Asymptotics of a Small Liqui Drop on a Cone an Plate Rheometer Vincent Cregan, Stephen B.G. O Brien, an Sean McKee Abstract A cone an a plate rheometer is a laboratory apparatus use to measure the viscosity an other relate parameters of a non-newtonian liqui subject to an applie force. A small rop, of orer millimetres, of the liqui is locate between the horizontal plate an the shallow cone of the rheometer. Rotation of the cone ensues, the liqui begins to flow an the plate starts to rotate. Liqui parameters are inferre base on the ifference in the applie rotational force an the resulting rotational force of the plate. To escribe the flow of the rop, the initial rop configuration, before rotation commences, must be etermine. The equilibrium rop profile is given by the solution to the well-known nonlinear Young Laplace equation. We formulate asymptotic solutions for the rop profile base on the small Bon number. The moelling of the rop exhibits a rich asymptotic structure consisting of five istinct scalings which are resolve via the metho matche asymptotics. 1 Introuction The stuy of surface tension an capillarity has long been an area of interest to both scientists an applie mathematicians. The importance of capillarity phenomena is highlighte by their abunance in both nature (self-cleaning behaviour of the lotus plant [1] an the water repellent properties of water striers [3]) an inustry (glass fabrication [4] an in the application of coatings to surfaces such as television screens [7]). V. Cregan ( ) S.B.G. O Brien MACSI, University of Limerick, Limerick, Irelan e-mail: vincent.cregan@ul.ie; stephen.obrien@ul.ie S. McKee University of Strathclye, Glasgow, Scotlan, UK e-mail: s.mckee@strath.ac.uk M. Günther et al. (es.), Progress in Inustrial Mathematics at ECMI 2010, Mathematics in Inustry 17, DOI 10.1007/978-3-642-25100-9 51, Springer-Verlag Berlin Heielberg 2012 449
450 V. Cregan et al. Early attempts at unerstaning surface tension inclue Leonaro a Vinci s a-hoc, intuitive explanation for capillary effects an Newton s experiments involving the rise of a liqui up a thin tube base on the attraction of the liqui to the tube [8]. In the early nineteenth century, the inepenent surface tension research of Young an Laplace resulte in the Young Laplace capillary equation p D 1 R 1 C 1 R 2 ; (1) which escribes the equilibrium profile of a static liqui gas interface. We observe that p is the pressure ifferenceacross the liqui interface, is the surface tension an R 1 an R 2 are the principal raii of curvature. A cone an plate rheometer is a laboratory evice use to stuy the way in which a non-newtonian flui flows an eforms subject to an applie force. A flui rop is place on the flat plate of the rheometer an the shallow cone is lowere towars an in to the rop (see Fig. 1a). Typically, the plate is rotate though in certain esigns the cone may rotate. The rotational force causes the flui to flow an thus, cone rotation ensues. On the basis of the ifference of the applie force an the resulting rotational force exerte on the cone, parameters such as flui viscosity can be establishe. To simulate the flui flow associate with the rop in contact with the cone an plate rheometer the initial, static rop profile must be etermine. The metho of matche asymptotics is use to erive expressions for the shape of the static rop profile. The perturbation approach is base on the small Bon number where surface tension ominates boy force terms an is similar to previous work on sessile rops an penant rops [5]. a b Upper Neck Cone r*, Upper Bounary O(ε 2 ) 0 φ Outer O(ε) O(ε) z*, Y Lower Bounary O(ε 2 ) Plate Lower Neck Fig. 1 (a) Schematic rop profile. (b) Drop asymptotic regions (neither rawn to scale.)
