Self-similar flow and contact line geometry at the rear of cornered drops

Physics of Fluids

Volumn 17, Issue 7, 2005, Pages 1-12

  Snoeijer, Jacco H ^a   Rio, Emmanuelle ^a   Le Grand, Nolwenn ^a   Limat, Laurent ^a  

^a UNIVERSITÉ PARIS DIDEROT   (France)

Author keywords

[No Author keywords available]

Indexed keywords

GEOMETRY; LUBRICATION; NUMERICAL ANALYSIS; VELOCITY MEASUREMENT; WETTING;

INCLINED PLANES; LUBRICATION EQUATIONS; PARTICLE IMAGE VELOCIMETRY; SELF-SIMILAR FLOW;

FLUID DYNAMICS;

FLUID FLOW;

EID: 23044450173     PISSN: 10706631     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.1946607     Document Type: Article

References (48)

1. C. Huh and L. E. Shriven, "Hydrodynamic model of steady movement of a solid/liquid/fluid contact line," J. Colloid Interface Sci.JCISA50021-979710.1016/0021-9797(71)90188-3 35, 85 (1971).
2. E. B. Dussan, V. Davis, and S. H. Davis, "On the motion of a fluid-fluid interface along a solid surface," J. Fluid Mech.JFLSA70022-1120 65, 71 (1974).
3. O. V. Voinov, "Hydrodynamics of wetting," Fluid Dyn.FLDYAH0015-462811, 714 (1976).
4. R. G. Cox, "The dynamics of the spreading of liquids on a solid surface," J. Fluid Mech.JFLSA70022-1120 168, 169 (1986).
5. L. M. Hocking, "The spreading of a thin drop by gravity and capillarity," Q. J. Mech. Appl. Math.QJMMAV0033-5614 36, 55 (1983).
6. T. D. Blake, J. De Coninck, and U. D'Ortuna, "Models of wetting: Immiscible lattice Boltzmann automata versus molecular kinetic theory," LangmuirLANGD50743-746310.1021/la00011a065 11, 4588 (1995).
7. L. M. Pismen and Y. Pomeau, "Disjoining potential and spreading of thin liquid layers in the diffuse-interface model coupled to hydrodynamics," Phys. Rev. EPLEEE81063-651X10.1103/PhysRevE.62.2480 62, 2480 (2000).
8. L. W. Schwartz, R. V. Roy, R. R. Eley, and S. Petrash, "Dewetting patterns in a drying liquid film," J. Colloid Interface Sci.JCISA50021-979710.1006/jcis.2000.7312 234, 363 (2001).
9. U. Thiele, K. John, and M. Bär, "Dynamical model for chemically driven running droplets," Phys. Rev. Lett. PRLTAO0031-900710.1103/PhysRevLett.93.027802 93, 027802 (2004).
10. P. Seppecher, "A numerical study of a moving contact line in Cahn-Hilliard theory," Int. J. Eng. Sci.IJESAN0020-722510.1016/ 0020-7225(95)00141-7 34, 977 (1996).
11. Y. D. Shikhmurzaev, "Moving contact lines in liquid/liquid/solid systems," J. Fluid Mech.JFLSA70022-112010.1017/S0022112096004569 334, 211 (1997).
12. Y. Pomeau, "Recent progress in the moving contact line problem: A review," C. R. Mec.CRMOC91631-072110.1016/S1631-0721(02)01445-6 330, 207 (2002).
13. S. F. Kistler and P. Schweizer, Liquid Film Coating-Scientific Principles and Their Technological Implications (Kluwer, Dordrecht, 1997).
14. P. G. de Gennes, "Deposition of Langmuir-Blodget layers," Colloid Polym. Sci.CPMSB60303-402X 264, 463 (1986).
15. L. M. Hocking, "Meniscus draw-up and draining," Eur. J. Appl. Math.ZZZZZZ0956-792510.1017/S0956792501004247 12, 195 (2001).
16. J. Eggers, "Hydrodynamic theory of forced dewetting," Phys. Rev. Lett.PRLTAO0031-900710.1103/PhysRevLett.93.094502 93, 094502 (2004).
17. D. Quéré, "On the minimal velocity of forced spreading in partial wetting," C. R. Acad. Sci., Ser. II: Mec., Phys., Chim., Sci. Terre UniversCRAMED0764-4450, 313, 313 (1991).
18. L. D. Landau and B. V. Levich, "Dragging of a liquid by a moving plate," Acta Physicochim. URSSACPYAR0365-1460 20, 17, 42 (1942)
19. B. V. Derjaguin, "On the thickness of the liquid film adhering to the walls of a vessel after emptying," Acta Physicochim. URSSACPYAR0365-1460 20, 349 (1943).
20. R. Golestanian and E. Raphael, "Relaxation of a moving contact line and the Landau-Levich effect," Europhys. Lett.EULEEJ0295-507510.1209/epl/i2001-00607-5 55, 228 (2001).
21. T. D. Blake and K. J. Ruschak, "A maximum speed of wetting," Nature (London)NATUAS0028-0836 282, 489 (1979).
22. U. Thiele, M. G. Velarde, K. Neuffer, M. Bestehorn, and Y. Pomeau, "Sliding drops in the diffuse interface model coupled to hydrodynamics," Phys. Rev. EPLEEE81063-651X10.1103/ PhysRevE.64.061601 64, 061601 (2001).
23. U. Thiele, K. Neuffer, M. Bestehorn, Y. Pomeau, and M. G. Velarde, "Sliding drops on an inclined plane," Colloids Surf., ACPEAEH0927-775710.1016/S0927-7757(02)00082-1 206, 87 (2002).
