Nonlinear evolution of thin free viscous films in the presence of soluble surfactant

The rupture of thin free viscous films is studied in the presence of soluble surfactant. In the limit of rapid surfactant bulk diffusion, higher-order long wavelength theory is used to derive a one-dimensional (1D) nonlinear model for the film thickness, tangential velocity, surfactant surface, and bulk concentrations, the latter being cross-sectionally averaged. For slow diffusion, an approximate (1D) model for the bulk concentration is derived; the predictions of this model in this limit are compared with those of the fully two-dimensional (2D) concentration model. Linear stability is investigated in detail for the 1D rapid diffusion model and numerical simulations of the 1D and 2D models for the symmetric (squeeze) mode are also conducted; this allows a parametric study of the nonlinear rupture time to be performed. Finally, self-similar scaling exponents for all flow variables as rupture is approached are extracted. Our results indicate that scaling exponents for rupture derived in the surfactant-free...

[1]  Eli Ruckenstein,et al.  Decay of standing foams: drainage, coalescence and collapse , 1997 .

[2]  B. U. Felderhof.  Dynamics of Free Liquid Films , 1968 .

[3]  Roy,et al.  Modeling Draining Flow in Mobile and Immobile Soap Films. , 1999, Journal of colloid and interface science.

[4]  Michael J. Miksis,et al.  The Dynamics of Thin Films II: Applications , 1998, SIAM J. Appl. Math..

[5]  M. Timmermans,et al.  The effect of surfactant on the stability of a liquid thread , 2002, Journal of Fluid Mechanics.

[6]  Sandra M. Troian,et al.  The development of transient fingering patterns during the spreading of surfactant coated films , 1999 .

[7]  A. Adamson.  Physical chemistry of surfaces , 1960 .

[8]  Richard F. Sincovec,et al.  Algorithm 540: PDECOL, General Collocation Software for Partial Differential Equations [D3] , 1979, TOMS.

[9]  Jean-Marc Chomaz,et al.  The dynamics of a viscous soap film with soluble surfactant , 2001, Journal of Fluid Mechanics.

[10]  Omar Matar,et al.  Surfactant transport on mucus films , 2000, Journal of Fluid Mechanics.

[11]  Andrew J. Bernoff,et al.  Dynamics of three-dimensional thin film rupture , 2000 .

[12]  Christo I. Christov,et al.  Nonlinear evolution equations for thin liquid films with insoluble surfactants , 1994 .

[13]  R. Craster,et al.  Pinchoff and satellite formation in surfactant covered viscous threads , 2002 .

[14]  R. Craster,et al.  Surfactant transport on highly viscous surface films , 2002, Journal of Fluid Mechanics.

[15]  Paul H. Muir,et al.  Algorithm 688: EPDCOL: a more efficient PDECOL code , 1991, TOMS.

[16]  J. Joosten.  Spectral analysis of light scattered by liquid films. I. General considerations , 1984 .

[17]  D. Gallez.  Non-linear stability analysis for animal cell adhesion to solid support , 1994 .

[18]  Mary C. Pugh,et al.  Long-wave instabilities and saturation in thin film equations , 1998 .

[19]  H. Stone.  A simple derivation of the time‐dependent convective‐diffusion equation for surfactant transport along a deforming interface , 1990 .

[20]  Howard Brenner,et al.  Interfacial transport processes and rheology , 1991 .

[21]  D. Hatziavramidis,et al.  STABILITY OF THIN EVAPORATING/COND ENSING FILMS IN THE PRESENCE OF SURFACTANTS , 1992 .

[22]  A. Sheludko,et al.  Thin liquid films , 1967 .

[23]  J. Berg,et al.  Surface Diffusion in Monolayers , 1969 .

[24]  M. P. Ida,et al.  The Dynamics of Thin Films I: General Theory , 1998, SIAM J. Appl. Math..

[25]  S. Naire,et al.  An insoluble surfactant model for a vertical draining free film with variable surface viscosity , 2001 .

[26]  John R. Lister,et al.  Similarity solutions for van der Waals rupture of a thin film on a solid substrate , 1999 .