Asymptotics of a Small Liqui Drop on a Cone an Plate Rheometer 451 2 Mathematical Moel an Nonimensionalisation Assuming that the contact angles are constant, the resulting rop is axisymmetric with profile z D z.r / with respect to a polar coorinate system aligne such that z D 0 is locate at the thinnest part of the upper neck of the rop with the z -axis pointing ownwars in the irection of gravity (see Fig. 1a). The hyrostatic pressure in the rop is given by p C gz where is the flui ensity, g is gravity an p is the unknown pressure at z D 0 where the profile becomes vertical. Thus, at the liqui gas interface, the hyrostatic pressure in the rop is balance by the capillary forces an it follows from (1)that z 00.1 C z 0 2 / 3=2 C z 0 r.1 C z 0 2 / 1=2 D p C gz ; (2) where ifferential geometry has been use to formulate expressions for R 1 an R 2. We aopt the stanar nonimensionalisation approach [5] an we nonimensionalise (2) via the funamental imensionless variables z D ay ; r D a ; p D gap; (3) where a p =.g/ is the liqui capillary length, to obtain Y 00 0.1 C Y 0 2 / C Y 3=2.1C Y 0 2 D P C Y: (4) / 1=2 From a numerical perspective a more convenient parametric formulation of (4)is s D cos ; Y s D sin ; s C sin D P C Y; (5) where is the inclination (see Fig. 1a) an s is the arclength. Finally, elimination of the arclength from (5) yiels D cos P C Y sin ; Y D sin P C Y sin ; (6) which is the starting point for our asymptotic analysis. We enote L to be the maximum raius of the rop (or rop half-with) in the main boy of the rop where its profile becomes vertical [9]. From previous work [5], we assume that the with of the neck is O." 3 / where " L=a p L 2 g= is the imensionless half-with an may also viewe as a Bon number. We consier solutions for " 1 (or L a) which represents the ominance of surface tension over boy force effects in etermining the rop profile. We begin the solutions from the point of minimum with in the upper neck where D 0, Y D 0 an D =2 an the corresponing bounary conitions are
452 V. Cregan et al.. D =2/ D " 3 an Y. D =2/ D 0 where is an O.1/ parameter which is foun via the asymptotic analysis. The imensionless half-with conition. D =2/ D " fixes the pressure P. In relation to the contact angles we aopt the strategy of previous authors whereby the contact angles are use to etermine the points at which the rop is in contact with the cone an the plate [5]. For example if the lower contact angle is =2 we truncate the solutions at the point in the main boy of the rop where the profile becomes vertical. The rop asymptotic structure consists of an upper neck, an upper bounary layer, an outer region (main boy), a lower bounary layer an a lower neck (see Fig. 1b) an is base on previous work on sessile rops an penant rops [5, 6, 8]. 3 Results To reflect the balance between the surface tension curvature terms, which are opposite in sign, in the upper neck of the rop we efine the rescale neck variables D " 3 u ; Y D " 3 v; P D p=" ; D O.1/; (7) which upon substituting into (6) leas to the leaing orer equations u D u cot ; v D u; (8) with bounary conitions u. D =2/ D an v. D =2/ D 0. The corresponing solutions are u D csc ; v D ln j tan =2j; (9) where we note the existence of a singularity as! 0 which implies that the upper rescaling is not appropriate an an alternative set of scale variables must be efine. The upper bounary layer provies a transitional layer between the curvature ominate terms of the upper neck region an the three term balance in the main boy of the rop. Moreover, Fig. 1b illustrates a change in sign in the curvature in the upper bounary layer which suggests the presence of a point of inflection. Consequently, (6) is rescale via D " 2 ; Y D " 3 ; P D p=" ; D " ; (10) to obtain the leaing equations D p ; D p : (11)
