Abstract
The influence of gravity on the motion of a two-dimensional droplet of a partially wetting fluid over a topographical substrate is considered. The spreading dynamics is modeled under the assumption of small contact angles in which case the long-wave expansion in the Stokes-flow regime can be employed to derive a single equation for the evolution of the droplet thickness. The relative importance of gravity to capillarity in the equation is measured by the Bond number which is taken to be low to moderate. In this regime, the flow in the vicinity of the contact line is matched asymptotically through a singular perturbation approach to the flow in the bulk of the droplet to yield a set of coupled integrodifferential equations for the location of the two droplet fronts. The matching procedure is verified through direct comparisons with numerical solutions to the full problem. The equations obtained by asymptotic matching are analyzed in the phase plane and the effects of Bond number on the droplet dynamics and its equilibria are scrutinized.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Blossey R (2003) Self-cleaning surfaces—virtual realities. Nat Mater 2: 301–306
Chu K-H, Xiao R, Wang EN (2010) Uni-directional liquid spreading on asymmetric nanostructured surfaces. Nat Mater 9(5): 413–417
Dussan VEB (1979) On the spreading of liquids on solid surfaces: static and dynamic contact lines. Annu Rev Fluid Mech 11: 371–400
de Gennes P-G (1985) Wetting: statics and dynamics. Rev Mod Phys 57: 827–863
Blake TD (1993) Dynamic contact angles and wetting kinetics. In: Berg JC (ed) Wettability, chap 5. Marcel Dekker Inc., New York, pp 251–310
Bonn D, Eggers J, Indekeu J, Meunier J, Rolley E (2009) Wetting and spreading. Rev Mod Phys 81: 739–805
Huh C, Scriven LE (1971) Hydrodynamic model of steady movement of a solid/liquid/fluid contact line. J Colloid Interface Sci 35: 85–101
Voinov OV (1976) Hydrodynamics of wetting. Fluid Dyn 11: 714–721
Greenspan HP (1978) On the motion of a small viscous droplet that wets a surface. J Fluid Mech 84: 125–143
Hocking LM (1983) The spreading of a thin drop by gravity and capillarity. Q J Mech Appl Math 36: 55–69
Hocking LM (1994) The spreading of drops with intermolecular forces. Phys Fluids 6: 3224–3228
Pismen LM, Eggers J (2008) Solvability condition for the moving contact line. Phys Rev E 78: 056304
Ehrhard P, Davis SH (1991) Non-isothermal spreading of liquid drops on horizontal plates. J Fluid Mech 229: 365–388
Schwartz LW, Eley RR (1998) Simulation of droplet motion on low-energy and heterogeneous surfaces. J Colloid Interface Sci 202: 173–188
Sodtke C, Ajaev VS, Stephan P (2008) Dynamics of volatile liquid droplets on heated surfaces: theory versus experiment. J Fluid Mech 610: 343–362
Wenzel RN (1936) Resistance of solid surfaces to wetting by water. Ind Eng Chem 28: 988–994
Shuttleworth R, Bailey GLJ (1948) The spreading of a liquid over a rough solid. Discuss Faraday Soc 3: 16–22
Johnson RE, Dettre RH (1964) Contact angle hysteresis. I. Study of an idealized rough surface. Adv Chem Ser 43: 112–135
Johnson RE, Dettre RH, Brandreth DA (1977) Dynamic contact angles and contact angle hysteresis. J Colloid Interface Sci 62: 205–212
Huh C, Mason SG (1977) Effects of surface roughness on wetting (theoretical). J Colloid Interface Sci 60: 11–38
Gramlich CM, Mazouchi A, Homsy GM (2004) Time-dependent free surface Stokes flow with a moving contact line. II. Flow over wedges and trenches. Phys Fluids 16: 1660–1667
Gaskell PH, Jimack PK, Sellier M, Thompson HM (2004) Efficient and accurate time adaptive multigrid simulations of droplet spreading. Int J Numer Methods Fluids 45: 1161–1186
Troian S, Herbolzheimer S, Safran S, Joanny J (1989) Fingering instabilities of driven spreading films. Europhys Lett 10: 25–39
Kalliadasis S (2000) Nonlinear instability of a contact line driven by gravity. J Fluid Mech 413: 355–378
Savva N, Kalliadasis S (2009) Two-dimensional droplet spreading over topographical substrates. Phys Fluids 21: 092102
Savva N, Kalliadasis S, Pavliotis GA (2010) Two-dimensional droplet spreading over random topographical substrates. Phys Rev Lett 104: 084501
McHale G, Newton MI, Rowan SM, Banerjee M (1995) The spreading of small viscous stripes of oil. J Appl Phys D Appl Phys 28: 1925–1929
Hocking LM (1981) Sliding and spreading of two-dimensional drops. Q J Mech Appl Math 34: 37–55
Reznik SN, Zussman E, Yarin AL (2002) Motion of an inclined plate supported by a sessile two-dimensional drop. Phys Fluids 14: 107–117
Lister JR, Morrison NF, Rallison JM (2006) Sedimentation of a two-dimensional drop towards a rigid horizontal plane. J Fluid Mech 552: 345–351
Zhang J, Miksis MJ, Bankoff SG (2006) Nonlinear dynamics of a two-dimensional viscous drop under shear flow. Phys Fluids 18: 072106
Nakaya C (1974) Spread of fluid drops over a horizontal plane. J Phys Soc Jpn 37: 539–543
Roux DCD, Cooper-White JJ (2004) Dynamics of water spreading on a glass surface. J Colloid Interface Sci 277: 424–436
Lauga E, Brenner MP, Stone HA (2008) Microfluidics: the no-slip boundary condition. In: Tropea C, Foss JF, Yarin A (eds) Springer handbook of experimental fluid mechanics, chap 19. Springer, New York
Cox RG (1983) The spreading of a liquid on a rough solid surface. J Fluid Mech 131: 1–26
Miksis MJ, Davis SH (1994) Slip over rough and coated surfaces. J Fluid Mech 273: 125–139
Saprykin S, Trevelyan PMJ, Koopmans RJ, Kalliadasis S (2007) Free-surface thin-film flows over uniformly heated topography. Phys Rev E 75: 026306
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Savva, N., Kalliadasis, S. Influence of gravity on the spreading of two-dimensional droplets over topographical substrates. J Eng Math 73, 3–16 (2012). https://doi.org/10.1007/s10665-010-9426-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10665-010-9426-4