zbMATH — the first resource for mathematics

A boundary integral formulation of quasi-steady fluid wetting. (English) Zbl 1213.76068
Summary: This paper considers the motion of a liquid droplet on a solid surface. When capillary relaxation is much faster than the motion of the contact line, the fluid geometry and its dynamical evolution can be characterized in terms of the contact line alone. This problem can be cast in terms of boundary integral equations involving a Dirichlet-Neumann map coupled to a volume conservation constraint. A computational method for this formulation is described which has two principal advantages over approaches which track the entire free surface: (1) only the curve which describes the contact line is computed and (2) the resulting method exhibits only mild numerical stiffness, obviating the need for implicit timestepping. Effects of both capillary and body forces are considered. Computational examples include surface inhomogeneities, topological transitions and cusp formation.

76D45 Capillarity (surface tension) for incompressible viscous fluids
76M25 Other numerical methods (fluid mechanics) (MSC2010)
PDF BibTeX Cite
Full Text: DOI
[1] Ajaev, V.S.; Davis, S.H., Boundary-integral simulations of containerless solidification, J. comp. phys., 187, 492-503, (2003) · Zbl 1061.76512
[2] Ben Amar, M.; Cummings, L.; Pomeau, Y., Points singuliers d’une ligne de contact mobile, C.R. acad. sci. Paris, 329, 277-282, (2001) · Zbl 1032.76015
[3] Bertsch, M.; Dal Passo, R.; Davis, S.H.; Giacomelli, L., Effective and microscopic contact angles in thin film dynamics, Eur. J. appl. math., 11, 181-201, (2000) · Zbl 0965.35125
[4] Blake, T.D.; Haynes, J.M., Kinetics of liquid/liquid displacement, J. colloid interf. sci., 30, 421-423, (1969)
[5] Brenner, M.; Bertozzi, A., Spreading of droplets on a solid surface, Phys. rev. lett., 71, 593-596, (1993)
[6] Cameron, A., Principles of lubrication, (1966), Longmans London
[7] Cox, R.G., The dynamics of the spreading of liquids on a solid surface. part 1. viscous flow, J. fluid mech., 168, 169-194, (1986) · Zbl 0597.76102
[8] Cox, R.G., Inertial and viscous effects on dynamic contact angles, J. fluid mech., 357, 249-278, (1998) · Zbl 0908.76026
[9] de Gennes, P.G., Wetting: statics and dynamics, Rev. mod. phys., 57, 827, (1985)
[10] de Gennes, P.G., Deposition of Langmuir-blodgett layers, Colloid polym. sci., 264, 463-465, (1986)
[11] Freund, J.B., The atomic detail of a wetting/dewetting flow, Phys. fluids, 15, L33, (2003)
[12] Glasner, K., Nonlinear preconditioning for diffuse interfaces, J. comp. phys., 174, 695-711, (2001) · Zbl 0991.65076
[13] Glasner, K.B., Spreading of droplets under the influence of intermolecular forces, Phys. fluids, 15, 1837-1842, (2003) · Zbl 1186.76194
[14] K.B Glasner, Variational models for moving contact lines and the quasi-static approximation (2004), preprint
[15] Greenbaum, A.; Greengard, L.; McFadden, G.B., Laplace’s equation and the Dirichlet-Neumann map in multiply connected domains, J. comput. phys., 105, 267-278, (1993) · Zbl 0769.65085
[16] Greenspan, H.P., On the motion of a small viscous droplet that wets a surface, J. fluid mech., 84, 125-143, (1978) · Zbl 0373.76040
[17] Grun, G.; Rumpf, M., Simulations of singularities and instabilities arising in thin film flow, Eur. J. appl. math., 12, 293-320, (2001) · Zbl 0991.76041
[18] Hadjiconstantinou, N.G., Hybrid atomistic-continuum formulations and the moving contact line problem, J. comp. phys., 154, 245-265, (1999) · Zbl 0935.81080
