We use the lubrication approximation to analyze the steady unidirectional flow of a thin rivulet on a vertical substrate subject to a prescribed uniform longitudinal shear stress at its free surface. In particular, we determine the quasisteady stability of a rivulet of prescribed flux, and calculate when it is energetically favorable for a rivulet to split into two narrower rivulets

Wilson, S.K.

English

University of Strathclyde Institutional Repository

Strathprints Institutional RepositoryWilson, S.K. (2005) Unidirectional flow of a thin rivulet on a vertical substrate subject to a prescribeduniform shear stress at its free surface. Physics of Fluids, 17 (10). ISSN 1070-6631Strathprints is designed to allow users to access the research output of the University of Strathclyde.Copyright c© and Moral Rights for the papers on this site are retained by the individual authorsand/or other copyright owners. You may not engage in further distribution of the material for anyprofitmaking activities or any commercial gain. You may freely distribute both the url (http://strathprints.strath.ac.uk/) and the content of this paper for research or study, educational, ornot-for-profit purposes without prior permission or charge.Any correspondence concerning this service should be sent to Strathprints administrator:mailto:strathprints@strath.ac.ukhttp://strathprints.strath.ac.uk/    Wilson, S K (2005) Unidirectional flow of a thin rivulet on a vertical substrate subject to a prescribed uniform shear stress at its free surface. Physics of Fluids, 17 (10). pp. 108105-108105. ISSN 1070-6631    http://eprints.cdlr.strath.ac.uk/2266/  This is a paper published in Physics of Fluid (ISSN: 0168-9274).  ©2005 American Institute of Physics   Strathprints is designed to allow users to access the research output of the University of Strathclyde. Copyright © and Moral Rights for the papers on this site are retained by the individual authors and/or other copyright owners. Users may download and/or print one copy of any article(s) in Strathprints to facilitate their private study or for non-commercial research. You may not engage in further distribution of the material or use it for any profit-making activities or any commercial gain. You may freely distribute the url (http://eprints.cdlr.strath.ac.uk) of the Strathprints website.    Any correspondence concerning this service should be sent to The Strathprints Administrator: eprints@cis.strath.ac.uk Unidirectional flow of a thin rivulet on a vertical substrate subjectto a prescribed uniform shear stress at its free surfaceS. K. Wilsona and B. R. DuffybDepartment of Mathematics, University of Strathclyde, Livingstone Tower, 26 Richmond Street,Glasgow G1 1XH, United KingdomReceived 8 October 2004; accepted 18 August 2005; published online 17 October 2005We use the lubrication approximation to analyze the steady unidirectional flow of a thin rivulet ona vertical substrate subject to a prescribed uniform longitudinal shear stress at its free surface. Inparticular, we determine the quasisteady stability of a rivulet of prescribed flux, and calculate whenit is energetically favorable for a rivulet to split into two narrower rivulets. © 2005 AmericanInstitute of Physics. DOI: 10.1063/1.2100987Rivulets of fluid subject to significant surface-shearforces occur in a variety of contexts, including the rivulets ofrainwater and/or deicing fluid that form on the wings of air-craft, the rivulets of condensate that frequently occur withinheat exchangers, and even the rivulets of rainwater that occuron the windscreen of a rapidly moving car on a rainy day.Various aspects of rivulet flow in the presence of significantsurface-shear forces have been investigated by, for example,Mikielewicz and Moszynski,1 Eres, Schwartz, and Roy,2 Wil-son, Duffy, and Hunt,3 Myers, Liang, and Wetton,4 and Saberand El-Genk.5 In this Brief Communication we use the lu-brication approximation to analyze the steady