Abstract.
A theory is formulated to describe the dynamics of a thin film flowing down an inclined plane laden with insoluble surfactant, present in dilute concentrations. Use of lubrication theory yields a coupled pair of partial differential equations for the film height and surfactant monolayer concentration. The contact line singularity is relieved by assuming the presence of a thin precursor layer ahead of the advancing film. Base flow solutions for a flow of constant flux are examined over various inclination angle, precursor-layer thickness, Peclet number, and capillary parameter ranges. Application of a transient growth analysis highlights the presence of an instability and the vulnerability of the flow to transverse disturbances of intermediate wavenumber. Our results reveal that several key features of the much-studied uncontaminated film flow, including stability, are modified qualitatively by the inclusion of surfactant.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
A. Oron S.H. Davis S.G. Bankoff (1997) ArticleTitleLong-scale evolution of thin liquid films Rev. Modern Phys. 69 931–980 Occurrence Handle10.1103/RevModPhys.69.931 Occurrence Handle1997RvMP...69..931O
O.K. Matar S.M. Troian (1997) ArticleTitleLinear stability analysis of an insoluble surfactant monolayer spreading on a thin liquid film Fluids A 9 3645–3657 Occurrence Handle1997PhFl....9.3645M
O.K. Matar S.M. Troian (1999) ArticleTitleSpreading of surfactant monolayer on thin liquid film: Onset and evolution of digitated structures Chaos 9 141–153 Occurrence Handle10.1063/1.166385 Occurrence Handle1999Chaos...9..141M
O.K. Matar S.M. Troian (1999) ArticleTitleThe development of transient fingering patterns during the spreading of surfactant coated films Phys. Fluids 11 3232–3246 Occurrence Handle10.1063/1.870185 Occurrence Handle1999PhFl...11.3232M
H. Huppert (1982) ArticleTitleFlow and instability of a viscous current down a slope Nature (London) 300 427–429 Occurrence Handle10.1038/300427a0
N. Silva E.B. Dussan (1985) ArticleTitleOn the rewetting of an inclined solid surface by a liquid Phys. Fluids 28 5–7 Occurrence Handle1985PhFl...28....5S
S.M. Troian E. Herbolzheimer S.A. Safran J.F. Joanny (1989) ArticleTitleFingering instabilities of driven spreading films.Europhys Lett. 10 25–30
R. Goodwin G.M. Homsy (1991) ArticleTitleViscous flow down a slope in the vicinity of a contact line Phys. Fluids A 3 515–528 Occurrence Handle10.1063/1.858113 Occurrence Handle1991PhFl....3..515G
A.L Bertozzi M.P Brenner (1997) ArticleTitleLinear stability and transient growth in driven contact lines Phys. Fluids 9 530–539 Occurrence Handle10.1063/1.869217 Occurrence Handle1997PhFl....9..530B Occurrence Handle97i:76061
M.F.G Johnson R.A Schwartz M.J Miksis S.G Bankoff (1999) ArticleTitleExperimental study of rivulet formation on an inclined plate by fluorescent imaging J. Fluid Mech 394 339–354 Occurrence Handle10.1017/S0022112099005765 Occurrence Handle1999JFM...394..339J
Y Ye H-C Chang (1999) ArticleTitleA spectral theory for fingering on a prewetted plane Phys. Fluids 11 2494–2515 Occurrence Handle10.1063/1.870114 Occurrence Handle1999PhFl...11.2494Y Occurrence Handle2000d:76058
M.H Eres L.W. Schwartz R.V Roy (2000) ArticleTitleFingering phenomena for driven coating films Phys. Fluids 12 1278–1295 Occurrence Handle10.1063/1.870382 Occurrence Handle2000PhFl...12.1278E Occurrence Handle1763680
