Axsymmetrc Drop Penetraton Into an Open Straght Capllary Poorya A. Ferdows and Markus Bussmann Department of Mechancal and Industral Engneerng Unversty of Toronto Emal: p.ferdows@utoronto.ca, bussmann@me.utoronto.ca ABSTRACT We consder the mbbton of a drop nto an open straght capllary. The drop sze compared to the hole sze, the equlbrum contact angle, and the geometry of the capllary determne whether the drop wll draw nto the capllary, or not wet the capllary at all. We have developed a numercal scheme based on the Volume of Flud (VoF) method to study ths phenomena. Here we focus on the numercal scheme to treat the contact lne moton near a sharp corner, and then present a few results of the dynamcs of wettng and dewettng. 1 INTRODUCTION Contact lne-drven flows on rough and porous surfaces occur n a varety of applcatons, ncludng nket prntng, enhanced ol recovery, water repellent fabrc and self-cleanng surface desgn, and spray coatng. Here we consder a drop atop an open sngle pore on such a surface, and examne the wettng and dewettng of the capllary. Ths fundamental analyss sheds lght on more complex phenomena such as the wettng and dryng of a real fabrc surface, structured surface, or porous meda. As shown n fgure 1, one can nfer three nterfacal flow regmes: the flow over the surface around the capllary, the mbbton of lqud nto the capllary, and the flow near the sharp corner. Whle the modellng of contact lne flows over flat surfaces (as n regons 1 and 2 n fgure 1) remans a topc of research, the focus here s on the development of a methodology for the contact lne moton and the dramatc nterface deformaton near the sharp corner (regon 3 n fgure 1). We have developed that methodology for the well-known Volume of Flud (VoF) method, and then used t to smulate the mbbton of a drop nto a capllary. Fgure 1: Interfacal flow regmes over and wthn a capllary: (1) The nterface moton on top of the surface, (2) the nterfacal flow wthn the capllary, and (3) contact lne pnnng and nterface deformaton near a sharp edge. The VoF method s a popular choce for trackng nterfaces because t can conserve mass exactly [14, 1]. In the VoF method appled to a two-phase flow, the volume fracton f s defned as the rato of the volume of one (reference) flud nsde a cell to the cell volume. f = 1 ndcates a full cell whle f = ndcates no reference flud n the cell. Interface cells are those wth < f < 1; n such cells, the nterface s geometrcally reconstructed based on f. The most common technque s the Pecewse Lnear Interface Calculaton (PLIC) method (e.g. [2, 12, 14, 1]) n whch the nterface s represented by a lnear segment wthn each nterface cell. PLIC requres a normal and an ntercept for reconstructng the nterface wthn a cell. The ntercept can be computed ether analytcally or teratvely [17], whle many methods have been suggested to compute normals (e.g. [6, 12, 13, 14, 1, 18]). In addton to reconstructon, for flow problems that nvolve surface tenson, the mean curvature must be calculated from the volume fractons (e.g. [3, ]). Fnally,
n a two-phase flow smulaton, f must be advected n a way that mantans a sharp nterface. Algebrac methods are often dffusve; thus the nterface s usually advected geometrcally (e.g. [14, 16, 18]). By reconstructng and advectng f, the new locaton of an nterface s determned. Much research has been conducted on the wettng and dewettng of a capllary [4, 8, 9, 1, 19, 2]. Mddleman [1] revews analytcal work on lqud mbbton nto a capllary, also known as capllary wckng, and penetraton dynamcs. When a capllary tube comes n contact wth an nfnte reservor, the nstantaneous capllary rse and wckng speed can be predcted by the Washburn-Rdeal-Lucas equaton (WRL) [19]. More closely related to the results presented here, Marmur [9] studed the penetraton of a sngle drop,.e. a fnte amount of lqud, nto a capllary, assumng that the drop s ntally pnned to the tube entrance and that gravty does not dstort the drop from a sphercal shape. Marmur determned that there s a crtcal value of the contact angle whch dstngushes between penetraton and nonpenetraton, that depends on the drop sze, and can be much greater than. Also, the smaller the drop, the faster t seeps nto the capllary. We have developed a VoF-based methodology for the behavour of a contact lne near a sharp corner, based on the concept of contact lne pnnng. Here we present detals of that methodology, and then some results of capllary wettng and dewettng to llustrate ts effcacy. α 1 α 2 α3 (a) ed 2 NUMERICAL TREATMENT OF CONTACT LINE MOTION NEAR A SHARP CORNER Consder the confguratons shown n fgure 2, of a contact lne traversng a sold surface at a corner. At the mcro scale shown n fgure 2(a), the corner s not sharp, and the contact angle s always the actual angle between the tangent lne to the sold surface and the nterface. However, at the macro level, the corner appears sharp, and we observe dfferent edge contact angles ed (e.g. +α 1, +α 2, +α 3 ) that depend on the poston of the contact lne on the sold at the mcro level. Accordng to Gbbs [7, 11], at equlbrum, the edge contact angle ed must le between the contact angles on ether sde of the corner. Now consder that the contact lne s movng slowly toward a sharp corner. At the mcro level, shown n fgure 2(a), the contact angle remans constant as the contact lne traverses the corner. The dstance traveled by the nterface s proportonal to the radus of curva- (b) Fgure 2: Contact lne moton near a corner at (a) the mcro scale and (b) the macro scale. α
1 1 (a) 1 ed (d) 1 1 (c) 1 (b) ed (e) the contact lne reaches the corner (fgure 3(b)), t pns there. The volume fracton n the cell (, ) at ths nstant, f p, can be calculated from smple geometry. Hereafter, the edge contact angle starts ncreasng as more lqud enters the cell (, ) (fgure 3(c)), so that f, > f p. Snce the contact lne s pnned, only the gas phase can be allowed to pass across the left face of the cell (, ), untl f, = 1., ed can be computed based on f,. Thereafter, the contact lne remans pnned untl 1, ed = +π/2 (fgure 3(e)) durng whch tme only the gas phase can be allowed to pass across the bottom face of the cell (, ), untl the contact lne releases (fgure 3(f)). From ths descrpton, t can be nferred that two VoF modfcatons are requred: to calculate, ed based on f, (reconstructon step), and to modfy the advecton scheme so that only one of the phases advects across nearby cell faces when the contact lne s pnned. These modfcatons were appled to the heght functon (HF) technque for calculatng nterface normals and curvatures [1, 3], and to the EI-LE advecton technque of Scardovell and Zalesk [18]. Detals wll be presented n a subsequent paper. (f) Fgure 3: Interface confguratons near a sharp corner, when the flow drecton s CCW about the corner. Cases (b) and (e) are two specal cases, when ed = ; n practce, an nterface confguraton wll lkely change from (a) to (c) (smlarly from (d) to (f)) from one tmestep to the next. ture of the corner. As the curvature ncreases, the dstance approaches zero. At the macro scale, the radus of curvature of the corner s neglgble compared to any other characterstc dmenson, e.g. a drop radus, and so the dstance travelled by the contact lne can be assumed to be zero. Therefore, at the macro level, when the contact lne reaches an deally sharp corner, t can be assumed to pn there and rotate about the