unidirectionalflow of a thin rivulet on a vertical substrate subject to aprescribed uniform longitudinal shear stress at its freesurface.Consider the steady unidirectional flow of a thin sym-metric rivulet with constant semiwidth a and constant vol-ume flux Q on a vertical substrate subject to a prescribeduniform longitudinal shear stress  at its free surface. Weassume that the fluid is Newtonian and has constant density, viscosity , and surface tension . We choose Cartesianaxes Oxyz with the x axis vertically downwards and the zaxis normal to the substrate z=0. The velocity u=uy ,zi andpressure p= px ,y ,z of the fluid are governed by the famil-iar mass-conservation and Navier-Stokes equations subject tothe usual normal and tangential stress balances and the kine-matic condition at the unknown free surface z=hy, andzero velocity at the substrate z=0. At the contact line y=a,where h=0, the contact angle takes the prescribed value ,where 0 is the nonzero static contact angle.Analytical progress can be made when the rivulet is thinwith, in particular 1 in which case it is appropriate tonondimensionalize y and a with l, z and h with l, u withU=g2l2 /, Q with l2U=g3l4 /, p− p with gl, and with gl, where l=  /g1/2 is the capillary length, g isgravitational acceleration, and p is atmospheric pressure.Henceforth all quantities are nondimensional unless it isstated otherwise. The leading-order problem can be immedi-ately solved to yield the simple solution p=−h=1/a, u= 2h−zz /2+z, h= a2−y2 /2a. Note that since the flow isunidirectional, the mass-conservation equation and the kine-matic condition are satisfied identically, and the solution isvalid for all values of the Reynolds number. In particular,the comment made by Myers et al.4 about the restriction onthe applicability of the thin-film approximation is erroneous.The maximum thickness of the rivulet occurs at y=0 and isequal to hm=h0=a /2. The flux Q is given byQ = −a+a0hyuy,zdzdy =4a4105+2a315. 1For 	0 the flux Q is a monotonically increasing function ofa, but for 0 the flux Q initially decreases monotonicallyto a minimum value of Q=Qmin=−4amin4 /315=−30874 /51200 at a=amin=−21 /8, before increasingmonotonically through the value Q=0 at a=a0=−7 /2.There are five possible types of flow pattern denoted astypes I–V as sketched in Fig. 1. When 0 the prescribedshear stress acts in the same direction as gravity. In this casethe velocity is always downwards i.e., u0 throughout therivulet, and the maximum velocity umax=aa+4 /80occurs on the free surface at y=0 and z=hm=a /2 type I.On the other hand, when 0 the prescribed shear stressopposes gravity, and this competition leads to more interest-ing behavior than in the case 0. In particular, the velocityis always upwards i.e., u0 near the edges of the rivulet,but it can be downwards i.e., u0 elsewhere. Whena−2 the velocity is always upwards throughout the rivu-let, and the minimum velocity umin=aa+4 /80 occurson the free surface at y=0 and z=hm=a /2 type V, butwhen a−2 there is a region of downwards flow in thecenter of the rivulet and the maximum velocity umax= a+22 /80 occurs within the flow at y=0 and z=hm+= a+2 /2, and the minimum velocity umin=−2 /20 oc-curs on the free surface at y= ±b= ±a1+2 /a1/2 andz=− type II when a−4, type III when a=−4 and typeIV when −2a−4. Note that both the solution a=a0=−7 /2 corresponding to Q=0 and the solutionaAuthor to whom correspondence should be addressed. Telephone: 440141 548 3820. Fax: 44 0141 548 3345. Electronic mail:s.k.wilson@strath.ac.ukbTelephone: 44 0141 548 3645. Fax: 44 0141 548 3345. Electronicmail: b.r.duffy@strath.ac.ukPHYSICS OF FLUIDS 17, 108105 20051070-6631/2005/1710/108105/4/$22.50 © 2005 American Institute of Physics17, 108105-1Downloaded 03 Nov 2006 to 130.159.248.222. Redistribution subject to AIP license or copyright, see http://pof.aip.org/pof/copyright.jspa=amin=−21 /8 corresponding to Q=Qmin have type IV be-havior. The different flow patterns that