Phys_Fluids_1_2009

Analytical solution for Stokes flow inside an evaporating sessile drop: Spherical and cylindrical cap shapes Hassan Masouda/H20850and James D. Felske Department of Mechanical and Aerospace Engineering, State University of New York at Buffalo, Buffalo, New York 14260, USA /H20849Received 18 December 2008; accepted 10 March 2009; published online 10 April 2009 /H20850 Exact analytical solutions are derived for the Stokes flows within evaporating sessile drops of spherical and cylindrical cap shapes. The results are valid for all contact angles. Solutions areobtained for arbitrary evaporative flux distributions along the free surface as long as the flux isbounded at the contact line. Specific results and computations are presented for evaporationcorresponding to uniform flux and to purely diffusive gas phase transport into an infinite ambient.Wetting and nonwetting contact angles are considered with the flow patterns in each case beingillustrated. For the spherical cap with evaporation controlled by vapor phase diffusion, when thecontact angle lies in the range 0 /H11349 /H9258c/H11021/H9266/2, the mass flux of vapor becomes singular at the contact line. This condition requires modification when solving for the liquid-phase transport. Droplets in allof the above categories are considered for the following two cases: the contact lines are eitherpinned or free to move during evaporation. The present viscous flow behavior is compared to theinviscid flow behavior previously reported. It is seen that the streamlines for viscous flow lie fartherfrom the substrate than the corresponding inviscid ones. © 2009 American Institute of Physics . /H20851DOI: 10.1063/1.3112002 /H20852 I. INTRODUCTION When a sessile droplet containing solid constituents dries on a substrate, the solute particles deposit on the sur-face in various patterns. The final deposition pattern and theability to control it are of high interest for many industrialand scientific processes. Examples are the creation of an or-dered array of semiconductor nanoparticles for electronics 1 or DNA molecules for gene expression analysis.2Also, evaporation-driven flow is used to assemble two-dimensional/H208492D /H20850crystals in protein crystallography. 3–5Moreover, a par- ticular deposition pattern—the ring shape—has importantimplications for DNA stretching, 6ink-jet printing,7–9and coatings. The deposition pattern produced depends on the flow within the drop. As shown previously,10the flow pattern strongly depends on the combination of evaporative flux,shape of the free surface, and behavior of the contact line.For instance, flow can be toward, away, or both toward andaway from the contact line under different combinations ofthese factors. Flow inside an evaporating sessile droplet has been stud- ied analytically, semianalytically, and numerically. Deeganet al. 11–13and Popov14utilized the vertically averaged veloc- ity in considering the behavior of velocities at small contactangles. Solutions in the limit of lubrication theory were ob-tained numerically by Fischer 15and semianalytically by Hu and Larson.16Stokes flow for contact angles ranging from 0° to 90° was evaluated numerically by Hu and Larson16and Widjaja et al.17The exact analytical solution for irrotationalflow was derived for hemispheres by Tarasevich,18for spherical caps by Masoud and Felske,10and for hemicylin- ders and cylindrical caps by Petsi and Burganos.19,20 The present study considers the exact analytical solu- tions for Stokes flows within evaporating, sessile, sphericaland cylindrical cap drops. The full range of contact angles/H208490/H11021 /H9258c/H11021/H9266/H20850are considered as well as arbitrary distributions of evaporative flux along the free surface. The contact line of the droplet is allowed to be either pinned or free to moveduring evaporation. Analytical expressions for the expansioncoefficients are given for the case of uniform flux. To thebest of our knowledge, the Stokes flow within an evaporat-ing, sessile, nonwetting drop has not been reported by anyapproach, analytical or numerical. II. MODEL DEVELOPMENT AND SOLUTIONS The droplets considered in the present study are /H110111m m from the axis of symmetry to the contact line /H20849R/H20850. For water drops of this size evaporating under room conditions, the characteristic velocity in the drop is /H110111/H9262m/s with a corre- sponding Reynolds number of /H1101110−3.16For such flows, in- ertial forces are negligible /H20849Stokes flow /H20850and the continuity, momentum, and vorticity equations become /H11612.V=0 , /H208491/H20850 /H9262/H116122V=/H11612p, /H208492/H20850 and /H116122/H9275=0 , /H208493/H20850 where Vis the fluid velocity, pis the difference between the thermodynamic /H20849P/H20850and hydrostatic /H20849ph/H20850pressures /H20849p=Pa/H20850Author to whom correspondence should be addressed. Electronic mail: [email protected] OF FLUIDS 21, 042102 /H208492009 /H20850 1070-6631/2009/21 /H208494/H20850/042102/11/$25.00 © 2009 American Institute of Physics 21, 042102-1 Author complimentary copy. Redistribution subject to AIP license or copyright, see http://phf.aip.org/phf/copyright.jsp −ph/H20850, and/H9275=/H11612/H11003Vis the vorticity. The application of these equations to the two geometries of interest will now be considered. A. Spherical cap shape 1. Geometry The shape acquired by a drop on a substrate is deter- mined by surface tension, gravity, and the fluid flows withinand over the surface of the droplet. For a stagnant externalfluid, the relative importance of the remaining quantities maybe assessed from the Bond and capillary numbers. The Bondnumber is the ratio of gravitational to surface tension forces:Bo= /H9267gRchmax //H9268, where Rc,hmax, and/H9268are, respectively, the droplet’s mean radius of curvature, the droplet’s maximumheight, and the surface tension. The capillary number is theratio of viscous to surface tension forces; Ca= /H9262V0//H9268, where /H9262is the viscosity and V0is a characteristic velocity. For millimeter size water drops, Bo= /H110110.04 and Ca= /H1101110−8. Hence, surface tension is the dominant influence on the drop-let’s shape, and the droplet becomes a spherical cap. It is alsonoted from Eq. /H208492/H20850that the pressure gradient /H20849/H11612p/H20850and the viscous forces /H20849 /H9262/H116122V/H20850are equal. Then, since Ca /Bo /H1101110−7 /H11011/H20849viscous /gravitational /H20850/H11011/H20849 /H11612p/gravitational /H20850, the thermo- dynamic pressure /H20849P/H20850variation throughout the drop is essen- tially equal to the hydrostatic pressure variation /H20849ph/H20850. The hemispherical cap geometry is a sphere cut by a plane. Since this shape and the boundary conditions beingconsidered are azimuthally independent, so is the flow withinthe drop. The natural coordinates are therefore toroidal. Fig-ure1shows a cross section of this geometry at a given azi- muthal angle /H9272. The metric coefficients for the toroidal co- ordinates are h/H9251=h/H9258=h/H9278/sinh/H9251=R/H20849cosh/H9251+ cos/H9258/H20850−1, /H208494/H20850 where, 0 /H11349/H9251/H11349/H11009,−/H9266/H11349/H9258/H11349/H9266, and 0 /H11349/H9272/H113492/H9266. In presenting results it is often clearer to use cylindrical coordinates /H20849r,/H9258/H20850. The relationships between the two systems are r/sinh/H9251=z/sin/H9258=R/H20849cosh/H9251+ cos/H9258/H20850−1. /H208495/H208502. Field equation For 2D and three-dimensional /H208493D /H20850axisymmetric flows, velocity components may be expressed in terms of a streamfunction. The stream function /H9274is defined to satisfy the con- tinuity equation, V/H9251=1 h/H9258h/H9272/H11509/H9274 /H11509/H9258=/H20849cosh/H9251+ cos/H9258/H208502 R2sinh/H9251/H11509/H9274 /H11509/H9258, /H208496/H20850 V/H9258=−1 h/H9272h/H9251/H11509/H9274 /H11509/H9251=−/H20849cosh/H9251+ cos/H9258/H208502 R2sinh/H9251/H11509/H9274 /H11509/H9251. /H208497/H20850 The field equation for /H9274follows from writing Eq. /H208493/H20850for axially symmetric flow. Substituting from above for V/H9251and V/H9258in terms of /H9274shows that the vorticity is related to the stream function by /H9275=/H20849eˆ/H9272/h/H9272/H20850E2/H9274, /H208498/H20850 where E2/H9274=sinh/H9251/H20849cosh/H9251+ cos/H9258/H20850 R2 /H20875/H11509 /H11509/H9251/H20873cosh/H9251+ cos/H9258 sinh/H9251/H11509/H9274 /H11509/H9251/H20874 +/H11509 /H11509/H9258/H20873cosh/H9251+ cos/H9258 sinh/H9251/H11509/H9274 /H11509/H9258/H20874/H20876. /H208499/H20850 Repeated application of the curl operator to Eq. /H208498/H20850re- sults in /H116122/H9275=/H11612/H11003/H20849/H11612/H11003/H9275/H20850=− /H20849eˆ/H9272/h/H9272/H20850E2/H20849E2/H9274/H20850. /H2084910/H20850 Consequently, the field equation for /H9274, which corresponds to /H116122/H9275=0, is E4/H9274=0 . /H2084911/H20850 3. Integration of E4/H9274=0 The analytical solution to E4/H9274=0 subject to appropriate boundary conditions is considered in this section. In toroidalcoordinates the integration of E 4/H9274=0 has been accomplished using different analytical techniques.21–23Stimson and Jefferey21first integrated E2/H20849/H9274/H20850and then transformed this result into cylindrical coordinates. They completed the inte- gration of E4/H20849/H9274/H20850=0 by exploiting the /H20849coordinate /H20850linearity of the cylindrical operator in the axial direction. Khuri and Wazwaz23employed the method of separation of variables and achieved a solution in terms of eigenfunctions in the /H9258-coordinate. The present study requires eigenfunctions in the/H9251-coordinate in order to represent arbitrary variations in the evaporative flux distribution. In Appendix A, using sepa-ration of variables, it is shown that the required eigenfunc-tions are Gegenbauer functions and that the general form ofthe solution is /H9274/H20849/H9251,/H9258/H20850=/H20849cosh/H9251+ cos/H9258/H20850−3 /2/H20885 0/H11009 /H20851A/H20849/H9270/H20850C1/2+i/H9270−1 /2/H20849cosh/H9251/H20850 +B/H20849/H9270/H20850C1/2+i/H9270/H11569−1 /2/H20849cosh/H9251/H20850/H20852 /H11003/H20853sinh /H20849/H9270/H9258/H20850/H20851C/H20849/H9270/H20850sin/H9258+D/H20849/H9270/H20850cos/H9258/H20852 + cosh /H20849/H9270/H9258/H20850/H20851E/H20849/H9270/H20850sin/H9258+F/H20849/H9270/H20850cos/H9258/H20852/H20854d/H9270 /H2084912/H20850 FIG. 1. Toroidal coordinates: lines of constant /H9251and/H9258; positive velocity components.042102-2 H. Masoud and J. D. Felske Phys. Fluids 21, 042102 /H208492009 /H20850 Author complimentary copy. Redistribution subject to AIP license or copyright, see http://phf.aip.org/phf/copyright.jsp where C1/2+i/H9270−1 /2and C1/2+i/H9270/H11569−1 /2are Gegenbauer functions of the first and second kinds, of order of /H110021/2. /H20849Note that Payne and Pell22used a different approach for obtaining the general solution. /H20850 To establish boundary conditions on the stream function /H9274, it is first noted that the velocity components normal to the axis of symmetry and normal to the solid surface vanish:V /H9251/H208490,/H9258/H20850=0 and V/H9258/H20849/H9251,0/H20850=0. Hence, the stream function is constant along the axis of symmetry and along the solid sur- face. Since these surfaces intersect, the constant must be thesame for both surfaces. It is arbitrarily set to zero since itsvalue does not affect the predicted velocities. The no slipcondition requires that the velocity component tangent to thesolid surface vanishes at the surface, V /H9251/H20849/H9251,0/H20850=0. Along the axis of symmetry, derivatives normal to the axis are zero; in particular, /H11509V/H9258//H11509/H9251/H20841/H9251=0=0. The corresponding boundary con- ditions on /H9274are /H9274/H208490,/H9258/H20850=0 , /H2084913/H20850 /H9274/H20849/H9251,0/H20850=0 , /H2084914/H20850 /H20879/H11509/H9274 /H11509/H9258/H20879 /H9258=0=0 , /H2084915/H20850 and /H11509 /H11509/H9251/H20875/H20849cosh/H9251+ cos/H9258/H208502 sinh/H9251/H11509/H9274 /H11509/H9251/H20876 /H9251=0=0 . /H2084916/H20850 Applying these conditions to Eq. /H2084912/H20850, it follows from Eq. /H2084913/H20850thatB/H20849/H9270/H20850=0;10from Eq. /H2084914/H20850, F/H20849/H9270/H20850=0 , /H2084917/H20850 and, from Eq. /H2084915/H20850, E/H20849/H9270/H20850+/H9270D/H20849/H9270/H20850=0 . /H2084918/H20850 Incorporating these into Eq. /H2084912/H20850, the stream function becomes /H9274/H20849/H9251,/H9258/H20850=/H20849cosh/H9251+ cos/H9258/H20850−3 /2 /H11003/H20885 0/H11009 K/H20849/H9258,/H9270/H20850C1/2+i/H9270−1 /2/H20849cosh/H9251/H20850d/H9270, /H2084919/H20850 where K/H20849/H9258,/H9270/H20850=k1/H20849/H9270/H20850sin/H9258sinh /H20849/H9270/H9258/H20850+k2/H20849/H9270/H20850/H20851cos/H9258sinh /H20849/H9270/H9258/H20850 −/H9270sin/H9258cosh /H20849/H9270/H9258/H20850/H20852. /H2084920/H20850 In the third and fourth boundary conditions in the /H9251-direction, in order for the integrals to exist the velocity components must be nonsingular /H20849finite /H20850at the contact line lim /H9251→/H11009/H20875/H20849cosh/H9251+ cos/H9258/H208502 sinh/H9251/H11509/H9274 /H11509/H9258/H20876= finite /H2084921/H20850 andlim /H9251→/H11009/H20875/H20849cosh/H9251+ cos/H9258/H208502 sinh/H9251/H11509/H9274 /H11509/H9251/H20876= finite. /H2084922/H20850 These two conditions and Eq. /H2084916/H20850are satisfied for physically meaningful distributions of evaporative flux at the freesurface. The unknown coefficients, k 1/H20849/H9270/H20850andk2/H20849/H9270/H20850, can be ob- tained from the distribution of the stream function at the free surface in conjunction with the zero shear stress boundarycondition, /H9270/H9251/H9258/H20849/H9251,/H9258c/H20850=0.K/H20849/H9258c,/H9270/H20850can be written in terms of the stream function at the free surface using the integral transform presented in Ref. 10, K/H20849/H9258c,/H9270/H20850=/H9270/H20849/H92702+1 /4/H20850tanh /H20849/H9266/H9270/H20850 /H11003/H20885 0/H11009/H9274/H20849/H9251,/H9258c/H20850/H20849cosh/H9251+ cos/H9258c/H208503/2 sinh/H9251 /H11003C1/2+i/H9270−1 /2/H20849cosh/H9251/H20850d/H9251. /H2084923/H20850 Using Eqs. /H208494–14.1 /H20850,/H208494–14.2 /H20850, and /H20849A-7.7 /H20850of Ref. 24, van- ishing of the shear stress at the free surface gives /H9270/H9251/H9258/H20841/H9258=/H9258c=−/H9262/H20877/H11509 /H11509/H9251/H20851/H20849cosh/H9251+ cos/H9258/H20850/H20849V/H9258/R/H20850/H20852 +/H11509 /H11509/H9258/H20851/H20849cosh/H9251+ cos/H9258/H20850/H20849V/H9251/R/H20850/H20852/H20878 /H9258=/H9258c=0 . /H2084924/H20850 Therefore, K˜/H20849/H9258c,/H9270/H20850=/H9270/H20849/H92702+1 /4/H20850tanh /H20849/H9266/H9270/H20850 /H11003/H20885 0/H11009/H9274˜/H20849/H9251,/H9258c/H20850 sinh/H9251C1/2+i/H9270−1 /2/H20849cosh/H9251/H20850d/H9251, /H2084925/H20850 in which /H9274˜/H20849/H9251,/H9258c/H20850=− /H20849cosh/H9251+ cos/H9258c/H20850−1 /2 /H11003/H20877R2sinh/H9251 cosh/H9251+ cos/H9258c/H11509 /H11509/H9251 /H11003/H20851/H20849cosh/H9251+ cos/H9258c/H20850V/H9258/H20849/H9251,/H9258c/H20850/H20852+3/H9274/H20849/H9251,/H9258c/H20850 2 /H11003/H20851cos/H9258c/H20849cosh/H9251+ cos/H9258c/H20850− sin2/H9258c/2/H20852/H20878 /H2084926/H20850 and K˜/H20849/H9258c,/H9270/H20850=/H115092 /H115092/H9258K/H20849/H9258c,/H9270/H20850 =k1/H20849/H9270/H20850/H20851/H20849/H92702−1 /H20850sin/H9258csinh /H20849/H9270/H9258c/H20850 +2/H9270cos/H9258ccosh /H20849/H9270/H9258c/H20850/H20852−k2/H20849/H9270/H20850/H20849/H92702+1 /H20850 /H11003/H20851cos/H9258csinh /H20849/H9270/H9258c/H20850+/H9270sin/H9258ccosh /H20849/H9270/H9258c/H20850/H20852./H2084927/H20850 Using Eqs. /H2084923/H20850and /H2084925/H20850, the coefficients k1/H20849/H9270/H20850andk1/H20849/H9270/H20850are found to be042102-3 Analytical solution for Stokes flow Phys. Fluids 21, 042102 /H208492009 /H20850 Author complimentary copy. Redistribution subject to AIP license or copyright, see http://phf.aip.org/phf/copyright.jsp k1/H20849/H9270/H20850=N2/H20849/H9270,/H9258c/H20850K/H20849/H9270,/H9258c/H20850+N1/H20849/H9270;/H9258c/H20850K˜/H20849/H9270,/H9258c/H20850 N2/H20849/H9270,/H9258c/H20850M1/H20849/H9270,/H9258c/H20850+N1/H20849/H9270,/H9258c/H20850M2/H20849/H9270,/H9258c/H20850/H2084928/H20850 and k2/H20849/H9270/H20850=M2/H20849/H9270,/H9258c/H20850K/H20849/H9270,/H9258c/H20850−M1/H20849/H9270,/H9258c/H20850K˜/H20849/H9270,/H9258c/H20850 N2/H20849/H9270,/H9258c/H20850M1/H20849/H9270,/H9258c/H20850+N1/H20849/H9270,/H9258c/H20850M2/H20849/H9270,/H9258c/H20850, /H2084929/H20850 where M1/H20849/H9270,/H9258c/H20850= sin/H9258csinh /H20849/H9270/H9258c/H20850, /H2084930/H20850 M2/H20849/H9270,/H9258c/H20850=/H20849/H92702−1 /H20850sin/H9258csinh /H20849/H9270/H9258c/H20850 +2/H9270cos/H9258ccosh /H20849/H9270/H9258c/H20850, /H2084931/H20850 N1/H20849/H9270,/H9258c/H20850= cos/H9258sinh /H20849/H9270/H9258/H20850−/H9270sin/H9258cosh /H20849/H9270/H9258/H20850, /H2084932/H20850 and N2/H20849/H9270,/H9258c/H20850=/H20849/H92702+1 /H20850/H20851cos/H9258csinh /H20849/H9270/H9258c/H20850 +/H9270sin/H9258ccosh /H20849/H9270/H9258c/H20850/H20852. /H2084933/H20850 The final boundary condition is the known distribution of evaporative flux along the free surface. This distribution isdetermined by the gas phase mass transport and is indepen-dent of the flow within the drop. The velocity and streamfunction at the free surface follow directly from this flux.The velocity is obtained from mass conservation, wherebythe mass flux in the liquid equals the mass flux in the gas. Interms of the liquid, the evaporation rate is J/H20849 /H9251/H20850=/H9267/H20851V/H9258/H20849/H9251,/H9258c/H20850−V/H9258,B/H20849/H9251/H20850/H20852, /H2084934/H20850 where J/H20849/H9251/H20850is the evaporative flux at the free surface, /H9267is the liquid density, and V/H9258,Bis the speed at which the boundary is moving in the direction normal to itself. From Eq. /H208497/H20850, the boundary condition on /H9274may then be written in terms of V/H9258/H20849/H9251,/H9258c/H20850, /H9274/H20849/H9251,/H9258c/H20850=−/H20885 0/H9251R2sinh/H9251/H11032 /H20849cosh/H9251/H11032+ cos/H9258c/H208502V/H9258/H20849/H9251/H11032,/H9258c/H20850d/H9251/H11032 =−/H20885 0/H9251R2sinh/H9251/H11032 /H20849cosh/H9251/H11032+ cos/H9258c/H208502/H20851J/H20849/H9251/H11032/H20850//H9267+V/H9258,B/H20849/H9251/H11032/H20850/H20852d/H9251/H11032. /H2084935/H20850 Finally, from /H9274/H20849/H9251,/H9258/H20850, the velocity distribution may be calculated. The velocity components in toroidal coordinates follow from Eqs. /H208496/H20850and /H208497/H20850, V/H9251/H20849/H9251,/H9258/H20850=/H20881cosh/H9251+ cos/H9258 R2sinh/H9251 /H11003/H20875/H208493 sin/H9258/2/H20850/H20881cosh/H9251+ cos/H9258/H9274/H20849/H9251,/H9258/H20850 +/H20885 0/H11009/H11509K/H20849/H9258,/H9270/H20850 /H11509/H9258C1/2+i/H9270−1 /2/H20849cosh/H9251/H20850d/H9270/H20876, /H2084936/H20850V/H9258/H20849/H9251,/H9258/H20850=/H20881cosh/H9251+ cos/H9258 R2/H208773/H20881cosh/H9251+ cos/H9258/H9274/H20849/H9251,/H9258/H20850 2 +/H20885 0/H11009 K/H20849/H9258,/H9270/H20850P−1 /2+i/H9270/H20849cosh/H9251/H20850d/H9270/H20878, /H2084937/H20850 where P−1 /2+i/H9270/H20849x/H20850is the conical function of the first kind. For visualizing and interpreting the flow field, the velocity com- ponents in cylindrical coordinates are useful. The radial andaxial components of the velocity are given by V r/H20849/H9251,/H9258/H20850=r−1/H20849/H11509/H9274//H11509z/H20850 =/H20849cosh/H9251+ cos/H9258/H20850−1/H20851V/H9251/H208491 + cosh /H9251cos/H9258/H20850 +V/H9258sinh/H9251sin/H9258/H20852, /H2084938/H20850 Vz/H20849/H9251,/H9258/H20850=−r−1/H20849/H11509/H9274//H11509x/H20850 =− /H20849cosh/H9251+ cos/H9258/H20850−1/H20851V/H9251sinh/H9251sin/H9258−V/H9258 /H11003/H208491 + cosh /H9251cos/H9258/H20850/H20852. /H2084939/H20850 B. Cylindrical cap shape 1. Geometry Similar to spherical drops, the sizes of the 2D liquid lines being considered are small enough that surface tensionis the dominant force defining their cross-sectional shape.Consequently, they have the cylindrical cap geometry whosefree surface is exactly mapped in bipolar coordinates. Bipo-lar coordinates /H20849 /H9251,/H9258/H20850are shown in Fig. 2along with Carte- sian coordinates /H20849x,y/H20850./H20849Note that the cross section of a cy- lindrical cap in bipolar coordinates is identical to the cross section of a spherical cap in toroidal coordinates, Fig. 1./H20850The metric coefficients for bipolar geometry are h/H9251=h/H9258=R/H20849cosh/H9251+ cos/H9258/H20850−1, /H2084940/H20850 where − /H11009/H11349/H9251/H11349/H11009and −/H9266/H11349/H9258/H11349/H9266. Bipolar and Cartesian co- ordinates are related by x/sinh/H9251=y/sin/H9258=R/H20849cosh/H9251+ cos/H9258/H20850−1. /H2084941/H20850 FIG. 2. Bipolar coordinates: lines of constant /H9251and/H9258; positive velocity components.042102-4 H. Masoud and J. D. Felske Phys. Fluids 21, 042102 /H208492009 /H20850 Author complimentary copy. Redistribution subject to AIP license or copyright, see http://phf.aip.org/phf/copyright.jsp 2. Field equation Based on the definition of the stream function /H9274, the velocity components are given by V/H9251=1 h/H9258/H11509/H9274 /H11509/H9258=cosh/H9251+ cos/H9258 R/H11509/H9274 /H11509/H9258, /H2084942/H20850 V/H9258=−1 h/H9251/H11509/H9274 /H11509/H9251=−cosh/H9251+ cos/H9258 R/H11509/H9274 /H11509/H9251. /H2084943/H20850 The field equation for /H9274follows from Eq. /H208492/H20850by substituting the above for V/H9251andV/H9258. For 2D flows, the vorticity is related to the stream function by /H9275=eˆz/H116122/H9274, /H2084944/H20850 where /H116122/H9274=/H20849cosh/H9251+ cos/H9258/H208502 R2 /H20873/H115092/H9274 /H11509/H92512+/H115092/H9274 /H11509/H92582/H20874. /H2084945/H20850 Hence, the field equation for /H9274, corresponding to /H116122/H9275=0, is /H116124/H9274=0 . /H2084946/H20850 3. Integration of /H116334/H9274=0 Jeffery25integrated of /H116124/H9274=0 in bipolar coordinates ob- taining a solution in terms of eigenfunctions in the /H9258-coordinate. In the present study, eigenfunctions are re- quired in the /H9251-coordinate. In Appendix B, using separation of variables, it is shown that the appropriate solution is /H9274/H20849/H9251,/H9258/H20850=/H20849cosh/H9251+ cos/H9258/H20850−1/H20885 0/H11009 /H20851A/H20849/H9270/H20850sin /H20849/H9270/H9251/H20850+B/H20849/H9270/H20850cos /H20849/H9270/H9251/H20850/H20852 /H11003/H20853sinh /H20849/H9270/H9258/H20850/H20851C/H20849/H9270/H20850sin/H9258+D/H20849/H9270/H20850cos/H9258/H20852 + cosh /H20849/H9270/H9258/H20850/H20851E/H20849/H9270/H20850sin/H9258+F/H20849/H9270/H20850cos/H9258/H20852/H20854d/H9270. /H2084947/H20850 The forms of the solutions for the 2D and 3D axisym- metric cases are similar and the boundary conditions are thesame. Consequently, the forms of the results for sphericalcaps /H20851Eqs. /H2084923/H20850–/H2084933/H20850/H20852apply to cylindrical caps as well. Using Eqs. /H2084923/H20850–/H2084933/H20850, the bipolar stream function, Eq. /H2084947/H20850, becomes /H9274/H20849/H9251,/H9258/H20850=/H20849cosh/H9251+ cos/H9258/H20850−1/H20885 0/H11009 K/H20849/H9258,/H9270/H20850sin /H20849/H9270/H9251/H20850d/H9270. /H2084948/H20850 Taking into account that sin /H20849/H9270/H9251/H20850is the eigenfunction of Eq. /H2084948/H20850, K/H20849/H9270,/H9258c/H20850=2 /H9266/H20885 0/H11009 /H9274/H20849/H9251,/H9258c/H20850/H20849cosh/H9251+ cos/H9258c/H20850sin /H20849/H9270/H9251/H20850d/H9251. /H2084949/H20850 Equation /H2084924/H20850is also valid in bipolar coordinates but the definitions of V/H9251andV/H9258in terms of the stream function are different. Consequently, the zero shear stress boundary con-dition yieldsK˜/H20849/H9270,/H9258c/H20850=2 /H9266/H20885 0/H11009 /H9274˜˜/H20849/H9251,/H9258c/H20850sin /H20849/H9270/H9251/H20850d/H9251, /H2084950/H20850 in which /H9274˜˜/H20849/H9251,/H9258c/H20850=−R cosh/H9251+ cos/H9258c/H11509 /H11509/H9251/H20851/H20849cosh/H9251+ cos/H9258c/H20850V/H9258/H20849/H9251,/H9258c/H20850/H20852 + cos/H9258c/H9274/H20849/H9251,/H9258c/H20850. /H2084951/H20850 Knowing K/H20849/H9270,/H9258c/H20850and K˜/H20849/H9270,/H9258c/H20850, the unknown coefficients, k1/H20849/H9270/H20850andk2/H20849/H9270/H20850, are then obtained from Eqs. /H2084928/H20850and /H2084929/H20850. Similar to the spherical cap case, the evaporation rate is related to the variation along the interface of the velocitycomponent normal to the interface, Eq. /H2084934/H20850. Using Eqs. /H2084934/H20850 and /H2084943/H20850, the distribution of the stream function along the surface of the droplet may be written in terms of V /H9258/H20849/H9251,/H9258c/H20850, /H9274/H20849/H9251,/H9258c/H20850=−/H20885 0/H9251Rsinh/H9251/H11032 cosh/H9251/H11032+ cos/H9258cV/H9258/H20849/H9251/H11032,/H9258c/H20850d/H9251/H11032 =−/H20885 0/H9251Rsinh/H9251/H11032 cosh/H9251/H11032+ cos/H9258c/H20851J/H20849/H9251/H11032/H20850//H9267+V/H9258,B/H20849/H9251/H11032/H20850/H20852d/H9251/H11032. /H2084952/H20850 In bipolar coordinates, the components of the velocity follow from Eqs. /H2084942/H20850and /H2084943/H20850, RV/H9251/H20849/H9251,/H9258/H20850= sin/H9258/H9274/H20849/H9251,/H9258/H20850+/H20885 0/H11009/H11509K/H20849/H9258,/H9270/H20850 /H11509/H9258sin /H20849/H9270/H9251/H20850d/H9270, /H2084953/H20850 RV/H9258/H20849/H9251,/H9258/H20850= sinh /H9251/H9274 /H20849/H9251,/H9258/H20850+/H20885 0/H11009 K/H20849/H9258,/H9270/H20850cos /H20849/H9270/H9251/H20850/H9270d/H9270./H2084954/H20850 In Cartesian coordinates the horizontal and vertical compo- nents of the velocity are Vx/H20849/H9251,/H9258/H20850=/H11509/H9274//H11509y =/H20849cosh/H9251+ cos/H9258/H20850−1/H20851V/H9251/H208491 + cosh /H9251cos/H9258/H20850 +V/H9258sinh/H9251sin/H9258/H20852, /H2084955/H20850 Vy/H20849/H9251,/H9258/H20850=−/H11509/H9274//H11509x =− /H20849cosh/H9251+ cos/H9258/H20850−1/H20851V/H9251sinh/H9251sin/H9258−V/H9258 /H11003/H208491 + cosh /H9251cos/H9258/H20850/H20852. /H2084956/H20850 III. PINNED CONTACT LINE When the droplet is “pinned” at the contact line, Ris constant and /H9258c=/H9258c/H20849t/H20850. This condition will be considered for spherical caps first followed by cylindrical caps. For each of these geometries, two different evaporative fluxes will beconsidered: uniform and gas-diffusion controlled.042102-5 Analytical solution for Stokes flow Phys. Fluids 21, 042102 /H208492009 /H20850 Author complimentary copy. Redistribution subject to AIP license or copyright, see http://phf.aip.org/phf/copyright.jsp A. Spherical cap shape As reported previously,10 d/H9258c dt=− /H208492/R/H20850/H208491 + cos /H9258c/H208502 /H11003/H20885 0/H11009sinh/H9251 /H20849cosh/H9251+ cos/H9258c/H208502J/H20849/H9251/H20850 /H9267d/H9251, /H2084957/H20850 V/H9258/H20849/H9251,/H9258c/H20850=R/H20849cosh/H9251+ cos/H9258c/H20850−1/H20849d/H9258c/dt/H20850+J/H20849/H9251/H20850//H9267, /H2084958/H20850 and /H9274/H20849/H9251,/H9258c/H20850=−d/H9258c dtR3 2/H208751 /H208491 + cos /H9258c/H208502−1 /H20849cosh/H9251+ cos/H9258c/H208502/H20876 −/H20885 0/H9251R2sinh/H9251/H11032 /H20849cosh/H9251/H11032+ cos/H9258c/H208502J/H20849/H9251/H11032/H20850 /H9267d/H9251/H11032. /H2084959/H20850 Any physically obtainable distribution of evaporative flux may be considered. Many are possible since differentdistributions result from different combinations of gas phasevelocities and degrees of vacuum into which the evaporationoccurs. Two specific cases are considered below. 1. Uniform evaporative flux In this case, the evaporative flux is uniform across the surface of the drop, J/H20849/H9251,/H9258c/H20850= const = J0. /H2084960/H20850 Therefore, d/H9258c dt=−2J0 R/H9267/H208491 + cos /H9258c/H20850, /H2084961/H20850 V/H9258/H20849/H9251,/H9258c/H20850=/H20849J0//H9267/H20850/H208511–2 /H208491 + cos /H9258c/H20850//H20849cosh/H9251+ cos/H9258c/H20850/H20852, /H2084962/H20850 and /H9274/H20849/H9251,/H9258c/H20850=R2J0 /H9267/H20849cosh/H9251+ cos/H9258c/H20850/H208731−1 + cos /H9258c cosh/H9251+ cos/H9258c/H20874. /H2084963/H20850 Consequently, /H9274˜/H20849/H9258c,/H9270/H20850=/H20849R2J0//H9267/H20850/H20875−/H20881cosh/H9251+ cos/H9258c +cos/H9258c/2 /H20881cosh/H9251+ cos/H9258c+4− /H20849cos/H9258c−3 /H208502/4 /H20849cosh/H9251+ cos/H9258c/H208503/2 −3 sin2/H9258c/H208491 + cos /H9258c/H20850/4 /H20849cosh/H9251+ cos/H9258c/H208505/2/H20876. /H2084964/H20850 Using Eqs. /H2084923/H20850and /H2084925/H20850in conjunction with Eq. /H20849A15 /H20850of Ref. 10, K/H20849/H9258c,/H9270/H20850=− /H20849R2J0//H9267/H20850/H20849/H208812 cosh /H9266/H9270/H20850−1/H208512/H9270cot /H20849/H9258c/2/H20850sinh/H9258c/H9270 + cosh /H9258c/H9270/H20852 /H2084965/H20850 andK˜/H20849/H9258c,/H9270/H20850=R2J0 /H9267/H208818 cosh /H9266/H9270sin2/H20849/H9258c/2/H20850/H208532/H9270sinh/H9258c/H9270cot /H20849/H9258c/2/H20850 /H11003/H20851/H92702/H20849cos/H9258c−1 /H20850− 2 cos /H9258c−1 /H20852 + cosh /H9258c/H9270/H20851/H92702/H208495 cos/H9258c+7 /H20850− cos/H9258c+1 /H20852/H20854. /H2084966/H20850 2. Diffusive evaporative flux A commonly considered flux distribution corresponds to diffusive mass transfer into a stagnant gas /H20849i.e., absent even the gas motion which occurs naturally due to the mass trans-fer/H20850. This model leads to Laplace’s equation in toroidal co- ordinates for the variation in vapor concentration throughoutthe gas phase. Solving this equation results in the nonphysi-cal /H20849singular /H20850evaporative flux at the contact line /H20849 /H9251→/H11009/H20850for cases of wetting contact angles: 0 /H11349/H9258c/H11021/H9266/2.14From previ- ous work,10it is known that the Gegenbauer transforms exist provided that lim /H9251→/H11009/H20851/H9274/H20849/H9251,/H9258c/H20850/H20849cosh/H9251+cos/H9258c/H20850/H20852=finite and lim/H9251→/H11009/H20851/H9274˜/H20849/H9251,/H9258c/H20850//H20881sinh/H9251/H20852=finite. These two constraints for obtaining well-behaved liquid-phase transport require the evaporation rate to approach a constant value as /H9251goes to infinity /H20849the contact line /H20850. Therefore, the singular behavior of the gas-diffusion solution needs to be remedied to enableuniformly valid, physical solutions to be obtained for theliquid motion. Fischer 15assumed that the evaporative flux decays expo- nentially near the contact line /H20849/H9251/H110227/H20850, J/H20849/H9251/H20850=/H20877J/H20849/H9251/H20850, /H9251/H113497 J/H208497/H20850exp /H208497−/H9251/H20850,/H9251/H110227./H20878 /H2084967/H20850 We use the above when /H9258c/H1102190° and refer to the boundary condition for these cases as modified diffusive evaporative flux. B. Cylindrical cap shape As reported by Petsi and Burganos,20 d/H9258c dt=−sin2/H9258c R/H9267/H208491−/H9258ccot/H9258c/H20850/H20885 0/H11009J/H20849/H9251/H20850 cosh/H9251+ cos/H9258cd/H9251, /H2084968/H20850 V/H9258/H20849/H9251,/H9258c/H20850=R/H20849cosh/H9251+ cos/H9258c/H20850−1/H20849d/H9258c/dt/H20850+J/H20849/H9251/H20850//H9267, /H2084969/H20850 and /H9274/H20849/H9251,/H9258c/H20850=−d/H9258c dtR2 sin2/H9258c/H20877sinh/H9251 cosh/H9251+ cos/H9258c+ cot/H9258c /H11003/H20875sin−1/H208731 + cosh /H9251cos/H9258c cosh/H9251+ cos/H9258c/H20874−/H9266/2/H20876/H20878 −R /H9267/H20885 0/H9251J/H20849/H9251/H11032/H20850 cosh/H9251/H11032+ cos/H9258cd/H9251/H11032. /H2084970/H20850042102-6 H. Masoud and J. D. Felske Phys. Fluids 21, 042102 /H208492009 /H20850 Author complimentary copy. Redistribution subject to AIP license or copyright, see http://phf.aip.org/phf/copyright.jsp 1. Uniform evaporative flux In this case, the evaporative flux is uniform across the surface of the drop, J/H20849/H9251,/H9258c/H20850= const = J0. /H2084971/H20850 Therefore, d/H9258c dt=−J0 R/H9267/H9258csin/H9258c 1−/H9258ccot/H9258c, /H2084972/H20850 V/H9258/H20849/H9251,/H9258c/H20850=/H20849J0//H9267/H20850/H208531−/H9258csin2/H9258c//H20851/H208491−/H9258ccot/H9258c/H20850/H20849cosh/H9251 + cos/H9258c/H20850/H20852/H20854, /H2084973/H20850 and /H9274/H20849/H9251,/H9258c/H20850=RJ0 /H9267/H20849sin/H9258c−/H9258ccos/H9258c/H20850/H20875/H9258csinh/H9251 cosh/H9251+ cos/H9258c + sin−1/H208731 + cosh /H9251cos/H9258c cosh/H9251+ cos/H9258c/H20874−/H9266/2/H20876. /H2084974/H20850 Consequently, /H9274˜˜/H20849/H9258c,/H9270/H20850=−RJ0 /H9267/H20849tan/H9258c−/H9258c/H20850/H20875tan/H9258csinh/H9251 cosh/H9251+ cos/H9258c + sin−1/H208731 + cosh /H9251cos/H9258c cosh/H9251+ cos/H9258c/H20874−/H9266/2/H20876. /H2084975/H20850 2. Diffusive evaporative flux Laplace’s equation for gas diffusion outside a cylindrical cap drop does not allow an analytical solution when the outerboundary is at infinity since the solution varies logarithmi-cally. Therefore, unlike the 3D axisymmetric case, a solutiondoes not exist for diffusive evaporative flux in 2D. IV. FREELY MOVING CONTACT LINE When the droplet contact line is free to move /H20849un- pinned /H20850, the radial distance to the contact line decreases with time, R=R/H20849t/H20850. One condition previously considered in this case is that the contact angle remains constant during evapo- ration /H9258c=const.10,20This will also be assumed here. Since diffusive evaporative flux does not lead to an acceptable so-lution /H20849see Secs. III A 2 and III B 2 /H20850only uniform flux will be considered as the boundary condition in what follows. A. Spherical cap shape As reported previously,10 dR dt=−/H208491 + cos /H9258c/H208502 sin/H9258c/H208491 + cos /H9258c/2/H20850 /H11003/H20885 0/H11009sinh/H9251 /H20849cosh/H9251+ cos/H9258c/H208502J/H20849/H9251/H20850 /H9267d/H9251, /H2084976/H20850 V/H9258/H20849/H9251,/H9258c/H20850= sin/H9258ccosh/H9251/H20849cosh/H9251+ cos/H9258c/H20850−1/H20849dR /dt/H20850 +J/H20849/H9251/H20850//H9267, /H2084977/H20850 and/H9274/H20849/H9251,/H9258c/H20850=−dR dtR2sin/H9258c/H208751 + cos /H9258c/2 /H208491 + cos /H9258c/H208502 −cosh/H9251+ cos/H9258c/2 /H20849cosh/H9251+ cos/H9258c/H208502/H20876 −/H20885 0/H9251R2sinh/H9251/H11032 /H20849cosh/H9251/H11032+ cos/H9258c/H208502J/H20849/H9251/H11032/H20850 /H9267d/H9251/H11032. /H2084978/H20850 1. Uniform evaporative flux In this case, the evaporative flux is uniform across the surface of the drop, J/H20849/H9251,/H9258c/H20850= const = J0. /H2084979/H20850 Therefore, dR dt=−J0 /H92671 + cos /H9258c sin/H9258c/H208491 + cos /H9258c/2/H20850, /H2084980/H20850 V/H9258/H20849/H9251,/H9258c/H20850=/H20849J0//H9267/H20850/H208531− /H20851/H208491 + cos /H9258c/H20850cosh/H9251/H20852/ /H20851/H208491 + cos /H9258c/2/H20850/H20849cosh/H9251+ cos/H9258c/H20850/H20852/H20854, /H2084981/H20850 and /H9274/H20849/H9251,/H9258c/H20850=−R2J0cos/H9258c /H9267/H20849cosh/H9251+ cos/H9258c/H20850/H208492 + cos /H9258c/H20850 /H11003/H208731−1 + cos /H9258c cosh/H9251+ cos/H9258c/H20874. /H2084982/H20850 Consequently, /H9274˜/H20849/H9258c,/H9270/H20850=R2J0cos/H9258c /H9267/H208492 + cos /H9258c/H20850/H20875/H20881cosh/H9251+ cos/H9258c −cos/H9258c/2 /H20881cosh/H9251+ cos/H9258c−4− /H20849cos/H9258c−3 /H208502/4 /H20849cosh/H9251+ cos/H9258c/H208503/2 +3 sin2/H9258c/H208491 + cos /H9258c/H20850/4 /H20849cosh/H9251+ cos/H9258c/H208505/2/H20876. /H2084983/H20850 Using Eqs. /H2084965/H20850and /H2084966/H20850, K/H20849/H9258c,/H9270/H20850=/H20849R2J0//H9267/H20850cos/H9258c/H20851/H208812/H208492 + cos /H9258c/H20850cosh/H9266/H9270/H20852−1 /H11003/H208512/H9270cot /H20849/H9258c/2/H20850sinh/H9258c/H9270+ cosh /H9258c/H9270/H20852 /H2084984/H20850 and K˜/H20849/H9258c,/H9270/H20850=−R2J0cos/H9258c//H208492 + cos /H9258c/H20850 /H9267/H208818 cosh /H9266/H9270sin2/H20849/H9258c/2/H20850 /H11003/H208532/H9270sinh/H9258c/H9270cot /H20849/H9258c/2/H20850 /H11003/H20851/H92702/H20849cos/H9258c−1 /H20850− 2 cos /H9258c−1 /H20852 + cosh /H9258c/H9270/H20851/H92702/H208495 cos/H9258c+7 /H20850− cos/H9258c+1 /H20852/H20854. /H2084985/H20850042102-7 Analytical solution for Stokes flow Phys. Fluids 21, 042102 /H208492009 /H20850 Author complimentary copy. Redistribution subject to AIP license or copyright, see http://phf.aip.org/phf/copyright.jsp B. Cylindrical cap shape As reported by Petsi and Burganos,20 dR dt=−sin2/H9258c /H9267/H20849/H9258c− sin/H9258ccos/H9258c/H20850/H20885 0/H11009J/H20849/H9251/H20850 cosh/H9251+ cos/H9258cd/H9251, /H2084986/H20850 V/H9258/H20849/H9251,/H9258c/H20850= sin/H9258ccosh/H9251/H20849cosh/H9251+ cos/H9258c/H20850−1/H20849dR /dt/H20850 +J/H20849/H9251/H20850//H9267, /H2084987/H20850 and /H9274/H20849/H9251,/H9258c/H20850=dR dtR sin2/H9258c/H20875sinh/H9251sin/H9258ccos/H9258c cosh/H9251+ cos/H9258c + sin−1/H208731 + cosh /H9251cos/H9258c cosh/H9251+ cos/H9258c/H20874−/H9266/2/H20876 −R /H9267/H20885 0/H9251J/H20849/H9251/H11032/H20850 cosh/H9251/H11032+ cos/H9258cd/H9251/H11032. /H2084988/H20850 1. Uniform evaporative flux In this case, the evaporative flux is uniform across the surface of the drop, J/H20849/H9251,/H9258c/H20850= const = J0. /H2084989/H20850 Therefore, dR dt=−J0/H9258csin/H9258c /H9267/H20849/H9258c− sin/H9258ccos/H9258c/H20850, /H2084990/H20850 V/H9258/H20849/H9251,/H9258c/H20850=/H20849J0//H9267/H20850/H208531−/H9258csin2/H9258ccosh/H9251/ /H20851/H20849/H9258c− sin/H9258ccos/H9258c/H20850/H20849cosh/H9251+ cos/H9258c/H20850/H20852/H20854,/H2084991/H20850 and /H9274/H20849/H9251,/H9258c/H20850=−J0cos/H9258c /H9267/H20849/H9258c− sin/H9258ccos/H9258c/H20850/H11003/H20875/H9258csinh/H9251 cosh/H9251+ cos/H9258c + sin−1/H208731 + cosh /H9251cos/H9258c cosh/H9251+ cos/H9258c/H20874−/H9266/2/H20876. /H2084992/H20850 Consequently, /H9274˜˜/H20849/H9258c,/H9270/H20850=RJ0cos2/H9258c /H9267/H20849/H9258c− sin/H9258ccos/H9258c/H20850/H20875tan/H9258csinh/H9251 cosh/H9251+ cos/H9258c + sin−1/H208731 + cosh /H9251cos/H9258c cosh/H9251+ cos/H9258c/H20874−/H9266/2/H20876. /H2084993/H20850 V. RESULTS AND DISCUSSION In this section, the velocity distributions for the present viscous flow analysis are compared to the previous distribu-tions obtained for inviscid flow. 10The cases of pinned and freely moving contact lines are computed for both the