Skip to main content
×
×
Home

Simulating surfactant spreading: Influence of a physically motivated equation of state

  • DINA SINCLAIR (a1), RACHEL LEVY (a1) and KAREN E. DANIELS (a2)
Abstract

In this paper, we present numerical simulations that demonstrate the effect of the particular choice of the equation of state (EoS) relating the surfactant concentration to the surface tension in surfactant-driven thin liquid films. Previous choices of the model EoS have been an ad-hoc decreasing function. Here, we instead propose an empirically motivated EoS; this provides a route to resolve some discrepancies and raises new issues to be pursued in future experiments. In addition, we test the influence of the choice of initial conditions and values for the non-dimensional groups. We demonstrate that the choice of EoS improves the agreement in surfactant distribution morphology between simulations and experiments, and influences the dynamics of the simulations. Because an empirically motivated EoS has regions with distinct gradients, future mathematical models may be improved by considering more than one timescale. We observe that the non-dimensional number controlling the relative importance of gravitational versus capillary forces has a larger influence on the dynamics than the other non-dimensional groups, but is nonetheless not a likely cause of discrepancy between simulations and experiments. Finally, we observe that the experimental approach using a ring to contain the surfactant could affect the surfactant and fluid dynamics if it disrupts the intended initial surfactant distribution. However, the fluid meniscus itself does not significantly affect the dynamics.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      Simulating surfactant spreading: Influence of a physically motivated equation of state
      Available formats
      ×
      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about sending content to Dropbox.

      Simulating surfactant spreading: Influence of a physically motivated equation of state
      Available formats
      ×
      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about sending content to Google Drive.

      Simulating surfactant spreading: Influence of a physically motivated equation of state
      Available formats
      ×
Copyright
Footnotes
Hide All

This work was funded by NSF grant DMS-FRG #096815 (RL and KED), Howard Hughes Medical Institute Undergraduate Science Education Program Award #52006301 (RL), and Research Corporation Cottrell Scholar Award #19788 (RL).