can occur when 0 can be understood by realizing that the local downwardsflux due to gravity varies with h like h3 while the local up-wards flux due to shear stress varies like h2. Thus when h− and, in particular, near the contact line the flow isalways upwards, while if h− the flow is downwards nearthe substrate and if h−2 the flow is downwards all theway across the rivulet.The present solution can be interpreted either as a rivuletof prescribed semiwidth a= a¯, or as a rivulet of prescribedflux Q=Q¯ . In the case of prescribed semiwidth a= a¯ there isa unique value of Q for all values of . Figure 2 shows asketch of Q as a function of a and summarizes when thedifferent types of flow pattern occur. The shape of the rivuletis independent of . Increasing decreasing  from zero hasthe effect of increasing decreasing u and hence increasingdecreasing Q. In the case of prescribed flux Q=Q¯ the pos-sible semiwidths are the positive solutions for a of Q=Q¯ .When 0 there is one solution for all positive values of Q¯ ,but no solution when Q¯ is zero or negative. On the otherhand, when 0 there is one solution for Q¯ 	0, two solu-tions for QminQ¯ 0 the narrower one satisfying 0aamin and the wider one aminaa0, one solution a=amin when Q¯ =Qmin, and no solution when Q¯ Qmin. Figure3 shows a sketch of a as a function of  and summarizeswhen the different types of flow pattern occur. The shape ofthe rivulet depends on  via a. When 0 the velocity isalways downwards and the effect of increasing the shearstress from zero is always to increase the local velocity of thefluid throughout the rivulet, i.e., u /0 everywhere.Hence, since the flux must remain constant, the rivulet al-ways becomes narrower and shallower, i.e., a /=−7a / 8a+210. When 0 the velocity is always up-wards near the edges of the rivulet, but may be downwardselsewhere. Moreover, the effect of decreasing the shear stressfrom zero can be either to increase or to decrease the localvelocity. However, the net effect of the changes to the size ofthe rivulet and to the velocity along it are that when aamin the rivulet always becomes narrower and shallower,i.e., a /0, whereas when aamin the rivulet always be-comes wider and deeper, i.e., a /0.A full stability analysis is beyond the scope of thepresent work, but in the case of prescribed flux Q=Q¯ we cangeneralize the quasisteady stability analysis of a purelyFIG. 1. Sketches of the flow in the cases a 0 type I, b 0 anda−4 type II, c 0 and a=−4 type III, d 0 and −2a−4 type IV, and e 0 and a−2 type V. In each part regions ofdownwards flow i.e., u0 are shaded in gray while regions of upwardsflow i.e., u0 are unshaded. In b, c, and d b=a1+2 /a1/2 and inb c=a1+4 /a1/2. The locations of the maximum and/or minimum veloc-ity are marked with dots  and/or open circles , respectively.FIG. 2. Sketch of Q as a function of a summarizing when the different typesof flow pattern occur for 0, =0, and 0.FIG. 3. Sketch of a as a function of  summarizing when the different typesof flow pattern occur for Q¯ 0, Q¯ =0, and Q¯ 0. Here 1=−−35Q¯ /161/4,max=−−5120Q¯ /30871/4, and 2=−105Q¯ /1281/4.108105-2 S. K. Wilson and B. R. Duffy Phys. Fluids 17, 108105 2005Downloaded 03 Nov 2006 to 130.159.248.222. Redistribution subject to AIP license or copyright, see http://pof.aip.org/pof/copyright.jspgravity-driven rivulet undertaken by Wilson and Duffy.6 Fol-lowing the earlier work we assume that the flow remainssymmetric and unidirectional and that the quasisteady mo-tion is driven entirely by that of the moving contact line y=A, where A=At. Furthermore, we assume that the speedof the moving contact line, A, and the dynamic contactangle, =t, are related by a general “Tanner law” in theform A=F, where the function F satisfies F1=0 andis monotonically increasing near =1. Perturbing about theconstant steady-state values of the semiwidth and the contactangle A=a and =1 by writing A=a+a1t and =1+1tyieldsa1 =Ma1mm!, 2where M = dmF /dm=10m=1,3 ,5 , . . .  