Hostname: page-component-5c6d5d7d68-vt8vv Total loading time: 0.001 Render date: 2024-08-22T21:31:56.088Z Has data issue: false hasContentIssue false
  • English
  • Français

The spreading of insoluble surfactant at the free surface of a deep fluid layer

Published online by Cambridge University Press:  26 April 2006

O. E. Jensen
O. E. Jensen
Affiliation:
Department of Mathematics and Statistics, University of Newcastle upon Tyne, Newcastle upon Tyne NE1 7RU, UK
Get access Rights & Permissions [Opens in a new window]

Abstract

The unsteady spreading of an insoluble monolayer containing a fixed mass of surface-active material over the initially horizontal free surface of a viscous fluid layer is investigated. A flow driving the spreading is induced by gradients in surface tension, which arise from the nonuniform surfactant distribution. Distinct phases in the flow's dynamics are distinguished by a time T = H02/v, where H0 is the fluid depth and v its viscosity. For times t [Lt ] T, i.e. before the lower boundary has any significant influence on the flow, a laminar sub-surface boundary-layer flow is generated. The effects of gravity, capillarity, surface diffusion or surface contamination may be weak enough for the flow to drive a substantial unsteady displacement of the free surface, upward behind the monolayer's leading edge and downward towards its centre. Similarity solutions are identified describing the spreading of a localized planar monolayer strip (which spreads like t1/2) or an axisymmetric drop (which spreads like t3/8); using the Prandtl transformation, the associated boundary-layer problems are solved numerically. Quasi-steady sub-layers are shown to exist at the centre and at the leading edge of the monolayer; that due to surface contamination, for example, may eventually grow to dominate the flow, in which case spreading proceeds like t3/4. Once t = O(T), vorticity created at the free surface has diffused down to the lower boundary and the flow changes character, slowing appreciably. The dynamics of this stage are modelled by reducing the problem to a single nonlinear diffusion equation. For a spreading monolayer strip or drop, the transition from an inertia-dominated (boundary-layer) flow to a viscosity-dominated (thin-film) flow is predicted to be largely complete once t ≈ 85 T.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 293 , 25 June 1995 , pp. 349 - 378
Copyright
© 1995 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ahmad, J. & Hansen, R. S. 1972 A simple quantitative treatment of the spreading of monolayers on thin liquid films. J. Colloid Interface Sci. 38, 601–604.Google Scholar
Banks, W. H. H. 1983 Similarity solutions of the boundary-layer equations for a stretching wall. J. Méc. Théor. Appl. 2, 375–392.Google Scholar
Banks, W. H. H. & Kuiken, M. B. 1986 Eigensolutions in boundary-layer flow adjacent to a stretching wall. IMA J. Appl. Maths 36, 263–273.Google Scholar
Borgas, M. S. & Grotberg, J. B. 1988 Monolayer flow on a thin film. J. Fluid Mech. 193, 151–170.Google Scholar
Bratukhin, Yu. K. & Maurin, L. N. 1967 Thermocapillary convection in a fluid filling a half-space. J. Appl. Math. Mech. 31, 577–580 (in Russian).Google Scholar
Buckmaster, J. 1973 Viscous-gravity spreading of an oil slick. J. Fluid Mech. 59, 481–491.Google Scholar
Camp, D. W. & Berg, J. C. 1987 The spreading of oil on water in the surface-tension regime. J. Fluid Mech. 184, 445–462.Google Scholar
Crane, L. J. 1970 Flow past a stretching plate. Z. Angew. Math. Phys. 21, 645–647.Google Scholar
Dagan, Z. 1984 Spreading of films of adsorption on a liquid surface. Physico Chem. Hydrodyn. 5, 43–51.Google Scholar
DiPietro, N. D. Huh, C. & Cox, R. G. 1978 The hydrodynamics of the spreading of one liquid on the surface of another. J. Fluid Mech. 84, 529–549.Google Scholar
Espinosa, F. F., Shapiro, A. H., Fredberg, J. J. & Kamm, R. D. 1993 Spreading of a small surfactant bolus on a thin film lining an airway. J. Appl. Physiol. 75, 2028–2039.Google Scholar
Foda, M. & Cox, R. G. 1980 The spreading of thin liquid films on a water-air interface. J. Fluid Mech. 101, 33–51.Google Scholar
Gaver, D. P. III & Grotberg, J. B. 1990 The dynamics of a localized surfactant on a thin film. J. Fluid Mech. 213, 127–148.Google Scholar
Gaver, D. P. III & Grotberg, J. B. 1992 Droplet spreading on a thin viscous film. J. Fluid Mech. 235, 399–414.Google Scholar
