made no attempt to optmze the code. However, we have found that even for the hghest resoluton unsteady flows, t only take around a day to run on standard laptops. The ssues related to optmzaton of the computatons can be consdered n future work where, n partcular, the developed framework could be appled to fully three-dmensonal phenomena. What follows s the mplementaton of the nterface formaton equatons nto our framework. For a more detaled descrpton of the basc components of the framework and a user-frendly step-by-step gude to mplementng dynamc wettng flows nto the fnte element method, the reader s referred to [32] and n partcular to the Appendx whch makes t possble for the nterested user to reproduce the results presented there. The Appendx n the present artcle provdes addtonal detals of the mplementaton and, n ths sense, complements the Appendx n [32], allowng one to reproduce the results of Secton 4 and Problem formulaton n the ALE scheme Consder how the equatons of Secton 2, wrtten n Euleran coordnates x, are formulated for an ALE system where the flow doman v ¼ vðx; tþ deforms n tme. Ths deformaton must be accounted for n the temporal dervatves of varables whose poston n the Euleran system s evolutonary, n partcular, n the Naver Stokes equatons where the materal dervatve D=Dt transforms as Du þ u ru þ ðu cþru: ð23þ v Here, cðv; v s the velocty of the ALE coordnates wth respect to the fxed reference frame [49,50]. It can be seen that, as should be expected, for c ¼ u we have a Lagrangan scheme whereas for c ¼ 0 the Euleran system s recovered. Before consderng temporal dervatves occurrng n the nterface formaton equatons, t s convenent to ntroduce the surface gradent r s, whch s the projecton of the usual gradent operator r onto the surface r s ¼ ði nnþr. An arbtrary surface vector a s s wrtten n terms of components normal and tangental to the surface as a s ¼ a s k þ as n n, where as n ¼ as n,so that for ts dvergence one has r a s ¼ r s a s k þ as n rs n. In partcular, as descrbed n [51], ponts on the surfaces move wth the normal surface velocty v s n,.e. accordng to the knematc Eq. (3), and an arbtrary tangental component cs k whch depends on the choce of mesh desgn. Then, on a gven surface

8 c s ¼ c s k þ vs n n; cs k I ð Þ: Therefore, n the ALE framework the left-hand sde of the surface mass balance Eqs. (8) and (14), s þ r q s cv s c c c s c rqs c þ r cv qs s c ; c v s c where c ¼ 1; 2 refers to the lqud gas and lqud sold nterface, respectvely, and v s c ¼ vs c ðx; t Þ are the correspondng coordnates. Then, (8) and (14) take the cs ck rs q s c þ rs q s cv s ck þ q s cv s cn rs n c þ q s c qs ce ¼ 0; ðc ¼ 1; 2Þ; ð24þ where we have used that n c r s q s c ¼ 0. Eq. (24) can be rearranged to obtan h v s ck cs ck þ q s cv s cn rs n c þ q s c qs ce ¼ 0; ðc ¼ 1; þ qs c rs c s ck þ rs q s c In the lmtng case, where the surface moves only normal to tself, so that c s ck ¼ 0, the usual Euleran equatons are þ rs q s cv s ck þ q s cv s cn rs n c þ q s c qs ce þ r cv qs s c þ q s c qs ce ¼ 0; ðc ¼ 1; 2Þ; whlst f the surface moves n a Lagrangan way, c s c ¼ vs c, then the term under the dvergence becomes dentcally zero, and we þ qs c rs v s ck þ qs cv s cn rs n c þ q s c qs ce þ qs c r vs c þ q s c qs ce ¼ 0; ðc ¼ 1; 2Þ: ð25þ Here, as should be expected, there s no term representng the convecton of surface densty by the surface velocty,.e. a term of the form v s rq s does not appear. Havng reformulated the equatons for the ALE system and ntroduced surface operators, we can now derve the approprate FEM resduals Formng the fnte element resduals J.E. Sprttles, Y.D. Shkhmurzaev / Journal of Computatonal Physcs 233 (2013) The defnng feature of the FEM s that the computatonal doman V s tessellated nto a fnte number of non-overlappng elements, each contanng a fxed number of nodes at whch the functons values are determned. Between these nodes the functons are approxmated usng nterpolatng functons whose functonal dependence on poston s chosen. In