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Surface-tension- and injection-driven spreading of a thin viscous film

  • K. B. Kiradjiev (a1), C. J. W. Breward (a1) and I. M. Griffiths (a1)

Abstract

We consider the spreading of a thin viscous droplet, injected through a finite region of a substrate, under the influence of surface tension. We neglect gravity and assume that there is a precursor layer covering the whole substrate and that the rate of injection is constant. We analyse the evolution of the film profile for early and late time, and obtain power-law dependencies for the maximum film thickness at the centre of the injection region and the position of an apparent contact line, which compare well with numerical solutions of the full problem. We relax the conditions on the injection rate to consider more general time-dependent and spatially varying forms. In the case of power-law injection of the form $t^{k}$ , we observe a switch in the behaviour of the evolution of the film thickness for late time from increasing to decreasing at a critical value of $k$ . We show that point-source injection can be treated as a limiting case of a finite-injection slot and the solutions exhibit identical behaviours for late time. Finally, we formulate the problem with thickness-dependent injection rate, discuss the behaviour of the maximum film thickness and the position of the apparent contact line and give power-law dependencies for these.

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Corresponding author

Email address for correspondence: kristian.kiradjiev@maths.ox.ac.uk

References

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Ajaev, V. S. 2012 Interfacial Fluid Mechanics: A Mathematical Modelling Approach. Springer.
Anderson, D. M. & Davis, H. 1995 The spreading of volatile liquid droplets on heated surfaces. Phys. Fluids 7 (2), 248265.
Benilov, E. S., Chapman, S. J., McLeod, J. B., Ockendon, J. R. & Zubkov, V. S. 2010 On liquid films on an inclined plate. J. Fluid Mech. 663, 5369.
Bonn, D., Eggers, J., Indekeu, J., Meunier, J. & Rolley, E. 2009 Wetting and spreading. Rev. Mod. Phys. 81 (2), 739805.
Carlson, A., Mandre, S. & Mahadevan, L.2015 Elastohydrodynamics of contact in adherent sheets. Preprint, 2015, arXiv:1508.06234.
Chilukuri, R., Aeling, D. & Middleman, S. 1984 Removal of a thin liquid film from a flat surface using an axisymmetric impinging jet. J. Fluids Engng 106 (2), 223226.
Cox, R. G. 1986 The dynamics of the spreading of liquids on a solid surface. Part 1. Viscous flow. J. Fluid Mech. 168, 169194.
Craster, R. V. & Matar, O. K. 2009 Dynamics and stability of thin liquid films. Rev. Mod. Phys. 81, 11311198.
Davis, S. H. & Hocking, L. M. 1999 Spreading and imbibition of viscous liquid on a porous base. Phys. Fluids 11 (1), 4857.
Davis, S. H. & Hocking, L. M. 2000 Spreading and imbibition of viscous liquid on a porous base. II. Phys. Fluids 12 (7), 16461655.
Diez, J. A., Kondic, L. & Bertozzi, A. 2001 Global models for moving contact lines. Phys. Rev. E 63, 113.
Duffy, B. R. & Moffatt, H. K. 1997 A similarity solution for viscous source flow on a vertical plane. Eur. J. Appl. Maths 8, 3747.
Duffy, B. R. & Wilson, S. K. 1997 A third-order differential equation arising in thin-film flows and relevant to tanner’s law. Appl. Math. Lett. 10 (3), 6368.
Ehrhard, P. & Davis, S. H. 1991 Non-isothermal spreading of liquid drops on horizontal plates. J. Fluid Mech. 229, 365388.
Fraysse, N. & Homsy, G. M. 1994 An experimental study of rivulet instabilities in centrifugal spin coating of viscous Newtonian and non-Newtonian fluids. Phys. Fluids 6 (4), 14911504.
Guzman, R. & Vasquez, D. A. 2016 Surface tension driven flow on a thin reaction front. Eur. Phys. J. Spec. Top. 225 (13–14), 25732580.
Halpern, D. & Grotberg, J. B. 1992 Fluid–elastic instabilities of liquid-lined flexible tubes. J. Fluid Mech. 244, 615632.
Hocking, G. C., Sweatman, W. L., Fitt, A. D. & Breward, C. 2011 Deformations during jet-stripping in the galvanizing process. J. Engng Maths 70 (1–3), 297306.
Hocking, L. M. 1980 Sliding and spreading of two-dimensional drops. Q. J. Mech. Appl. Maths 34 (1), 3755.
Hocking, L. M. 1983 The spreading of a thin drop by gravity and capillarity. Q. J. Mech. Appl. Maths 36 (1), 5569.
Hocking, L. M. 1992 Rival contact-angle models and the spreading of drops. J. Fluid Mech. 239, 671681.
Howell, P. D. 2010 Surface-tension-driven flow on a moving curved surface. J. Engng Maths 45 (3), 283308.
Howell, P. D., Robinson, J. & Stone, H. A. 2013 Gravity-driven thin-film flow on a flexible substrate. J. Fluid Mech. 732, 190213.
Howison, S. D., Moriarty, J. A., Ockendon, J. R., Terrill, E. L. & Wilson, S. K. 1997 A mathematical model for drying paint layers. J. Engng Maths 32 (4), 377394.
