Cox-Voinov theory with slip. (English) Zbl 1460.76262
Summary: Most of our understanding of moving contact lines relies on the limit of small capillary number \(Ca\). This means the contact line speed is small compared to the capillary speed \(\gamma /\eta\), where \(\gamma\) is the surface tension and \(\eta\) the viscosity, so that the interface is only weakly curved. The majority of recent analytical work has assumed in addition that the angle between the free surface and the substrate is also small, so that lubrication theory can be used. Here, we calculate the shape of the interface near a slip surface for arbitrary angles, and for two phases of arbitrary viscosities, thereby removing a key restriction in being able to apply small capillary number theory. Comparing with full numerical simulations of the viscous flow equation, we show that the resulting theory provides an accurate description up to \(Ca \approx 0.1\) in the dip coating geometry, and a major improvement over theories proposed previously.
MSC:
| 76D45 | Capillarity (surface tension) for incompressible viscous fluids |
Keywords:
contact linesReferences:
| [1] | Barrat, J.-L. & Bocquet, L.1999Large slip effect at a nonwetting fluid-solid interface. Phys. Rev. Lett.82, 4671-4674. |
| [2] | Benkreira, H. & Ikin, J. B.2010Dynamic wetting and gas viscosity effects. Chem. Engng Sci.65, 1790-1796. |
| [3] | Benkreira, H. & Khan, M. I.2008Air entrainment in dip coating under reduced air pressures. Chem. Engng Sci.63, 448-459. |
| [4] | Blake, T. D. & Ruschak, K. J.1979A maximum speed of wetting. Nature282, 489-491. |
| [5] | Bonn, D., Eggers, J., Indekeu, J., Meunier, J. & Rolley, E.2009Wetting and spreading. Rev. Mod. Phys.81, 739-805. |
| [6] | Chan, T. S., Srivastava, S., Marchand, A., Andreotti, B., Biferale, L., Toschi, F. & Snoeijer, J. H.2013Hydrodynamics of air entrainment by moving contact lines. Phys. Fluids25, 074105. |
| [7] | Cottin-Bizonne, C., Cross, B., Steinberger, A. & Charlaix, E.2005Boundary slip on smooth hydrophobic surfaces: intrinsic effects and possible artifacts. Phys. Rev. Lett.94, 056102. |
| [8] | Cox, R. G.1986The dynamics of the spreading of liquids on a solid surface. Part 1. Viscous flow. J. Fluid Mech.168, 169-194. · Zbl 0597.76102 |
| [9] | Hocking, L. M.1977A moving fluid interface. Part 2. The removal of the force singularity by a slip flow. J. Fluid Mech.79, 209-229. · Zbl 0355.76023 |
| [10] | Hocking, L. M.1983The spreading of a thin drop by gravity and capillarity. Q. J. Mech. Appl. Maths36, 55-69. · Zbl 0507.76100 |
| [11] | Hocking, L. M. & Rivers, A. D.1982The spreading of a drop by capillary action. J. Fluid Mech.121, 425-442. · Zbl 0492.76101 |
| [12] | Huh, C. & Scriven, L. E.1971Hydrodynamic model of steady movement of a solid/liquid/fluid contact line. J. Colloid Interface Sci.35, 85-101. |
| [13] | Kamal, C., Sprittles, J., Snoeijer, J. H. & Eggers, J.2018Dynamic drying transition via free-surface cusps. J. Fluid Mech.858, 760-786. · Zbl 1415.76641 |
| [14] | Landau, L. D. & Lifshitz, E. M.1984Fluid Mechanics. Pergamon. |
| [15] | Lauga, E., Brenner, M. P. & Stone, H. A.2008Microfluidics: the no-slip boundary condition. In Springer Handbook of Experimental Fluid Mechnaics (ed. Tropea, C., Foss, J. F. & Yarin, A.), pp. 1219-1240. Springer. |
| [16] | Marchand, A., Chan, T. S., Snoeijer, J. H. & Andreotti, B.2012Air entrainment by contact lines of a solid plate plunged into a viscous fluid. Phys. Rev. Lett.108, 204501. |
| [17] | Sibley, D. N., Nold, A. & Kalliadasis, S.2015The asymptotics of the moving contact line: cracking an old nut. J. Fluid Mech.764, 445-462. · Zbl 1335.76012 |
| [18] | Snoeijer, J. H.2006Free surface flows with large slopes: beyond lubrication theory. Phys. Fluids18, 021701. |
| [19] | Snoeijer, J. H. & Andreotti, B.2013Moving contact lines: scales, regimes, and dynamical transitions. Annu. Rev. Fluid Mech.45, 269-292. · Zbl 1359.76320 |
| [20] | Sprittles, J. E.2015Air entrainment in dynamic wetting: Knudsen effects and the influence of ambient air pressure. J. Fluid Mech.769, 444-481. · Zbl 1431.76111 |
| [21] | Vandre, E., Carvalho, M. S. & Kumar, S.2012Delaying the onset of dynamic wetting failure through meniscus confinement. J. Fluid Mech.707, 496-520. · Zbl 1275.76086 |
| [22] | Vandre, E., Carvalho, M. S. & Kumar, S.2014Characteristics of air entrainment during dynamic wetting failure along a planar substrate. J. Fluid Mech.747, 119-140. |
| [23] | Voinov, O. V.1976Hydrodynamics of wetting [english translation]. Fluid Dyn.11, 714-721. |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