what follows, N p s the total number of nodes n V at whch the pressure s determned, N u s the number of nodes at whch the velocty components are to be found, N 1 s the number of nodes on the free surface A 1 ; N 2 the number of nodes on the sold surface A 2, and N c the number of nodes along the contact lne. The procedure of generatng the fnte element equatons s well known and a detaled explanaton of how ths s acheved for dynamc wettng flows descrbed by the conventonal model s gven n [32], so that here we just gve the man detals. Functons are approxmated as a lnear combnaton of nterpolatng functons each weghted by the correspondng nodal value. In partcular, we use mxed nterpolaton so that the Ladyzhenskaya Babus ka Brezz [52] condton s satsfed 2 wth lnear bass functons w j to represent pressure and quadratc ones / j for velocty: p ¼ XNp p j w j ðxþ; j¼1 u ¼ XNu u j / j ðxþ; j¼1 ðx 2 VÞ: In the Galerkn fnte element method, the bass functons whch are used to dscretze the functons of the problem are also used as weghtng functons to create the weak form of the problem, see [Secton 3 of 32] for specfc detals. Note that here we use Roman letters for the ndces (; j, etc) to refer to the nodal values and approxmatng functons spannng the whole doman (globally); n the Appendx, where all the numercal detals are gven, these ndces wll be used alongsde talczed ones (; j, etc), whch wll refer to local, element-based, values and functons. Surface varables are also approxmated quadratcally wth bass functons on surface A c ðc ¼ 1; 2Þ denoted by / c;j ðs 1 ; s 2 Þ. So, all of the nterface formaton varables, represented by an arbtrary surface varable a s c, as well as the shape of the free surface x 1, are approxmated as 2 Equal-order methods may also be used wth stablzaton, e.g. [53,54], but these ssues le beyond the scope of ths paper.

9 42 J.E. Sprttles, Y.D. Shkhmurzaev / Journal of Computatonal Physcs 233 (2013) a s c ¼ XNc a s c;j / c;jðs 1 ; s 2 Þ; x 1 ¼ XN 1 x 1;j / 1;j ðs 1 ; s 2 Þ: j¼1 j¼1 To determne the free surface shape,.e. the nodal values x 1;j, a functon h ¼ hðs 1 ; s 2 Þ s found as part of the soluton at every free-surface node, so that h j ¼ h j ðs 1 ; s 2 Þ for j ¼ 1;...; N 1, whch ponts n a drecton lnearly ndependent from both s 1 and s 2 at each node,.e. out of the free surface. For example, n the smplest case of a Cartesan coordnate system one could have the free surface at ðx; y; zþ ¼ðs 1 ; s 2 ; hðs 1 ; s 2 ÞÞ, so that h s the heght above the ðx; yþ-plane, or, n a two-dmensonal example, one may have a polar coordnate system wth the free surface gven by ðh; rþ ¼ðs 1 ; hðs 1 ÞÞ, n whch case h s the dstance of the free surface from the orgn for every angle h. The bass functons used to approxmated the varables are now used to derve the weak form of the problem,.e. the fnte element equatons. From (1), the contnuty of mass (ncompressblty of the flud) resduals R C are ¼ w r u dv ð ¼ 1;...; N p Þ: ð26þ R C V After projectng the momentum Eq. (1) onto the unt bass vectors of the coordnate system e a ða ¼ 1; 2; 3Þ and usng (23), our momentum resduals R M;a take the form R ¼ / e a þ ðu cþru r P St F dv ð ¼ 1;...; N u Þ: ð27þ V Integratng by parts and usng the dvergence theorem, as shown n [32], one can rewrte (27) n terms of volume and surface contrbutons: R M;a ¼ R M;a þ R M;a ð ¼ 1;...; N u Þ; V A R ¼ / e a Re V þ ðu c Þ ru St F þ rð/ e a Þ : P dv; ¼ / e a P n da: ð28þ R M;a A A In (28), only when node s on the surface A wll / be non-zero,.e. t s nodes on the surface of V whch contrbute to the momentum resduals va (28). Ths term allows us to ncorporate stress boundary condtons naturally,.e. by addng them as contrbutons to the momentum equatons at the surfaces, whch s a well known advantage of the fnte element method over other numercal approaches where specal boundary approxmatons have to be constructed. To ncorporate our free-surface stress boundary condtons nto (28), Eqs. (4) and (5) are rewrtten nto the computatonally favourable