is the first non-zero derivative of F evaluated at =1 and = − QAQA=a,=1 = − 8a + 212a3a + 7 . 3Equation 2 can be immediately solved for a1 and this so-lution shows that, whatever the specific form of F, therivulet is unstable to small perturbations when 0 andstable when 0. Hence when 	0 and when 0 andQ¯ 	0 the rivulet is always stable, while when 0 andQminQ¯ 0 the wider rivulet is always stable and the nar-rower rivulet is stable when 0a−7 /3 but unstable when−7 /3aamin=−21 /8.We can also address the question posed by Schmuki andLaso7 and Myers et al.4 of whether or not it is ever energeti-cally favorable for a rivulet to split into two or more subrivu-lets. The total energy of the rivulet is the sum of its kineticenergy and surface energy. Momentarily reverting todimensional variables, the kinetic energy per unit length isgiven by2−a+a0hu2dzdy 4and the surface energy, or, more precisely, the difference be-tween the surface energy of the rivulet and the surface en-ergy of the dry substrate per unit length is given by−a+a1 + h21/2dy − 2a cos 	 . 5Thus, if we nondimensionalize energy per unit length withU2l2=3g25l6 /2 then the leading-order energy of therivulet, E, is given byE = −a+a h515+5h424+2h36dy +1W12−a+ah2dy + a	 ,6which can be evaluated explicitly to yieldE =16a6103 95+2a5189+22a4105+4a3W, 7where W=lU2 /=23 /gl2 is an appropriately definedWeber number.8 It can readily be demonstrated that for afixed value of  E is a monotonically increasing function ofa. Thus wider rivulets always have more energy than nar-rower ones, and hence it is never energetically favorable fora rivulet to split into one or more wider rivulets. However, itcan be energetically favorable for a rivulet to split into one ormore narrower rivulets. Specifically, it is energetically favor-able for a rivulet with semiwidth a and flux Q to split intotwo rivulets, one with flux Q and the other with flux 1−Q, where 01/2, if the difference between the en-ergies of the two states, E, is positive.In the special case of a purely gravity-driven rivulet i.e.,the case =0 the semiwidth is given by a= 105Q /41/4, andwhen aac then E0 for all 01/2, but when a=ac then E=0 at =1/2, and when aac then E0 at=1/2. Thus for aac or equivalently for QQc it isenergetically favorable for the rivulet to split into two equalnarrower rivulets, where the critical values of the semiwidth,ac, and the flux, Qc, are given byac = 346523/4 − 141 − 2−1/2W 	1/5  4.5805W1/5 , 8Qc =4105346523/4 − 141 − 2−1/2W 	4/5  16.7703W4/5 .This result agrees qualitatively with Myers et al.,4 who cal-culated ac and Qc numerically for a range of values of .In the limit of large positive shear stress → whenQ0, and for the narrower rivulet in the limit of large nega-tive shear stress →− when Q0, the leading order semi-width is given by a= 15Q /21/3, and the behavior of E isqualitatively the same as that in the case =0 describedabove, where ac and Qc are now given byac =  7022/3 − 11 − 2−1/32W	1/3  5.84132/3W1/3 , 9Qc =2822/3 − 131 − 2−1/3W26.5750W.In particular, this result shows that the conjecture proposedby Myers et al.4 that it is never energetically favorable for apurely shear-stress-driven rivulet to split is not correct.When both gravity and shear-stress effects are significantanalytical progress is harder. However, we can still makeprogress numerically. Before doing this it is convenient toremove W from the problem by scaling a with W−1/5,  withW−1/5, Q with W−4/5, and E with W−6/5. When Q0 the be-havior is qualitatively the same as that in the case =0.When QminQ0 which is possible only when 0 thesituation is somewhat more complicated. In this case thereare always two possible rivulets with the same flux the nar-rower one satisfying 0aamin and the wider one aminaa0. As we have already seen, it is always energeticallyfavorable for the wider rivulet to “split” into the narrower108105-3 A rivulet with prescribed shear stress Phys. Fluids 17, 108105 2005Downloaded 