Goldstein, S. 1965 On backward boundary layers and flow in converging passages. J. Fluid Mech. 21, 33–45.Google Scholar
Grotberg, J. B. 1994 Pulmonary flow and transport phenomena. Ann. Rev. Fluid Mech. 26, 529–571.Google Scholar
Grotberg, J. B., Halpern, D. & Jensen, O. E. 1995 The interaction of exogenous and endogenous surfactant: spreading-rate effects. J. Appl. Physiol. 78, 750–756.Google Scholar
Harper, J. F. 1992 The leading edge of an oil slick, soap film, or bubble stagnant cap in Stokes flow. J. Fluid Mech. 237, 23–32.Google Scholar
Harper, J. F. & Dixon, J. N. 1974 The leading edge of a surface film on contaminated water. Proc. Fifth Australasian Conf. Hydraulics Fluid Mech., Christchurch, NZ Vol. 2, pp. 499–505.
Hiemenz, K. 1911 Die Grenzschicht an einem in den gleichförmigen Flüssigkeitsstrom eingetauchten geraden Kreiszylinder. Dinglers Polytech. J. 326, 321.Google Scholar
Hoult, D. P. 1972 Oil spreading on the sea. Ann. Rev. Fluid Mech. 4, 341–368.Google Scholar
Jensen, O. E. 1994 Self-similar, surfactant-driven flows. Phys. Fluids 6, 1084–1094.Google Scholar
Jensen, O. E. & Grotberg, J. B. 1992 Insoluble surfactant spreading on a thin viscous film: shock evolution and film rupture. J. Fluid Mech. 240, 259–288 (herein called JG).Google Scholar
Joos, P. & Hunsel, J. van 1985 Spreading of aqueous surfactant solutions on organic liquids. J. Colloid Interface Sci. 106, 161–167.Google Scholar
Joos, P. & Pintens, J. 1977 Spreading kinetics of liquids on liquids. J. Colloid Interface Sci. 60, 507–513.Google Scholar
Karkare, M. V., La, H. T. & Fort, T. 1993 Criteria for effectiveness of surfactants as water-moving agents in “unsaturated” wet sand. Langmuir 9, 1684–1690.Google Scholar
Kuiken, H. K. 1981 On boundary layers in fluid mechanics that decay algebraically along stretches of wall that are not vanishingly small. IMA J. Appl. Maths 27, 387–405.Google Scholar
Lacey, A. A., Ockenden, J. R. & Tayler, A. B. 1982 Waiting-time solutions of a nonlinear diffusion equation. SIAM J. Appl. Maths 42, 1252–1264.Google Scholar
Lister, J. R. 1992 Viscous flows down an inclined plane from point and line sources. J. Fluid Mech. 242, 631–653.Google Scholar
Lister, J. R. & Kerr, R. C. 1989 The propagation of two-dimensional and axisymmetric viscous gravity currents at a fluid interface. J. Fluid Mech. 203, 215–249.Google Scholar
Ma, P. K. H. & Hui, W. H. 1990 Similarity solutions of the two-dimensional unsteady boundary-layer equations. J. Fluid Mech. 216, 537–559.Google Scholar
McCutchen, C. W. 1970 Surface films compacted by moving water: demarcation lines reveal film edges. Science 170, 61–64.Google Scholar
Mockros, L. F. & Krone, R. B. 1968 Hydrodynamics effects on an interfacial film. Science 161, 361–363.Google Scholar
Rajeswari, V., Kumari, m. & Nath, G. 1993 Unsteady three-dimensional boundary layer flow due to a stretching surface. Acta Mechanica 98, 123–141.Google Scholar
Scott, J. C. 1982 Flow beneath a stagnant film on water: the Reynolds ridge. J. Fluid Mech. 116, 283–296.Google Scholar
Shtern, V. & Hussain, F. 1993 Azimuthal instability of divergent flows. J. Fluid Mech. 256, 535–560.Google Scholar
Smith, P. C. 1973 A similarity solution for slow viscous flow down an inclined plane. J. Fluid Mech. 58, 275–288.Google Scholar
Smith, S. H. 1994 An exact solution of the unsteady Navier-Stokes equations resulting from a stretching surface. Trans. ASME E: J. Appl. Mech. 61, 629–633.Google Scholar
Stuart, J. T. 1966 Double boundary layers in oscillatory viscous flow. J. Fluid Mech. 24, 673–687.Google Scholar
Surma Devi, C. D., Takhar, H. S. & Nath, G. 1986 Unsteady, three-dimensional, boundary-layer flow due to a stretching surface. Intl J. Heat Mass Transfer 29, 1996–1999.Google Scholar
Troian, S. M., Herbolzheimer, E. & Safran, S. A. 1990 Model for the fingering instability of spreading surfactant drops. Phys. Rev. Lett. A 65, 333–336.Google Scholar
Tuck, E. O. & Bentwich, M. 1983 Sliding sheets: lubrication with comparable viscous and inertia forces. J. Fluid Mech. 135, 51–69.Google Scholar
Van Dyke, M. 1975 Perturbation Methods In Fluid Mechanics. Academic Press.
Wang, C. Y. 1971 Effect of spreading of material on the surface of a fluid — an exact solution. Intl. J. Nonlin. Mech. 6, 255–262.Google Scholar
Wang, C. Y. 1984 The three-dimensional flow due to a stretching flat surface. Phys. Fluids 27, 1915–1917.Google Scholar