uni-form evaporative flux and the modified diffusive evaporativeflux boundary conditions. Flow patterns are illustrated for /H9258c/H1102190° /H20849wetting /H20850,/H9258c=90°, and /H9258c/H1102290° /H20849nonwetting /H20850.The results are presented in terms of the dimensionless stream function, /H9274/H11569=/H9274//H92740, and the dimensionless velocity, V/H11569=V/V0. For axially symmetric flow, /H92740=R2J0//H9267, /H2084994/H20850 and for planar flow, /H92740=RJ0//H9267. /H2084995/H20850 For both flows, V0=J0//H9267. /H2084996/H20850 where J0is the characteristic evaporative flux. For diffusive evaporation, J0is defined as J0=/H9267gD/H20849Ys−Y/H11009/H20850/R, /H2084997/H20850 in which /H9267gis the density of the vapor-air mixture, Dis the binary diffusion coefficient for vapor through air, Ysis the mass fraction of vapor in the gas phase at the droplet surface/H20849saturation /H20850, and Y /H11009is the mass fraction of vapor in the far field. A. Viscous versus inviscid solutions Figure 3compares, for the pinned contact, the viscous and inviscid flows within hemispherical, Figs. 3/H20849a/H20850and3/H20849b/H20850, and hemicylindrical, Figs. 3/H20849c/H20850and3/H20849d/H20850, drops. For this case /H20849/H9258c=/H9266/2/H20850, the diffusive flux is uniform for both geometries. It is seen that in both cases, streamlines for viscous flow lie FIG. 3. Contours of nondimensional stream function for pinned contact lines with uniform=diffusive evaporative flux at /H9258c=90°: /H20849a/H20850inviscid, spherical cap, /H20849b/H20850viscous, spherical cap /H20849/H9274a,b/H11569=0.005, 0.03, 0.08, 0.15, and 0.22 /H20850,/H20849c/H20850inviscid, cylindrical cap, and /H20849d/H20850viscous, cylindrical cap /H20849/H9274c,d/H11569 =0.03, 0.01, 0.17, 0.24, and 0.3 /H20850.042102-8 H. Masoud and J. D. Felske Phys. Fluids 21, 042102 /H208492009 /H20850 Author complimentary copy. Redistribution subject to AIP license or copyright, see http://phf.aip.org/phf/copyright.jsp farther from the substrate than the corresponding streamlines for inviscid flow. On the other hand, since both viscous andinviscid flows follow the same boundary conditions at thefree surface, the flows behave similarly near the dropletsurface. For the modified diffusive evaporative flux, Fig. 4com- pares the vertical and radial velocities for inviscid flow tothose in viscous flow for pinned contacts with /H9258c=40°. As r/Rincreases, significant differences are observed in the ra- dial velocities due to the no slip condition at the solid sur-face. The vertical velocities are qualitatively the same al-though they differ in magnitude. In Fig. 4the contact angle is set to 40° so that the present results could be compared directly with the numerical solu-tion by Hu and Larson. 16Their solution was only presented graphically. Laying Fig. 4of this study over Fig. 5 of Ref. 16 showed that the lines were congruent.B. Uniform flux Since spherical and cylindrical cap drops exhibit the same behavior for various evaporative fluxes, the results forcylindrical cap drops will be used to illustrate the flow struc-tures corresponding to a uniform evaporative flux. The flowpatterns computed for different contact angles and contactline conditions are illustrated in Fig. 5. It can be seen that when the contact line is pinned, the flow is from the center ofthe drop to its edge. On the other hand, when the contact lineis free to move, distinctively different flow patterns are ob- served for wetting /H20849 /H9258c/H11021/H9266/2/H20850and nonwetting /H20849/H9258c/H11022/H9266/2/H20850 conditions—see Figs. 5/H20849a/H20850and5/H20849c/H20850./H20849This is consistent with the behavior in inviscid flow.10,19/H20850 C. Modified diffusive evaporative flux It is shown in Ref. 9that /H20849i/H20850for/H9258c/H1102290°, the evaporative flux decays to zero as /H20849r/R/H20850→1,/H20849ii/H20850for/H9258c=90°, the evapo- ration rate is uniform over the free surface, and /H20849iii/H20850for/H9258c /H1102190°, the evaporation rate diverges at the contact line. In FIG. 4. Nondimensional velocities vs vertical position at different radial positions /H20849r/R=0.1, 0.3, 0.5, 0.7, 0.9 /H20850for a pinned contact line, modified diffusive evaporative flux, and /H9258c=40°. The dash lines are inviscid flow /H20849Ref. 10/H20850; the solid lines are viscous flow. /H20849a/H20850Radial velocity; /H20849b/H20850vertical velocity. FIG. 5. Contours of nondimensional stream function for viscous flow insidea cylindrical cap exposed to uniform flux /H20849a/H20850 /H9258c=60°, freely moving contact line /H20849/H9274/H11569=−0.06, /H110020.045, /H110020.03,/H110020.015, and /H110020.004 /H20850,/H20849b/H20850/H9258c=60°, pinned contact line /H20849/H9274/H11569=0.015, 0.06, 0.115, 0.17, and 0.22 /H20850,/H20849c/H20850/H9258c=120°, freely moving contact line /H20849/H9274/H11569=0.03, 0.07, 0.11, 0.15, and 0.185 /H20850,a n d /H20849d/H20850/H9258c =120°, pinned contact line /H20849/H9274/H11569=0.07, 0.16, 0.26, 0.36, and 0.46 /H20850.042102-9 Analytical solution for Stokes flow Phys. Fluids 21, 042102 /H208492009 /H20850 Author complimentary copy. Redistribution subject to AIP license or copyright, see http://phf.aip.org/phf/copyright.jsp order to prevent the nonphysical divergence when /H9258c/H1102190°, modified diffusive evaporative flux has been used as theboundary condition. Figure 6presents the four cases considered: /H9258c=60° and 120° for both pinned and freely moving contact lines. When the contact line is pinned, the flow is directed from the centerof the drop to its edges /H20849“coffee-ring” phenomenon /H20850. The flow behavior remains the same even for contact anglesgreater than 90° where the evaporative flux distribution isquite different. On the other hand, when the contact line isfreely moving, the flow pattern is more intricate. In a givendrop, fluid flows both toward and away from the edge, mak-ing it unlikely that a colloidal drop would deposit a coffee-ring pattern of particles. ACKNOWLEDGMENTS The importance of this problem was brought to our at- tention by Professor R. C. Wetherhold.APPENDIX A: E4/H9274=0, TOROIDAL COORDINATES— SEPARATION OF VARIABLES SOLUTION Similar to Khuri and Waswaz,23an “R-separable” form is assumed to separate variables for E4/H9274=0 in toroidal coor- dinates. The form assumed