Numerical solutions for thin film flow down the outside and inside of a vertical cylinder

We consider a model for thin film flow down the outside and inside of a vertical cylinder. Our focus is the effect the curvature of the cylinder has on the gravity driven instability of the advancing contact line and to simulate the resulting fingering patterns that form due to this instability. The governing partial differential equation is fourth order with a nonlinear degenerate diffusion term that represents the stabilising effect of surface tension. We present numerical solutions obtained by implementing an efficient alternating direction implicit scheme. When compared to the problem of flow down a vertical plane, we find that increasing substrate curvature tends to increase the fingering instability for flow down the outside of the cylinder, whereas flow down the inside of the cylinder substrate curvature has the opposite effect. Further, we demonstrate the existence of nontrivial travelling wave solutions which describe fingering patterns that propagate down the inside of a cylinder at constant speed without changing form. These solutions are perfectly analogous to those found previously for thin film flow down an inclined plane. References A. L. Bertozzi and M. P. Brenner, Linear stability and transient growth in driven contact lines, Phys. Fluids, 9:530--539, 1997. doi:10.1063/1.869217 R. V. Craster and O. K. Matar, Dynamics and stability of thin liquid films, Rev. Mod. Phys., 81:1131--1198, 2009. doi:10.1103/RevModPhys.81.1131 R. V. Craster and O. K. Matar, On viscous beads flowing down a vertical fibre, J. Fluid Mech., 553:85--105, 2006. doi:10.1017/S0022112006008706 L. M. Hocking, Spreading and instability of a viscous fluid sheet, J. Fluid Mech., 211:373--392, 1990. doi:10.1017/S0022112090001616 H. E. Huppert, Flow and instability of a viscous current down a slope, Nature, 300:427--429, 1982. doi:10.1038/300427a0 I. L. Kliakhandler, S. H. Davis and S. G. Bankoff, Viscous beads on vertical fibre, J. Fluid Mech., 429:381--390, 2001. doi:10.1017/S0022112000003268 L. Kondic and J. Diez, Pattern formation in the flow of thin films down an incline: Constant flux configuration, Phys. Fluids, 13:3168--3184, 2001. doi:10.1063/1.1409965 L. Kondic and J. Diez, On nontrivial travelling waves in thin film flows including contact lines, Physica D, 209:135--144, 2005. doi:10.1016/j.physd.2005.06.029 T.-S. Lin and L. Kondic, Thin films flowing down inverted substrates: Two dimensional flow, Phys. Fluids, 22:052105, 2010. doi:10.1063/1.3428753 T.-S. Lin, L. Kondic and A. Filippov, Thin films flowing down inverted substrates: Three dimensional flow, Phys. Fluids, 24:022105, 2012. doi:10.1063/1.3682001 J. R. Lister, J. M. Rallison and S. J. Rees, The nonlinear dynamics of pendent drops on a thin film coating the underside of a ceiling, J. Fluid Mech., 647:239--264, 2010. doi:10.1017/S002211201000008X L. C. Mayo, S. W. McCue and T. J. Moroney, Gravity-driven fingering simulations for a thin liquid film flowing down the outside of a vertical cylinder, Phys. Rev. E, 87:053018, 2013. doi:10.1103/PhysRevE.87.053018. T. G. Myers, Thin films with high surface tension, SIAM Rev., 40:441--462, 1998. doi:10.1137/S003614459529284X A. Oron, S. H. Davis and S. G. Bankoff, Long-scale evolution of thin liquid films, Rev. Mod. Phys., 69:931--980, 1997. doi:10.1103/RevModPhys.69.931 W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P Flannery, Numerical Recipes in C 2nd ed., Cambridge University Press, Chap. 2, Sec. 7, 1992. L. B. Smolka and M. SeGall, Fingering instability down the outside of a vertical cylinder, Phys. Fluids, 23:092103, 2011. doi:10.1063/1.3633530 L. W. Schwartz, Hysteretic effects in droplet motions on heterogeneous substrates: Direct numerical simulation, Langmuir, 14:3440--3453, 1998. doi:10.1021/la971407t S. M. Troian, E. Herbolzheimers, A. Safran and J. F. Joanny, Fingering instabilities of driven spreading films, Europhys. Lett., 10:25--30, 1989. doi:10.1209/0295-5075/10/1/005 T. P. Witelski and M. Bowen, adi schemes for higher-order nonlinear diffusion equations, Appl. Numer. Math., 45:331--351, 2003. doi:10.1016/S0168-9274(02)00194-