03 Nov 2006 to 130.159.248.222. Redistribution subject to AIP license or copyright, see http://pof.aip.org/pof/copyright.jsprivulet with the same flux. On the other hand, the behavior ofthe narrower rivulet is somewhat similar to that of the rivu-lets in the case Q0 described previously. Specifically, forc0, where c−2.5038W−1/5, it is never energeticallyfavorable for the smaller rivulet to split, but for c thereis a critical value of the semiwidth ac 0acamin abovewhich and a critical value of the flux Qc QminQc0below which it is energetically favorable for the rivulet tosplit into two narrower rivulets, and the critical situationis again that of splitting into two equal rivulets eachwith half the flux of the original, i.e., the case =1/2.Note that ac=aminc6.5725 W−1/5 and Qc=Qminc−23.6954 W−4/5. Figures 4 and 5 show W1/5a and W4/5Q,respectively, plotted as functions of W1/5 and indicate whenit is energetically favorable or unfavorable for a rivulet tosplit.Note that, although perhaps slightly obscured by thechoice of nondimensionalization, the results described in thepresent work include the very simple special case of a purelyshear-stress-driven rivulet. Specifically, if we revert to di-mensional variables and set g=0 then we have u=z / andQ=22a3 /15. In this case there is always single rivuletfor each value of a¯ 0 or Q¯ provided that it has the samesign as , and the flow is always in the same direction as theprescribed shear stress throughout the rivulet i.e., the flowpattern is always of type I when 0 and always of type Vwhen 0.1J. Mikielewicz and J. R. Moszynski, “An improved analysis of breakdownof thin liquid films,” Arch. Mech. 30, 489 1978.2M. H. Eres, L. W. Schwartz, and R. V. Roy, “Fingering phenomena fordriven coating films,” Phys. Fluids 12, 1278 2000.3S. K. Wilson, B. R. Duffy, and R. Hunt, “A slender rivulet of a power-lawfluid driven by either gravity or a constant shear stress at the free surface,”Q. J. Mech. Appl. Math. 55, 385 2002.4T. G. Myers, H. X. Liang, and B. Wetton, “The stability and flow of arivulet driven by interfacial shear and gravity,” Int. J. Non-Linear Mech.39, 1239 2004.5H. H. Saber and M. S. El-Genk, “On the breakup of a thin liquid filmsubject to interfacial shear,” J. Fluid Mech. 500, 113 2004.6S. K. Wilson and B. R. Duffy, “On the gravity-driven draining of a rivuletof viscous fluid down a slowly varying substrate with variation transverseto the direction of flow,” Phys. Fluids 10, 13 1998.7P. Schmuki and M. Laso, “On the stability of rivulet flow,” J. Fluid Mech.215, 125 1990.8The expression for the kinetic energy in 7 is in agreement with theappropriate limit of the corresponding result given by Myers et al. seeRef. 4, Eq. 40 if the typographical error in their coefficient a2 is cor-rected specifically, their mL should be C. The expression for the surfaceenergy in 7 corrects the corresponding result given by Myers et al. seeRef. 4, Eq. 41, who mistakenly omitted a contribution due to the surfaceenergy of the free surface.FIG. 4. Plot of W1/5a as a function of W1/5 indicating when it is energeti-cally favorable denoted by “F” or unfavorable denoted by “UF” for arivulet to split. The curves a=amin=−21 /8 and a=a0=−7 /2 are indicatedwith dashed lines.FIG. 5. Plot of W4/5Q as a function of W1/5 indicating when it is energeti-cally favorable denoted by “F” or unfavorable denoted by “UF” for arivulet to split. The curve Q=Qmin=−30874 /5120 in 0 is indicated witha dashed line. Note that when 0 and QminQ0 the region in which itis unfavorable to split applies only to the narrower rivulet.108105-4 S. K. Wilson and B. R. Duffy Phys. Fluids 17, 108105 2005Downloaded 03 Nov 2006 to 130.159.248.222. Redistribution subject to AIP license or copyright, see http://pof.aip.org/pof/copyright.jsp

Unidirectional flow of a thin rivulet on a vertical substrate subject to a prescribed uniform shear stress at its free surface

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