L Kondic J Diez (2001) ArticleTitlePattern formation in the flow of thin films down an incline: Constant flux configuration Phys.Fluids 13 3168–3184 Occurrence Handle10.1063/1.1409965 Occurrence Handle2001PhFl...13.3168K
L Kondic J Diez (2001) ArticleTitleFlow on films with patterned surfaces: Controlling the instability Phys. Rev E 65 1–4
L Kondic J Diez (2003) ArticleTitleFlow of thin films on patterned surfaces Colloids Surf. A 214 1–11 Occurrence Handle10.1016/S0927-7757(02)00387-4
L.W. Schwartz R.A Cairncross D.E Weidner (1996) ArticleTitleAnomalous behaviour during leveling of thin coating layers with surfactant Phys. Fluids 8 1693–1695 Occurrence Handle10.1063/1.868952 Occurrence Handle1996PhFl....8.1693S
M.S Borgas J.B Grotberg (1988) ArticleTitleMonolayer flow on a thin film J. Fluid Mech 193 151–170 Occurrence Handle1988JFM...193..151B
D.P Gaver III J.B Grotberg (1990) ArticleTitleThe dynamics of a localized surfactant on a thin film J. Fluid Mech 213 127–148 Occurrence Handle1990JFM...213..127G
D.P Gaver III J.B. Grotberg (1992) ArticleTitleDroplet spreading on a thin viscous film. J. Fluid Mech. 235 399–414 Occurrence Handle1992JFM...235..399G
O.E Jensen J.B Grotberg (1992) ArticleTitleInsoluble surfactant spreading on a thin viscous film: shock evolution and film rupture J. Fluid Mech 240 259–288 Occurrence Handle1992JFM...240..259J Occurrence Handle93f:76026
D Halpern J.B Grotberg (1992) ArticleTitleDynamics and transport of a localised soluble surfactant on a thin film J. Fluid Mech 237 1–11 Occurrence Handle1992JFM...237....1H
O.E Jensen J.B Grotberg (1993) ArticleTitleThe spreading of heat or soluble surfactant along a thin liquid film Phys. Fluids 5 58–68 Occurrence Handle1993PhFl....5...58J
O.E Jensen D Halpern J.B Grotberg (1994) ArticleTitleTransport of passive solute by surfactant-driven flows Chem. Engng.Sci 49 1107–1117 Occurrence Handle10.1016/0009-2509(94)85083-6
Y.L Zhang O.K Matar R.V Craster (2002) ArticleTitleA theoretical study of chemical delivery within the lung using exogenous surfactant Med. Engng. Phys 25 115–132
M.R.E Warner R.V Craster O.K Matar (2002) ArticleTitleUnstable van der Waals driven line rupture in Marangoni driven thin viscous films Phys. Fluids 14 1642–1654 Occurrence Handle10.1063/1.1511734 Occurrence Handle2002PhFl...14.1642W Occurrence Handle2003b:76015
M.R.E Warner R.V Craster O.K Matar (2004) ArticleTitleFingering phenomena associated with insoluble surfactant spreading on thin liquid films J. Fluid Mech 510 169–200 Occurrence Handle10.1017/S0022112004009437 Occurrence Handle2004JFM...510..169W
B.J Fischer S.M Troian (2003) ArticleTitleThinning and disturbance growth in liquid films mobilized by continuous surfactant delivery Phys. Fluids 15 3837–3845 Occurrence Handle10.1063/1.1623489 Occurrence Handle2003PhFl...15.3837F
B.J Fischer S.M. Troian (2003) ArticleTitleGrowth and decay of localized disturbances at the leading edge of a surfactant monolayer spreading on a thin viscous film Phys. Rev. E 67 1–11 Occurrence Handle10.1103/PhysRevE.67.016309
M.A Spaid G.M Homsy (1996) ArticleTitleStability of Newtonian and viscoelastic dynamic contact lines Phys. Fluids 8 460–478 Occurrence Handle10.1063/1.868800 Occurrence Handle1996PhFl....8..460S Occurrence Handle96j:76070
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Edmonstone, B.D., Matar, O.K. & Craster, R.V. Flow of surfactant-laden thin films down an inclined plane. J Eng Math 50, 141–156 (2004). https://doi.org/10.1007/s10665-004-3689-6
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/s10665-004-3689-6