form Ca n 1 p g þ n 1 P þ r s ½r 1 ði n 1 n 1 ÞŠ ¼ 0; where r 1 ði n 1 n 1 Þs the surface stress, playng the same role on the surface as P does n the bulk. Then (28) can be rewrtten on the free surface as / 1; e a P n 1 da 1 ¼ 1 / A 1 Ca 1; e a r s ½r 1 ði n 1 n 1 ÞŠþCap g n 1 da1 : A 1 It was ntally suggested by Ruschak [55], and generalzed for three-dmensonal problems n [56,57], that, by usng the surface dvergence theorem, one could lower the hghest dervatves and thus both reduce the constrants on the dfferentablty of the nterpolatng functons whch are used to approxmate the free surface shape and gve a natural way to mpose boundary condtons on the shape of the surface where t meets other boundares. Ths s acheved by, frst, usng the product rule: / 1; e a P n 1 da 1 ¼ 1 r s / A 1 Ca 1; r 1 e a ði n 1 n 1 Þ r1 r s / 1; e a þ Capg / 1; ðe a n 1 Þ da1 ; A 1 and then usng the surface dvergence theorem [58, p. 224], whch for a surface vector a s wth no normal component, a s ¼ a s k, s gven by r s a s k da ¼ a s k mdc; ð29þ A C where the unt vector m s the nwardly facng normal to the contour C that confnes A (Fg. 1), wth a s k ¼ / 1;r 1 e a ði n 1 n 1 Þ, so that / 1; e a P n 1 da 1 ¼ 1 r 1 r s / A 1 Ca 1; e a Capg / 1; ðe a n 1 Þ da1 þ 1 / Ca 1; r 1 e a m 1 dc 1 : C 1 A 1

10 J.E. Sprttles, Y.D. Shkhmurzaev / Journal of Computatonal Physcs 233 (2013) Thus, on the free surface, the term (28) n the momentum resdual s now replaced by a dfferent surface contrbuton and a lne contrbuton R M;a ¼ 1 r 1 r s / A 1 Ca 1; e a Capg / 1; ðe a n 1 Þ da1 ; R M;a ¼ 1 / C 1 Ca 1; r 1 e a m 1 dc 1 : ð30þ C 1 A 1 The same procedure of ntegratng by parts and usng the dvergence theorem has been used on both the surface stress term r s ½r 1 ði n 1 n 1 ÞŠ and the bulk stress term r P. In both cases, ths has created contrbutons from the boundary of that term s doman,.e. the confnng contour and surface, respectvely. Consequently, the momentum resdual now contans a cascade of scales R M;a ¼ R M;a þ R M;a þ R M;a ; ð31þ V A C whch represent, respectvely, the volume, surface and lne contrbutons. In partcular, part of the contour C 1 whch bounds the free surface s the contact lne C cl where the free surface meets the sold. Other boundares to the free surface further away from the contact lne, for example an axs or plane of symmetry, are treated n the same way but are not to be formulated untl specfc problems are consdered n Secton 4. At the contact lne, t s useful to rearrange the term n the ntegrand of the contour ntegral n (30) by representng the vector m 1 n terms of a lnear combnaton of ts components parallel to n 2 and m 2 (Fg. 1): m 1 ¼ ðm 1 m 2 Þm 2 þ ðm 1 n 2 Þn 2 : Ths dentty can be used to make the contrbuton to (31) comng from the contact lne dependent on the known vector n 2, the vector m 2, varyng along the contact lne but ndependent of the free-surface shape, and h d, defned by (16) and determned by Young s Eq. (17), so that R M;a C cl ¼ 1 Ca r 1 / 1; e a ðm 2 cos h d þ n 2 sn h d ÞdC cl C cl ð ¼ 1;...; N c Þ: If the contact lne s tangent vector s t c, then m 2 ¼t c n 2, wth the sgn chosen to ensure the nward facng vector m 2 s selected. Thus, Eq. (16), whch defnes the contact angle, can be appled n a natural way, that s wthout needng to drop another equaton n order to make room for an equaton that would fx the shape of the free surface at the contact lne. The contact angle tself s determned from Young s Eq. (17), whch, when put n resdual form as an ntegral over the contact lne contour, gves ¼ ð ÞdC cl ð ¼ 1;...; N c Þ: R Y C cl / 1; r 1 cos h d þ r 2 At the lqud sold nterface the approach developed n [32] s used. Instead of droppng the momentum equaton normal to the sold to mpose a Drchlet condton on the normal velocty (13), we use t to determne the normal stress actng on the lqud sold nterface whch allows the contact lne, where boundary condtons of dfferent type meet, to be treated n a manner consstent wth standard FEM deology. Specfcally, we ntroduce