[19] Hocking, L.M.; Miksis, M.J., Stability of a ridge of fluid, J. fluid mech., 247, 157-177, (1993) · Zbl 0767.76019
[20] Hocking, L.M.; Rivers, A.D., J. fluid mech., 121, 425, (1982)
[21] Hou, T.Y.; Lowengrub, J.S.; Shelley, M.J., Removing the stiffness from interfacial flows with surface tension, J. comp. phys., 114, 312, (1994) · Zbl 0810.76095
[22] Hou, T.Y.; Lowengrub, J.S.; Shelley, M.J., Boundary integral methods for multicomponent fluids and multiphase materials, J. comput. phys., 169, 302-362, (2001) · Zbl 1046.76029
[23] Huh, C.; Scriven, L.E., Hydrodynamic model of steady movement of a solid/liquid/fluid contact line, J. colloid interf. sci., 35, 85-101, (1971)
[24] Juric, D.; Tryggvason, G., A front tracking method for dendritic solidification, J. comp. phys., 123, 127, (1996) · Zbl 0843.65093
[25] Kress, R., On the numerical solution of a hyper-singular integral equation in scattering theory, J. comp. appl. math., 61, 345-360, (1995) · Zbl 0839.65119
[26] Lopez, P.G.; Miksis, M.J.; Bankoff, S.G.S., Stability and evolution of a dry spot, Phys. fluids, 13, 1601-1614, (2001) · Zbl 1184.76332
[27] Nikolayev, V.S.; Beysens, D.A., Relaxation of nonspherical sessile drops towards equilibrium, Phys. rev. E, 65, 046135, (2002)
[28] Oron, A.; Bankoff, S.G., Dynamics of a condensing liquid film under conjoining/disjoining pressures, Phys. fluids, 13, 1107-1117, (2001) · Zbl 1184.76408
[29] Oron, A.; Davis, S.H.; Bankoff, S.G., Long-scale evolution of thin liquid films, Rev. mod. phys., 69, 931-980, (1997)
[30] Pismen, L.; Pomeau, Y., Nonlocal diffuse interface theory of thin films and the moving contact line, Phys. rev. E, 62, 2840, (2000)
[31] Podgorski, T.; Flesselles, J.-M.; Limat, L., Corners, cusps and pearls in running drops, Phys. rev. lett., 87, 036102, (2001)
[32] Pomeau, Y., Recent progress in the moving contact line problem: a review, C.R. acad. sci., 330, 207-222, (2002) · Zbl 1177.76021
[33] Schwartz, L., Unsteady simulation of viscous thin-layer flows, (1997), Computational Mechanics Publications Boston, pp. 203-233
[34] Schwartz, L.W.; Roy, R.V.; Eley, R.R.; Petrash, S., Dewetting patterns in a drying liquid film, J. colloid interf. sci., 234, 363-374, (2001)
[35] Seppecher, P., Moving contact lines in the Cahn-Hilliard theory, Int. J. eng. sci., 34, 977-992, (1996) · Zbl 0899.76042
[36] Shikhmurzaev, Y.D., Spreading of drops on solid surfaces in a quasi-static regime, Phys. fluids, 9, 266, (1997)
[37] Sidi, A.; Israeli, M., Quadrature methods for singular and weakly Fredholm integral equations, J. sci. comp., 3, 201-231, (1988) · Zbl 0662.65122
[38] Stone, H.; Limat, L.; wilson, S.; Flesselles, J.-M.; Podgorski, T., Gingularite angulese d’nue ligne de contact en mouvement sur un subsrat solide, C.R. physique, 103, 103-110, (2002)
[39] Tanner, L., The spreading of silicone oil drops on horizontal surfaces, J. phys. D, 12, 1473-1484, (1979)
[40] Verheijen, H.J.J.; Prins, M.W.J., Reversible electrowetting and trapping of charge: model and experiments, Langmuir, 15, 6616-6620, (1999)
[41] Voinov, O.V., Hydrodynamics of wetting (English translation), Fluid dynam., 11, 714-721, (1976)
[42] Weidner, D.E.; Schwartz, L.W., Contact-line motion of shear-thinning liquids, Phys. fluids, 6, 3535, (1994) · Zbl 0843.76003
[43] Zhornitskaya, L.; Bertozzi, A.L., Positivity perserving numerical schemes for lubrication-type equations, SIAM J. numer. anal., 37, 523-555, (2000) · Zbl 0961.76060
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.