Footnotes
References
Hide All
[2] Angelini T. E., Roper M., Kolter R., Weitz D. A. & Brenner M. P. (2009) Bacillus subtilis spreads by surfing on waves of surfactant. Proc. Natl. Acad. Sci. 106 (43), 1810918113.
[3] Barrett J. W., Garcke H. & Nürnberg R. (2003) Finite element approximation of surfactant spreading on a thin film. SIAM J. Numer. Anal. 41 (4), 14271464.
[4] Borgas M. S. & Grotberg J. B. (1988) Monolayer flow on a thin film. J. Fluid Mech. 193, 151170.
[5] Braun R. J. (2012) Dynamics of the tear film. Annu. Rev. Fluid Mech. 44, 267297.
[6] Bull J. & Grotberg J. (2003) Surfactant spreading on thin viscous films: Film thickness evolution and periodic wall stretch. Exp. Fluids 34 (1), 115.
[7] Conti C., Autry E. A., Kronmiller G. & Levy R. (2013) The effects of spatial and temporal grids on simulations of thin films with surfactant. SIURO 916, 8193.
[8] Craster R. & Matar O. (2000) Surfactant transport on mucus films. J. Fluid Mech. 425, 235258.
[9] Craster R. & Matar O. (2009) Dynamics and stability of thin liquid films. Rev. Mod. Phys. 81 (3), 1131.
[10] De Wit A., Gallez D. & Christov C. (1994) Nonlinear evolution equations for thin liquid films with insoluble surfactants. Phys. Fluids (1994-present) 6 (10), 32563266.
[11] Espinosa F., Shapiro A., Fredberg J. & Kamm R. (1993) Spreading of exogenous surfactant in an airway. J. Appl. Phys. 75 (5), 20282039.
[12] Fallest D. W., Lichtenberger A. M., Fox C. J. & Daniels K. E. (July 2010) Fluorescent visualization of a spreading surfactant. New J. Physics 12 (7), 73029.
[13] Garcke H. & Wieland S. (2006) Surfactant spreading on thin viscous films: Nonnegative solutions of a coupled degenerate system. SIAM J. Math. Anal. 37 (6), 20252048.
[14] Gaver D. P. & Grotberg J. B. (1990) The dynamics of a localized surfactant on a thin film. J. Fluid Mech. 213, 127148.
[15] Gaver D. P. & Grotberg J. B. (1992) Droplet spreading on a thin viscous film. J. Fluid Mech. 235, 399414.
[16] Halpern D. & Grotberg J. (1993) Surfactant effects on fluid-elastic instabilities of liquid-lined flexible tubes: A model of airway closure. J. Biomech. Eng. 115 (3), 271277.
[17] Heidari A. H., Braun R. J., Hirsa A. H., Snow S. A. & Naire S. (2002) Hydrodynamics of a bounded vertical film with nonlinear surface properties. J. Colloid Interface Sci. 253 (2), 295307.
[19] Claridge J., Levy R. & Wong J. https://github.com/claridge/implicit_solvers/
[20] Jensen O. (1994) Self-similar, surfactant-driven flows. Phys. Fluids (1994-present) 6 (3), 10841094.
[21] Jensen O. & Grotberg J. (1992) Insoluble surfactant spreading on a thin viscous film: Shock evolution and film rupture. J. Fluid Mech. 240, 259288.
[22] Jensen O. & Grotberg J. (1993) The spreading of heat or soluble surfactant along a thin liquid film. Phys. Fluids A: Fluid Dyn. (1989–1993) 5 (1), 5868.
[23] Kaganer V., Möhwald H. & Dutta P. (April 1999) Structure and phase transitions in Langmuir monolayers. Rev. Mod. Phys. 71 (3), 779819.
[24] Karsa D. (1999) Industrial Applications of Surfactants IV. Special Publication/Royal Society of Chemistry. Vol. 230, Elsevier Science.
[25] Levy R. (2005) Partial differential equations of thin liquid films: Analysis and numerical simulation, thesis, North Carolina State University.
[26] Levy R., Hill D. B., Forest M. G. & Grotberg J. B. (2014) Pulmonary fluid flow challenges for experimental and mathematical modeling. Integrative Comparative Biol. 54 (6), 9851000.
[27] Levy R. & Shearer M. (2006) The motion of a thin liquid film driven by surfactant and gravity. SIAM J. Appl. Math. 66 (5), 15881609.
[28] Levy R., Shearer M. & Witelski T. P. (2007) Gravity-driven thin liquid films with insoluble surfactant: Smooth traveling waves. Eur. J. Appl. Math. 18 (06), 679708.
[29] Naire S., Braun R. J. & Snow S. A. (2004) A 2+1 dimensional insoluble surfactant model for a vertical draining free film. J. Comput. Appl. Math. 166 (2), 385410.
[30] Oron A., Davis S. H. & Bankoff S. G. (1997) Long-scale evolution of thin liquid films. Rev. Mod. Phys. 69 (3), 931.
[31] Otis D., Johnson M., Pedley T. & Kamm R. (1993) Role of pulmonary surfactant in airway closure: A computational study. J. Appl. Physiology 75 (3), 13231333.
[32] Pereira A., Trevelyan P., Thiele U. & Kalliadasis S. (2007) Dynamics of a horizontal thin liquid film in the presence of reactive surfactants. Phys. Fluids (1994-present) 19 (11), 112102.
[33] Peterson E. R. (2010) Flow of thin liquid films with surfactant: Analysis, numerics, and experiment.
[34] Peterson E. R. & Shearer M. (2011) Radial spreading of a surfactant on a thin liquid film. Appl. Math. Res. Express 2011 (1), 122.
[35] Reis P., Holmberg K., Watzke H., Leser, M. E. & Miller R. (2009) Lipases at interfaces: A review. Adv. Colloid Interface Sci. 147–148, 237250.
[36] Renardy M. (1996) A singularly perturbed problem related to surfactant spreading on thin films. Nonlinear Anal.: Theory, Methods Appl. 27 (3) (1996), 287296.
[37] Shrive J. D. A., Brennan J. D., Brown R. S. & Krull U. J. (1995) Optimization of self-quenching response of nitrobenzoxadiazole dipalmitoylphosphatidylethanolaminein phospholipid membranes for biosensor development. Appl. Spectrosc. 49, 304313.
[38] Strickland S. L. Surfactant dynamics: Spreading and wave induced dynamics of a monolayer.
[39] Strickland S. L., Hin M., Sayanagi M. R., Gaebler C., Daniels K. E., Levy R. & Conti C. (April 2014) Self-healing dynamics of surfactant coatings on thin viscous films. Phys. Fluids 26 (4), 042109.
[40] Swanson E. R., Strickland S. L., Shearer M. & Daniels K. E. (2015) Surfactant spreading on a thin liquid film: Reconciling models and experiments. J. Eng. Math. 94, 6379.
[41] Tiberg F. & Cazabat A.-M. (1994) Spreading of thin films of ordered nonionic surfactants – Origin of the stepped shape of the spreading precursor. Langmuir 10 (7), 23012306.
[42] Troian S. M., Herbolzheimer E., Safran S. A., Joanny J. F., Wu X. L. & Safran S. A. (1989) Fingering instability in thin wetting films. Phys. Rev. Lett. 62, 14961499.
[43] Troian S. M., Herbolzheimer E. & Safran S. A. (1990) Model for the fingering instability of spreading surfactant drops. Phys. Rev. Lett. 65, 333336.
[44] Warner M., Craster R. & Matar O. (2004) Fingering phenomena associated with insoluble surfactant spreading on thin liquid films. J. Fluid Mech. 510, 169200.
[45] Witelski T. P., Shearer M. & Levy R. (2006) Growing surfactant waves in thin liquid films driven by gravity. Appl. Math. Res. Express 2006, 15487.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

European Journal of Applied Mathematics
  • ISSN: 0956-7925
  • EISSN: 1469-4425
  • URL: /core/journals/european-journal-of-applied-mathematics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Keywords:

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 58 *
Loading metrics...

Abstract views

Total abstract views: 179 *
Loading metrics...

* Views captured on Cambridge Core between 9th March 2017 - 17th December 2017. This data will be updated every 24 hours.