a new unknown K 3 on the lqud sold nterface whch s defned by the equaton K ¼ n 2 P n 2 : ð32þ It should be ponted out that ths partcular mplementaton smplfes the fnte element procedure ndependently of the dynamc wettng model chosen and s partcularly useful when consderng a surface non-algned wth a coordnate axs. In ths case, the procedure of rotatng the momentum equatons to algn wth the coordnate axes [59] s cumbersome whereas our approach s ndependent of both the free surface and the sold s shape, that s the curved nature of a surface s as easy to handle as, say, a planar surface algned wth coordnate axes. On the lqud sold nterface the contrbuton to the momentum equatons from the stress on the surface, whch contans contrbutons from both the generalzed Naver condton (11) for the tangental stress and (32) for the normal stress gves R M;a ¼ A 2 A 2 / 2; b Kðe a n 2 Þþ Ca e 1 a u k U k 2Ca e a r s r 2 da 2 ð ¼ 1;...; N 2 Þ: ð33þ where / 2; s an nterpolatng functon for the lqud sold nterface correspondng to the th node. In addton to the boundary condtons nvolvng stress on each nterface, we have an addtonal equaton nvolvng the velocty normal to each nterface. On the lqud gas free surface ths s the knematc condton (3) whose resduals R K are gven by R 1 ¼ / 1; A vs 1 n 1 da 1 ð ¼ 1;...; N 1 Þ; ð34þ 3 Labelled k n [32].

11 44 J.E. Sprttles, Y.D. Shkhmurzaev / Journal of Computatonal Physcs 233 (2013) whlst on the lqud sold nterface we have a condton of mpermeablty of the sold (10) wth resduals R I gven by R I A ¼ / 2; v s 2 U n2 da 2 ð ¼ 1;...; N 2 Þ: ð35þ 2 In keepng wth the framework presented n [32], all momentum equatons are appled at both the lqud gas free surface and at the lqud sold nterface and hence, once the two boundares meet at the contact lne, the contact lne condtons can be mplemented naturally, wthout droppng any of the equatons there. A crude, but useful, nterpretaton s to thnk of the momentum equatons as determnng the bulk veloctes, the knematc condton on the free surface as determnng the unknowns that specfy ts poston, and the mpermeablty condton, whch s the geometrc constrant of the prescrbed shape, as determnng the extra unknown K,.e. the normal stress. Thus far, the equatons are assumed to determne the bulk velocty, the shape of the free surface and the normal stress on the lqud sold nterface (K). In addton to these equatons we have (6) (9) on the free surface and (12) (15) on the lqud sold nterface whch determne the surface velocty v s, surface densty q s and surface tenson r. In partcular, we can thnk of the Darcy-type Eqs. (6) and (12) as determnng the surface velocty tangental to the surface v s k, whch n a fully threedmensonal flow wll have two components n the e s1 and e s2 drecton, where e s m s a bass vector n the drecton of ncreasng s m ; m ¼ 1; 2. The resduals R vs ;m 1k and R vs ;m 2k are R vs 1k ;m R vs 2k ;m ¼ / 1; e s m A 1 ¼ / 2; e s m A 2 v s 1k u k 1 þ 4a b 4 b v s 2k 1 2 u k þ U k ar s r 2 r s r 1 da 1 ð ¼ 1;...; N 1 Þ; ð36þ da 2 ð ¼ 1;...; N 2 Þ: ð37þ Eqs. (7) and (13) can be thought of as determnng the normal component of the surface velocty on the surface A c (c ¼ 1; 2) and the resduals R vs cn from these equatons take the same form on each nterface: R vs cn ¼ / c; A c h u v s c The correspondng resduals R qs c R qs c ¼ Ac / þ rs q s c n c Q q s c qs ce da c ð ¼ 1;...; N c ; c ¼ 1; 2Þ: ð38þ from Eqs. (8) and (14) to determne the evoluton of the surface densty are h v s ck cs ck þ q s c rs c s ck þ qs c ðvs c n cþr s n c þ q s c qs ce da c ð ¼ 1;...; N c ; c ¼ 1; 2Þ: Usng the standard FEM deology, whch wll gve us a method for applyng boundary condtons on the surface flux q s v s k where the surface meets a boundary, we ntegrate the dvergence term n (39) by parts to obtan R qs c ¼ q s c v s ck cs ck r s / c; þ / c þ qs c rs c s ck þ qs c ðvs c n cþr s n c þ q s c qs ce da c h þ r s / c; q s c v s ck cs ck da c ð ¼ 1;...; N c ; c ¼ 1; 2Þ: ð40þ A c Then, usng the surface dvergence theorem (29) wth a