[27]  P. Bisch,et al.  Long-Scale Evolution of Thin Liquid Films Bounded by a Viscous Phase with Diffusing Charged Surfactants , 2001 .

[28]  E. H. Lucassen-Reynders,et al.  Properties of capillary waves , 1970 .

[29]  H. Squire.  Investigation of the instability of a moving liquid film , 1953 .

[30]  N. Clark,et al.  Dynamics of freely suspended lyotropic films. I. An inelastic light scattering study of thermal surface fluctuations , 1981 .

[31]  Eli Ruckenstein,et al.  Spontaneous rupture of thin liquid films , 1974 .

[32]  M. Prévost,et al.  Nonlinear rupture of thin free liquid films , 1986 .

[33]  J. Eggers.  Nonlinear dynamics and breakup of free-surface flows , 1997 .

[34]  Symmetry and self-similarity in rupture and pinchoff: a geometric bifurcation , 2001, European Journal of Applied Mathematics.

[35]  Braun,et al.  An Insoluble Surfactant Model for a Vertical Draining Free Film. , 2000, Journal of colloid and interface science.

[36]  Andrew J. Bernoff,et al.  Stability of Self-similar Solutions for Van der Waals Driven Thin Film Rupture , 1999 .

[37]  Geoffrey Ingram Taylor,et al.  The dynamics of thin sheets of fluid II. Waves on fluid sheets , 1959, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[38]  Lopez,et al.  Surfactant-Influenced Gas-Liquid Interfaces: Nonlinear Equation of State and Finite Surface Viscosities. , 2000, Journal of colloid and interface science.

[39]  Lopez,et al.  Direct Determination of the Dependence of the Surface Shear and Dilatational Viscosities on the Thermodynamic State of the Interface: Theoretical Foundations. , 1998, Journal of colloid and interface science.

[40]  Oliver E. Jensen,et al.  The spreading of heat or soluble surfactant along a thin liquid film , 1993 .

[41]  D. Gallez,et al.  Pattern formation in thin liquid films with insoluble surfactants , 1998 .

[42]  P. Bisch,et al.  Nonlinear dynamics and rupture of lipid bilayers , 1993 .

[43]  John R. Lister,et al.  Rupture of thin viscous films by van der Waals forces: Evolution and self-similarity , 2001 .

[44]  Hwang,et al.  A Nonlinear Rupture Theory of Thin Liquid Films with Soluble Surfactant. , 2000, Journal of colloid and interface science.

[45]  Ashutosh Sharma,et al.  An Analytical Nonlinear Theory of Thin Film Rupture and Its Application to Wetting Films , 1986 .

[46]  R. Craster,et al.  Unstable van der Waals driven line rupture in Marangoni driven thin viscous films , 2002 .

[47]  Paul A. Zegeling,et al.  Algorithm 731: A moving-grid interface for systems of one-dimensional time-dependent partial differential equations , 1994, TOMS.

[48]  Stephen H. Davis,et al.  Nonlinear theory of film rupture , 1982 .

[49]  S. G. Bankoff,et al.  Nonlinear stability of evaporating/condensing liquid films , 1988, Journal of Fluid Mechanics.

[50]  S. Bankoff,et al.  Long-scale evolution of thin liquid films , 1997 .

[51]  R. K. Jain,et al.  Stability of stagnant viscous films on a solid surface , 1976 .

[52]  Oliver E. Jensen,et al.  Insoluble surfactant spreading on a thin viscous film: shock evolution and film rupture , 1992, Journal of Fluid Mechanics.

[53]  R. Craster,et al.  Models for Marangoni drying , 2001 .

[54]  Thomas Erneux,et al.  Nonlinear rupture of free films , 1993 .

[55]  Wu,et al.  External dissipation in driven two-dimensional turbulence , 2000, Physical review letters.

[56]  R. J. Braun,et al.  Limiting Cases of Gravitational Drainage of a Vertical Free Film for Evaluating Surfactants , 2000, SIAM J. Appl. Math..