corner, untl ed reaches the contact angle assocated wth the other sde, as llustrated n fgure 2(b). A VoF treatment of a contact lne at a sharp corner requres modfcaton of both the nterface reconstructon algorthm (.e. the computaton of ed ) and the advecton scheme. For the sake of smplcty, we assume that < π 2. Consder that a lqud moves counter clockwse about a corner, and that the contact lne approaches the sharp corner from the rght, as depcted n fgure 3(a). When 3 RESULTS We now present the results of two smulatons, one for a wettng contact angle of, and the other for a nonwettng angle of. In both cases, we ntally place a drop partally wthn a capllary n the axsymmetrc coordnate system. The drop s assgned the propertes of water; the surroundng gas s ar. For the frst case presented here, the radus of the drop s 1 µm, the radus of the capllary s 18.7 µm and the ntal penetraton depth s 17 µm. Based on these parameters the Ohnesorge number, that here reflects the rato of vscous to surface tenson forces, s Oh =.34. We nvestgate the problem by ntroducng two parameters. Denotng the velocty vector n cell (, ) by V = (u,v ), the average seepng velocty v s of flud wthn the capllary s: v s = f Ω v n hole n hole f Ω (1) where Ω s the 3D axsymmetrc volume of the cell (, ). When there s no lqud n the capllary, v s =. The second parameter reflects the centrod of the flud
n the z drecton, whch s computed as: z m = f Ω z all cells all cells f Ω (2) Fgure 4 shows the poston of the mbbed drop at 1 µs ncrements; fgure shows close-ups of the contact lne whle pnned at the corner. (Note that for all results presented here, the nterfaces are the actual pecewse lnear segments from the VoF PLIC reconstructon, and not smply a contour plot of f = 1/2.) The varaton of v s versus tme s plotted n fgure 6, and z m n fgure 7. As shown n fgure 4(a), the drop s ntally postoned partally wthn the capllary, but mmedately begns to be mbbed by the capllary. Sometme between 7. and 8 µs the upper contact lne reaches the edge of the capllary openng and s pnned at the sharp edge. The nstant when the nterface s pnned corresponds to the maxmum seepng velocty v s shown n fgure 6, whch reflects that the drop accelerates from t = untl t reaches the sharp corner. The pnnng duraton,.e. the tme that the upper menscus remans pnned, s about 3 µs. It s durng ths nterval that the edge contact angle changes from that assocated wth the top surface to that assocated wth the capllary, as flud rapdly enters the capllary. Due to the hgh velocty as well as the contnuous change n the edge contact angle, the upper menscus deforms dramatcally durng those 3 µs. After t 9. µs, no lqud remans above the capllary. After the release of the contact lne from the sharp corner, sometme between 11 and 11. µs, vscous dampng and the surface tenson force decelerate the column of flud sldng nto the capllary. The asymptotc behavor of z m shown n fgures 6 and 7 llustrates that by t µs the column of lqud has reached equlbrum (z m 78 µm). At t = µs, h = 18.293 µm s the penetraton depth measured from the top sold surface. Now consder the non-wettng case, when the contact angle s. Fgure 8 shows the poston of the drop n 2 µs ncrements. The varaton of v s wth tme s shown n fgure 9, and z m s plotted n fgure 1. Fgure 8(a) shows the drop ntally postoned partally wthn the capllary, smlar to the prevous case. The exact equlbrum confguraton, based on a thermodynamc analyss presented elsewhere, s ndcated by the dashed lne; at t >, lqud begns to rse. Around 24 µs the nterface reaches the edge of the capllary openng. Ths s the tme when z m and v s manfest oscllatons due to the pnnng effect. Note 1 (a) µs 1 1 (e) 4 µs 1 (b) 1 µs 1 1 1 (f) µs 1 (c) 2 µs 1 1 1 (g) µs 1 (d) 3 µs Fgure 4: Case I, = : drop mbbton nto a capllary. By t = µs the flud has almost reached equlbrum. The dmensons are n µm. 1 1