Asymptotics of a Small Liqui Drop on a Cone an Plate Rheometer 453 We note the solution D 1 2 ( p 2 C 4C); (12) where the positive root applies before the point of inflection, locate at D " p 4C, an the negative root applies after inflection. The unknown integration constants are foun via asymptotic matching [2]. In the main boy of the rop, the basic shape is nearly spherical an the funamental balance is between the curvature terms an the pressure term P. To highlight this balance we rescale (6) by the outer variables to attain the outer equations x D z D Lx ; r D Ly; (13) x cos " 2 xy C xp sin ; y D x sin " 2 xy C xp sin ; (14) with the conition x. D =2/ D 1. The presence of the " 2 terms in (14) suggest O." 2 / asymptotic expansions in x, y an p. The leaing orer solutions are x 0 D sin ; y 0 D 1 cos ; p 0 D 2; (15) which represents a circular rop profile. Proceeing to O." 2 / we have x 1 D 1 6.1 3 / cos 2 2 cos 3 C 1 3 ; (16) sin which upon inspection reveals a singularity as! an thus an alternative rescaling for near D is require. At the base of the rop we encounter a lower bounary layer analogous to the upper bounary layer an we rescale via D " 2 ; Y D 2" C " 3 ; P D p=" ; D " ; (17) which upon introuction into (6) leas to a system of equations ientical to (11). Noteworthy is the solution D D ; (18) where via asymptotic matching we fin D D 2=3. From(18) it is evient that if D<0(an thus >2=3)then >0an it follows that <.Thisleasto another point of inflection in the lower bounary layer an the beginning of a new rop. Hence, the magnitue of an thus the sign of D has a profoun effect on the structure of the rop profile. Accoringly, we rescale (6) via a set of lower bounary layer an lower neck variables (analogous to (7) an(10), respectively) procee to
454 V. Cregan et al. leaing orer an obtain the relevant solutions. The theoretical results pertaining to the lower bounary layer ( >2=3) an the presence of another rop structure nee to be valiate experimentally. In reality, the static solutions outline here may be quite ifficult to achieve if the appropriate experimental configuration is not calibrate correctly. Other authors have reporte on systems which exhibit a similar type of multiple rop structure as outline here [8]. a x 10 3 0 1 Upper Composite Numerical b 0.05 Y 2 Y 0.1 Outer Numerical 0.15 3 0 0.002 0.004 0.006 0.008 0.01 0.2 0 0.02 0.04 0.06 0.08 0.1 Fig. 2 Comparison of numerical solution (arclength formulation (5)) with associate asymptotic solutions in (a) upper neck an (b) outer region. " D 0:1, D 1 0.19 0.195 Numerical Outer Neck Lower Bounary Layer 0.2 0.205 0.21 0 0.01 0.02 0.03 0.04 0.05 Fig. 3 Comparison of numerical solution (arclength formulation 5)) with lower asymptotic solutions. " D 0:1, D 1
Asymptotics of a Small Liqui Drop on a Cone an Plate Rheometer 455 4 Conclusion The metho of matche asymptotic expansions has been use to erive asymptotic solutions for the profile of a liqui rop in contact with a cone an plate rheometer. A number of rescalings an bounary layers were require to fully escribe the rop profile. As inicate by Fig. 2 an Fig. 3 the asymptotic solutions isplay excellent agreement with the corresponing numerical solutions. Acknowlegements We gratefully acknowlege the financial support of the Mathematics Applications Consortium for Science an Inustry (MACSI) supporte by a Science Founation Irelan mathematics grant 06/MI/005 an an Embark Initiative postgrauate awar RS/2006/41. References 1. Dupuis, A., Yeomans, Y.: Moeling roplets on superhyrophobic surfaces: equilibrium states an transitions. Langmuir 21(6), 2264 2629 (2005) 2. Dyke, M.V.: Perturbation Methos in Flui Mechanics. Acaemic Press, New York (Annotate eition from Parabolic Press, Stanfor) (1975) 3. Hu, D., Bush, J.: The hyroynamics of water-walking arthropos. J. Flui Mech. 644, 5 33 (2010) 4. Nemchimsky, V.: Size an shape of the liqui roplet at the molten tip of an arc electroe. J. Appl. Phys. 27(7), 1433 1442 (1994) 5. O Brien, S.: On the shape of small sessile an penant rops by singular perturbation techniques. J. Flui Mech. 233, 519 537 (1991) 6. O Brien, S.: Asymptotic solutions for ouble penant an extene sessile rops. Q. Appl. Math. 52(1), 43 48 (1994) 7. O Brien, S.: The meniscus near a small sphere an its relationship to line pinning of contact lines. J. Colloi Interface Sci. 183(1), 51 56 (1996) 8. O Brien, S.: Asymptotics of a sequence of penant rops. SIAM J. Appl. Math. 62(5), 1569 1580 (2002) 9. Paay, J.: The profiles of axially symmetric menisci. Phil. Trans. Roy. Soc. Lon. A Math. Phys. Sci. 269(1197), 265 293 (1971)