here is /H9274/H20849/H9251,/H9258/H20850=/H20849cosh/H9251+ cos/H9258/H20850−3 /2/H9278/H20849/H9251,/H9258/H20850. /H20849A1 /H20850 /H20849Khuri and Waswaz23assumed Rsinh/H9251times the above. /H20850 Substituting Eq. /H20849A1 /H20850into Eq. /H208499/H20850and then repeating the E2 operation leads to /H115094/H9278 /H11509/H92514+2/H115094/H9278 /H11509/H92512/H11509/H92582+/H115094/H9278 /H11509/H92584− 2 coth /H9251/H115093/H9278 /H11509/H92513 − 2 coth /H9251/H115093/H9278 /H11509/H9251/H11509/H92582+/H208493 coth2/H9251− 3.5 /H20850/H115092/H9278 /H11509/H92512 +/H208495/4/H20850/H115092/H9278 /H11509/H92582+ coth /H9251/H208494.5 − 3 coth2/H9251/H20850/H11509/H9278 /H11509/H9251+/H208499/16/H20850/H9278 =0 . /H20849A2 /H20850 Assuming a product form for separating variables in Eq. /H20849A2 /H20850, /H9278/H20849/H9251,/H9258/H20850=f/H20849/H9251/H20850g/H20849/H9258/H20850, /H20849A3 /H20850 leads to g/H208494/H20850+2/H20875f/H208492/H20850 f− coth /H9251f/H208491/H20850 f+5 /4/H20876g/H208492/H20850 +/H20875f/H208494/H20850 f− 2 coth /H9251f/H208493/H20850 f+/H208493 coth2/H9251− 3.5 /H20850f/H208492/H20850 f + coth /H9251/H208494.5 − 3 coth2/H9251/H20850f/H208491/H20850 f+9 /16/H20876g=0 . /H20849A4 /H20850 To be consistent with the assumed separable form, Eq. /H20849A3 /H20850, the two bracketed coefficients in Eq. /H20849A4 /H20850must be constants. Denoting the constant for the first bracket as /H208491−/H92702/H20850/H20849in order to obtain an established differential equation for the eigen- functions in /H9251/H20850, the Gegenbauer equation is obtained f/H208492/H20850− coth /H9251f/H208491/H20850+/H20849/H92702+1 /4/H20850f=0 . /H20849A5 /H20850 The solution to Eq. /H20849A5 /H20850is f/H20849/H9251/H20850=A/H20849/H9270/H20850C1/2+i/H9270−1 /2/H20849cosh/H9251/H20850+B/H20849/H9270/H20850C1/2+i/H9270/H11569−1 /2/H20849cosh/H9251/H20850. Using Eq. /H20849A5 /H20850and its first and second derivatives, the sec- ond bracketed coefficient in Eq. /H20849A4 /H20850is readily shown to be f/H208494/H20850 f− 2 coth /H9251f/H208493/H20850 f+/H208493 coth2/H9251− 3.5 /H20850f/H208492/H20850 f + coth /H9251/H208494.5 − 3 coth2/H9251/H20850f/H208491/H20850 f+9 /16 = /H20849/H92702+1 /H208502. /H20849A6 /H20850 Inserting the above constant values for the first and second brackets, Eq. /H20849A4 /H20850becomes FIG. 6. Contours of nondimensional stream function for viscous flow inside a spherical cap exposed to modified diffusive evaporative flux /H20849a/H20850/H9258c=60°, freely moving contact line /H20849/H9274/H11569=−0.0007, /H110020.0002, 0.001, 0.006, and 0.02 /H20850, /H20849b/H20850/H9258c=60°, pinned contact line /H20849/H9274/H11569=0.005, 0.03, 0.08, 0.15, and 0.22 /H20850,/H20849c/H20850 /H9258c=120°, freely moving contact line /H20849/H9274/H11569=−0.007, /H110020.001, 0.0003, 0.003, 0.01, 0.02, and 0.03 /H20850, and /H20849d/H20850/H9258c=120°, pinned contact line /H20849/H9274/H11569=0.005, 0.03, 0.08, 0.15, and 0.22 /H20850.042102-10 H. Masoud and J. D. Felske Phys. Fluids 21, 042102 /H208492009 /H20850 Author complimentary copy. Redistribution subject to AIP license or copyright, see http://phf.aip.org/phf/copyright.jsp d4g d/H92584+2 /H208491−/H92702/H20850d2g d/H92582+/H20849/H92702+1 /H208502g=0 . /H20849A7 /H20850 The solution to this constant coefficient equation follows di- rectly and is given by g/H20849/H9258/H20850= sinh /H20849/H9270/H9258/H20850/H20851C/H20849/H9270/H20850sin/H9258+D/H20849/H9270/H20850cos/H9258/H20852 + cosh /H20849/H9270/H9258/H20850/H20851E/H20849/H9270/H20850sin/H9258+F/H20849/H9270/H20850cos/H9258/H20852, /H20849A8 /H20850 where, C/H20849/H9270/H20850,D/H20849/H9270/H20850,E/H20849/H9270/H20850, and F/H20849/H9270/H20850are real functions. Since 0/H11349/H9251/H11021/H11009, the associated eigenvalues are continuously dis- tributed: 0 /H11349/H9270/H11021/H11009. The general solution is then /H9274/H20849/H9251,/H9258/H20850=/H20849cosh/H9251+ cos/H9258/H20850−3 /2/H20885 0/H11009 /H20851A/H20849/H9270/H20850C1/2+i/H9270−1 /2/H20849cosh/H9251/H20850 +B/H20849/H9270/H20850C1/2+i/H9270/H11569−1 /2/H20849cosh/H9251/H20850/H20852 /H11003/H20853sinh /H20849/H9270/H9258/H20850/H20851C/H20849/H9270/H20850sin/H9258+D/H20849/H9270/H20850cos/H9258/H20852 + cosh /H20849/H9270/H9258/H20850/H20851E/H20849/H9270/H20850sin/H9258+F/H20849/H9270/H20850cos/H9258/H20852/H20854d/H9270. /H20849A9 /H20850 APPENDIX B: /H116334/H9274=0, BIPOLAR COORDINATES— SEPARATION OF VARIABLES SOLUTION Following the analysis of Jeffery,25the biharmonic equa- tion /H116124/H9274=0 /H20850may be R-separated in bipolar coordinates by assuming /H9274/H20849/H9251,/H9258/H20850=/H20849cosh/H9251+ cos/H9258/H20850−1/H9278/H20849/H9251,/H9258/H20850. /H20849B1 /H20850 Substituting Eq. /H20849B1 /H20850into Eq. /H2084945/H20850and then repeating the /H116122 operation leads to /H115094/H9278 /H11509/H92514+2/H115094/H9278 /H11509/H92512/H11509/H92582+/H115094/H9278 /H11509/H92584−2/H115092/H9278 /H11509/H92512+2/H115092/H9278 /H11509/H92582+/H9278=0 . /H20849B2 /H20850 Assuming the variables are separable, /H9278/H20849/H9251,/H9258/H20850=f/H20849/H9251/H20850g/H20849/H9258/H20850, Eq. /H20849B2 /H20850becomes g/H208494/H20850+2/H20873f/H208492/H20850 f+1/H20874g/H208492/H20850+/H20873f/H208494/H20850 f−2f/H208492/H20850 f+1/H20874g=0 . /H20849B3 /H20850 Since gwas assumed to be independent of /H9251, the coefficients ofg/H208492/H20850andgmust be constants. Setting the coefficient of g/H208492/H20850 equal to 2 /H208491−/H92702/H20850, f/H208492/H20850/f=−/H92702, /H20849B4 /H20850 where the form of the constant was chosen to obtain eigen- functions in /H9251. The solution of Eq. /H20849B4 /H20850is f/H20849/H9251/H20850=A/H20849/H9270/H20850sin /H20849/H9270/H9251/H20850+B/H20849/H9270/H20850cos /H20849/H9270/H9251/H20850. /H20849B5 /H20850 Substituting this into the coefficient of gyields f/H208494/H20850 f−2f/H208492/H20850 f+1= /H20849/H92702+1 /H208502. /H20849B6 /H20850 Consequently, the differential equation for g/H20849/H9258/H20850is d4g d/H92584+2 /H208491−/H92702/H20850d2g d/H92582+/H20849/H92702+1 /H208502g=0 . /H20849B7 /H20850 Note that this equation in bipolar coordinates is identical to the corresponding equation in toroidal coordinates, Eq. /H20849A7 /H20850. 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