2. 7. 11 11 11 1 1 1 1 12. 1 17. 2 2. 7. 1 12. 1 17. 2 z m (µm) 2. 7. 1 12. 1 17. 2 (a) 7. µs (b) 8 µs (c) 8. µs 11 11 11 2 4 Fgure 7: Case I, = : lqud poston versus tme. 1 1 1 2. 7. 1 12. 1 17. 2 2. 7. 1 12. 1 17. 2 2. 7. 1 12. 1 17. 2 11 1 2. (d) 9 µs 7. 1 12. 1 (g) 1. µs 17. 2 11 1 2. (e) 9. µs 7. 1 12. 1 (h) 11 µs 17. 2 11 1 2. (f) 1 µs 7. 1 12. 1 () 11. µs Fgure : Case I, = : contact lne transton from slppng to pnned and agan slppng (releasng) near a sharp corner. The dmensons are n µm.. 1 1. v s (m/s) 17. 2 that n ths case due to the ntal amount of lqud wthn the capllary, the velocty of the lower menscus, when t reaches the sharp corner for the frst tme, s not enough to force the release of the pnned contact lne from the corner. There can be cases that depend on the ntal confguraton, when the velocty of the lower menscus s enough to force the release of the contact lne from the edge, only to pn agan to the edge and to oscllate between these states, untl equlbrum s reached. Although the contact lne s pnned, because of the nerta of the movng lqud, the vscous and surface tenson forces cannot mmedately arrest the moton at the nstant of frst pnnng. As a result the lqud moves slghtly upward untl the resultant force that acts n the opposte drecton stops the lqud at t = 237 µs (the frst tme when v s = on fgure 9). Snce the resultant force s not zero, the lqud accelerates back nto the capllary and ed decreases. Such oscllatons are observed untl the lqud comes to rest. Even at t = 4 µs, the upper nterface stll vbrates whle v s (see fgure 9), because of the perodc deformatons of the upper menscus wth an area much bgger than that of the lower menscus. The edge contact angle at the end of the smulaton s (4) ed = 11.1, whch s only.28% dfferent than the thermodynamc predcton. 2 2. 2 4 Fgure 6: Case I, = : varaton of the seepng velocty versus tme. 4 CONCLUSION We have presented a VoF-based treatment for contact lne moton around a sharp corner, nvolvng modfcatons both to the reconstructon and advecton steps, and appled an axsymmetrc VoF code to the predcton of drop mbbton nto a capllary. Examples of both wettng and non-wettng behavour llustrate very nterestng dynamcs of a contact lne pnned to the capllary edge.
v s (m/s).6.4.2.2.4 4 4 4 4 3 3 3 3 2 2 2 2 1 1 2 2 3 (a) µs 1 1 2 2 3 (b) µs 1 1 2 2 3 (c) 1 µs 1 1 2 2 3 (d) 17 µs 4 4 4 4 3 3 3 3 2 2 2 2 1 1 1 2 2 3 3 4 Fgure 9: Case II, = : varaton of the seepng velocty versus tme. z m (µm) 78 76 74 72 1 2 2 3 3 4 Fgure 1: Case II, = : lqud poston versus tme. 2 2 (e) 2 µs 3 1 1 2 2 (f) 22 µs 3 1 1 2 2 (g) 2 µs 3 1 1 2 2 (h) 4 µs Fgure 8: Case II, = : the drop pulls out of the capllary. By t = 4 µs the bottom menscus has almost reached equlbrum but the upper nterface stll oscllates. The dashed lne s the exact equlbrum confguraton. All dmensons are n µm. 3 REFERENCES [1] S. Afkham and M. Bussmann. Heght functons for applyng contact angles to 2D VOF smulatons. Internatonal Journal for Numercal Methods n Fluds, 7(4):43 472, 28. [2] N. Ashgrz and J. Y. Poo. FLAIR: Flux lnesegment model for advecton and nterface reconstructon. Journal of Computatonal Physcs, 93(2):449 468, 1991. [3] S. J. Cummns, M. M. Francos, and D. B. Kothe. Estmatng curvature from volume fractons. Computers and Structures, 83(6-7):42 434, 2. [4] A. Delbos, E. Lorenceau, and O. Ptos. Forced mpregnaton of a capllary tube wth drop mpact. Journal of Collod and Interface Scence, 341(1):171 177, 21.
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