Huh, C. & Scriven, L. E. 1971 Hydrodynamic model of steady movement of a solid/liquid/fluid contact line. J. Colloid Interface Sci. 35 (1), 85101.
Huppert, H. E. 1982 The propagation of two-dimensional and axisymmetric viscous gravity currents over a rigid horizontal surface. J. Fluid Mech. 121, 4358.
King, J. R. & Bowen, M. 2001 Moving boundary problems and non-uniqueness for the thin film equation. Eur. J. Appl. Maths 12 (3), 321356.
Lacey, A. A. 1982 The motion with slip of a thin viscous droplet over a solid surface. Stud. Appl. Maths 67 (3), 217230.
Lister, J. R. 1992 Viscous flows down an inclined pane from point and line sources. J. Fluid Mech. 242, 631653.
Lister, J. R., Peng, G. G. & Neufeld, J. A. 2013 Viscous control of peeling an elastic sheet by bending and pulling. Phys. Rev. Lett. 111 (15), 15.
Mason, D. P. & Momoniat, E. 2004 Axisymmetric spreading of a thin liquid drop with suction or blowing at the horizontal base. Intl J. Non-Linear Mech. 39, 10131026.
McEwan, A. D. & Taylor, G. I. 1966 The peeling of a flexible strip attached by a viscous adhesive. J. Fluid Mech. 26 (1), 115.
McKinley, I. S., Wilson, S. K. & Duffy, B. R. 1999 Spin coating and air-jet blowing of thin viscous drops. Phys. Fluids 11 (1), 3047.
Momoniat, E. & Mason, D. P. 2007 Spreading of a thin film with suction or blowing including surface tension effects. Intl J. Comput. Maths Appl. 53 (2), 198208.
Momoniat, E., Ravindran, R. & Roy, S. 2010 The influence of slot injection/suction on the spreading of a thin film under gravity and surface tension. Acta Mech. 211 (1–2), 6171.
Murphy, E. A. & Lee, W. T. 2017 Mathematical modelling of contact lens moulding. IMA J. Appl. Maths 82 (3), 473495.
Myers, T. G. 1998 Thin films with high surface tension. SIAM Rev. 40 (3), 441462.
Oliver, J. M., Whiteley, J. P., Saxton, M. A., Vella, D., Zubkov, V. S. & King, J. R. 2015 On contact-line dynamics with mass transfer. Eur. J. Appl. Maths 26 (5), 671719.
Oron, A., Davis, S. H. & Bankoff, S. G. 1997 Long-scale evolution of thin liquid films. Rev. Mod. Phys. 69 (3), 931980.
Savva, N. & Kalliadasis, S. 2009 Two-dimensional droplet spreading over topographical substrates. Phys. Fluids 21 (9), 92102.
Savva, N. & Kalliadasis, S. 2011 Dynamics of moving contact lines: a comparison between slip and precursor film models. Europhys. Lett. 94 (6), 64004.
Savva, N. & Kalliadasis, S. 2013 Droplet motion on inclined heterogeneous substrates. J. Fluid Mech. 725, 462491.
Saxton, M. A., Vella, D., Whiteley, J. P. & Oliver, J. M. 2017 Kinetic effects regularize the mass-flux singularity at the contact line of a thin evaporating drop. J. Engng Maths 106 (1), 4773.
Saxton, M. A., Whiteley, J. P., Vella, D. & Oliver, J. M. 2016 On thin evaporating drops: when is the d2-law valid? J. Fluid Mech. 792, 134167.
Schwartz, L. W. & Michaelides, L. E. 1988 Gravity flow of a viscous liquid down a slope with injection. Phys. Fluids 31 (1988), 397399.
Sibley, D. N., Nold, A., Savva, N. & Kalliadasis, S. 2015 A comparison of slip, disjoining pressure, and interface formation models for contact line motion through asymptotic analysis of thin two-dimensional droplet spreading. J. Engng Maths 94 (1), 1941.
Smith, P. C. 1973 A similarity solution for slow viscous flow down an inclined plane. J. Fluid Mech. 58 (2), 275288.
Tanner, L. H. 1979 The spreading of silicone oil drops on horizontal surfaces. J. Phys. D 12, 14731484.
Thompson, A. B., Tseluiko, D. & Papageorgiou, D. T. 2015 Falling liquid films with blowing and suction. J. Fluid Mech. 787, 292330.
Tuck, E. O. & Schwartz, L. W. 1990 A numerical and asymptotic study of some third-order ordinary differential equations relevant to draining and coating flows. SIAM Rev. 32 (3), 453469.
Voinov, O. V. 1976 Hydrodynamics of wetting. Fluid Dyn. 11 (5), 714721.
Wilson, S. D. R. 1982 The drag-out problem in film coating theory. J. Engng Maths 16 (3), 209221.
Wilson, S. K., Hunt, R. & Duffy, B. R. 2000 The rate of spreading in spin coating. J. Fluid Mech. 413, 6588.
Yatim, Y. M., Duffy, B. R. & Wilson, S. K. 2012 Similarity solutions for unsteady shear-stress-driven flow of Newtonian and power-law fluids: slender rivulets and dry patches. J. Engng Maths 73 (1), 5369.
Yatim, Y. M., Duffy, B. R. & Wilson, S. K. 2013 Travelling-wave similarity solutions for a steadily translating slender dry patch in a thin fluid film. Phys. Fluids 25 (5), 52103.
Yatim, Y. M., Wilson, S. K. & Duffy, B. R. 2010 Unsteady gravity-driven slender rivulets of a power-law fluid. J. Non-Newtonian Fluid Mech. 165 (21–22), 14231430.
Zheng, Z., Fontelos, M. A., Shin, S., Dallaston, M. C., Tseluiko, D., Kalliadasis, S. & Stone, H. A. 2018 Healing capillary films. J. Fluid Mech. 838, 404434.
Zheng, Z., Griffiths, I. M. & Stone, H. A. 2015 Propagation of a viscous thin film over an elastic membrane. J. Fluid Mech. 784, 443464.
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