24. T. Podgorski, J. M. Flesselles, and L. Limat, "Corners, cusps and pearls in running drops," Phys. Rev. Lett.PRLTAO0031-900710.1103/ PhysRevLett.87.036102 87, 036102 (2001).
25. A. Daerr, N. Le Grand, L. Limat, and H. A. Stone, "Drops sliding along an inclined plane: Experiments versus 3D hydrodynamical model," Proceedings of the Fifth European Coating Symposium, 17-19 September 2003, Polytype Converting, Friburg, 2004, edited by P. M. Schweizer.
26. U. Thiele and E. Knobloch, "Front and back instability of a liquid film on a slightly inclined plate," Phys. FluidsPHFLE61070-663110.1063/1.1545443 15, 892 (2003).
27. A. G. Gonzales, J. Diez, J. Gomba, R. Gratton, and L. Kondic, "Spreading of a thin two-dimensional strip of fluid on a vertical plane: Experiments and modeling," Phys. Rev. EPLEEE81063-651X10.1103/PhysRevE.70.026309 70, 026309 (2004).
28. G. Reiter and A. Sharma, "Auto-optimization of dewetting rates by rim instabilities in slipping polymer films," Phys. Rev. Lett.PRLTAO0031-900710.1103/PhysRevLett.87.166103 87, 166103 (2001).
29. H. A. Stone, L. Limat, S. K. Wilson, J.-M. Flesselles, and T. Podgorski, "Corner singularity of a contact line moving on a solid surface," C. R. Phys.CRPOBN1631-070510.1016/S1631-0705(02)01288-4 3, 103 (2002).
30. L. Limat and H. A. Stone, "Three-dimensional lubrication model of a contact line corner singularity," Europhys. Lett.EULEEJ0295-507510.1209/epl/i2003-10096-0 65, 365 (2004).
31. X. D. Shi, M. P. Brenner, and S. R. Nagel, "A cascade structure in a drop falling from a faucet," ScienceSCIEAS0036-8075 265, 157 (1994)
32. M. Brenner, J. R. Lister and H. A. Stone, "Pinching threads, singularities and the number 0.0304...," Phys. FluidsPHFLE61070-663110.1063/1.869086 8, 2827 (1996)
33. J. Eggers, "Nonlinear dynamics and breakup of free-surface flows," Rev. Mod. Phys.RMPHAT0034-686110.1103/RevModPhys.69.865 69, 865 (1997).
34. I. Cohen and S. R. Nagel, "Scaling at a selective withdrawal transition through a tube suspended above the fluid surface," Phys. Rev. Lett.PRLTAO0031-900710.1103/PhysRevLett.88.074501 88, 074501 (2002)
35. W.W. Zhang, "Viscous entrainment from a nozzle: Singular liquid spouts," Phys. Rev. Lett.PRLTAO0031-900710.1103/ PhysRevLett.93.184502 93, 184502 (2004).
36. J. Eggers, "Air entrainment through free-surface cusps," Phys. Rev. Lett.PRLTAO0031-900710.1103/PhysRevLett.86.4290 86, 4290 (2001); E. Lorenceau, F. Restagno, and D. Quéré, "Fracture of a viscous liquid," Phys. Rev. Lett.PRLTAO0031-900710.1103/ PhysRevLett.90.184501 90, 184501 (2003).
37. E. Rio, A. Daerr, B. Andreotti, and L. Limat, "Boundary condition in the vicinity of a dynamic contact line," Phys. Rev. Lett.PRLTAO0031-900710.1103/PhysRevLett.94.024503 94, 024503 (2005).
38. M. Ben Amar, L. J. Cummings, and Y. Pomeau, "Transition of a moving contact line from smooth to angular," Phys. FluidsPFLDAS0031-917110.1063/1.1604778 15, 2949 (2003).
39. Recent experiments show that as the corners develops, there is a tremedous increase of contact line curvature in a very narrow range of Ca. These results will be published in a forthcoming paper, by the same authors as of Ref. 24.
40. Note that the curve H(ζ) vs F(ζ) is different from the one presented in Ref. 29, which had a problem with numerical precision. However, the conclusions remain unaltered.
41. 0.
42. The solutions never become purely parabolic; Eq. (12) shows that the higher-order derivatives become dominant close to the contact line. However, our numerical curves indicate that deviations from a simple parabola become increasingly localized of contact line when φ→0
43. J. Eggers, "Toward a description of contact line motion at higher capillary numbers," Phys. FluidsPHFLE61070-663110.1063/1.1776071 16, 3491 (2004).
44. plate from 0.1 to 10.
45. α1/ln(l/1)
46. 33α is the inclination angle of the plane, while U is simply the normal velocity of the contact line.
47. M. Bestehorn and K. Neuffer, "Surface patterns of laterally extended thin liquid films in three dimensions," Phys. Rev. Lett.PRLTAO0031-900710.1103/PhysRevLett.87.046101 87, 046101 (2001).
48. P. G. de Gennes, X. Hua, and P. Levinson, "Dynamics of wetting: Local contact angles," J. Fluid Mech.JFLSA70022-1120 212, 55 (1990).