s k ¼ / 1;q s v s k cs k we have a contrbuton from both the surface and the contour boundng that surface, so that R qs c ¼ R qs c R qs c ¼ A c R qs c ¼ Cc A c þ Ac Cc R qs c ; C c / c v s c n c r s n c þ q s c qs ce da c þ qs c rs c s ck þ cv qs s cn rs n c þ q s c ð42þ / c; q s c v s ck cs ck m c dc c ð ¼ 1;...; N c ; c ¼ 1; 2Þ: ð43þ Consequently, we are able to apply any boundary condtons on surface mass flux by replacng the contour contrbuton (43) wth the approprate value. In partcular, at the contact lne contour C cl we have the surface mass flux contnuty condton (18). Ths condton could potentally be appled by usng t to replace ether the contour contrbuton to the free surface or the lqud sold nterface equatons, and to determne whch way s correct we study the structure of the nterface formaton equatons. Specfcally, an asymptotc approach to the dynamc wettng problem for small Ca and recaptulated n Secton 2.2 shows that t s the flux of surface mass nto the contact lne from the free surface, whch depends on the global flow va the velocty n the far feld on the free surface u f ðh d Þ, that determnes the relaxaton process on the sold surface. Therefore, n a numercal mplementaton of ths problem one should allow the free surface equatons to determne the flux nto ð39þ ð41þ

12 J.E. Sprttles, Y.D. Shkhmurzaev / Journal of Computatonal Physcs 233 (2013) the contact lne and then use the surface mass contnuty condton (18) on the lqud sold sde of the contact lne to take ths flux as the surface mass supply nto the lqud sold nterface. Therefore, on the free surface we have R qs 1 ¼ / 1; q s 1 v s 1k cs 1k m 1 dc cl ; C cl C cl ð44þ whlst on the lqud sold nterface, usng (18) to rewrte the flux q s 2 v s 2k cs 2k m 2 nto ths nterface n terms of the flux nto the contact lne from the free surface, we have R qs 2 ¼ / 2; q s 1 v s 1k cs 1k m 1 dc cl : C cl C cl ð45þ Then, the flux nto the lqud sold nterface s gven n terms of the flux that goes nto the contact lne from the free surface. The fnal resduals from the surface equatons, R rc, are obtaned from the surface equaton of state (9) and (15), whch express the surface tenson as an algebrac functon of the surface densty: ¼ da c ð ¼ 1;...; N c ; c ¼ 1; 2Þ: ð46þ R rc Ac / c; r c k 1 q s c In ths secton, we have derved the fnte element resduals requred to solve dynamc wettng flows usng the nterface formaton model. In partcular we have the followng couplng of unknowns u e a ; p ; h ; K ;v s ck; e m;v s cn; ; qs c; ; r c;; h d a ¼ 1; 2; 3 c ¼ 1; 2 m ¼ 1; 2; ð47þ each of whch has ts correspondng resdual R M;a ; R C ; RK ; RI ; ck;m Rvs ; R vs cn ; R qs c ; R rc ; R Y a ¼ 1; 2; 3 c ¼ 1; 2 m ¼ 1; 2: ð48þ The subscrpts above wll have dfferent lmts that are the same n both the varable and ts correspondng resdual,.e. the same number of equatons and unknowns has been assured. 4. Valdaton of the code and some benchmark calculatons for Problem A Computatons are performed for cases whch are axsymmetrc or two-dmensonal n smple geometres so that calculatons may be easly reproduced, thus gvng benchmark results for future nvestgators. In what follows, we consder a menscus rsng aganst gravty through a cylndrcal capllary of radus R. The computatonal doman s a regon n the ðr; zþ-plane, and the free surface s parameterzed by arclength s. Moton s assumed to be axsymmetrc about the z-axs whch runs vertcally through the centre of the capllary wth the radal axs perpendcular and located at the base of the capllary (Fg. 2). Frst, we consder the steady propagaton of a menscus through a capllary (hereafter Problem A) to check convergence of our code as the mesh s refned and to compare our computatons to the asymptotc results summarzed n Secton 2.2. Then, we wll examne Problem A from the vew pont of analyzng the nfluence of the radus R of the capllary Fg. 2. Illustraton showng the computatonal doman for flow through a capllary wth the bulk doman V, lqud gas free surface A 1, lqud sold surface A 2 and contact lne C cl all ndcated.