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

Published online by Cambridge University Press:  09 March 2017

DINA SINCLAIR ,
RACHEL LEVY  and
KAREN E. DANIELS
DINA SINCLAIR
Affiliation:
Mathematics Department, Harvey Mudd College, Claremont, CA, USA emails: [email protected], [email protected]
RACHEL LEVY
Affiliation:
Mathematics Department, Harvey Mudd College, Claremont, CA, USA emails: [email protected], [email protected]
KAREN E. DANIELS
Affiliation:
Department of Physics, North Carolina State University, Raleigh, NCUSA email: [email protected]
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Abstract

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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.

Keywords

AMS classifications: 35Q35 PDEs in connection with fluid mechanics76A20 thin fluid films76M12 finite volume methods76B45 capillarity (surface tension)
Type
Papers
Information
European Journal of Applied Mathematics , Volume 29 , Issue 1 , February 2018 , pp. 30 - 54
Copyright
Copyright © Cambridge University Press 2017 

Footnotes

†

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).

References

[1] Avanti Polar Lipids. www.avantilipids.com/index.php?option=com_content&view=article&id=1028&Itemid=220&catnumber=810131 Google Scholar
[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), 18109–18113.Google Scholar
[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), 1427–1464.Google Scholar
[4] Borgas, M. S. & Grotberg, J. B. (1988) Monolayer flow on a thin film. J. Fluid Mech. 193, 151–170.CrossRefGoogle Scholar
[5] Braun, R. J. (2012) Dynamics of the tear film. Annu. Rev. Fluid Mech. 44, 267–297.Google Scholar
[6] Bull, J. & Grotberg, J. (2003) Surfactant spreading on thin viscous films: Film thickness evolution and periodic wall stretch. Exp. Fluids 34 (1), 1–15.CrossRefGoogle Scholar
[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, 81–93.Google Scholar
[8] Craster, R. & Matar, O. (2000) Surfactant transport on mucus films. J. Fluid Mech. 425, 235–258.CrossRefGoogle Scholar
[9] Craster, R. & Matar, O. (2009) Dynamics and stability of thin liquid films. Rev. Mod. Phys. 81 (3), 1131.CrossRefGoogle Scholar
[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), 3256–3266.CrossRefGoogle Scholar
[11] Espinosa, F., Shapiro, A., Fredberg, J. & Kamm, R. (1993) Spreading of exogenous surfactant in an airway. J. Appl. Phys. 75 (5), 2028–2039.Google Scholar
[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.CrossRefGoogle Scholar
[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), 2025–2048.CrossRefGoogle Scholar
[14] Gaver, D. P. & Grotberg, J. B. (1990) The dynamics of a localized surfactant on a thin film. J. Fluid Mech. 213, 127–148.Google Scholar
[15] Gaver, D. P. & Grotberg, J. B. (1992) Droplet spreading on a thin viscous film. J. Fluid Mech. 235, 399–414.Google Scholar
[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), 271–277.Google Scholar
[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), 295–307.CrossRefGoogle ScholarPubMed
[18] Claridge, J., Levy, R. & Wong, J. https://github.com/claridge/implicit_solvers/blob/master/ImplicitSolversPaper.pdf Google Scholar
[19] Claridge, J., Levy, R. & Wong, J. https://github.com/claridge/implicit_solvers/ Google Scholar
[20] Jensen, O. (1994) Self-similar, surfactant-driven flows. Phys. Fluids (1994-present) 6 (3), 1084–1094.Google Scholar
[21] Jensen, O. & Grotberg, J. (1992) Insoluble surfactant spreading on a thin viscous film: Shock evolution and film rupture. J. Fluid Mech. 240, 259–288.Google Scholar
[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), 58–68.CrossRefGoogle Scholar
[23] Kaganer, V., Möhwald, H. & Dutta, P. (April 1999) Structure and phase transitions in Langmuir monolayers. Rev. Mod. Phys. 71 (3), 779–819.Google Scholar
[24] Karsa, D. (1999) Industrial Applications of Surfactants IV. Special Publication/Royal Society of Chemistry. Vol. 230, Elsevier Science.Google Scholar
[25] Levy, R. (2005) Partial differential equations of thin liquid films: Analysis and numerical simulation, thesis, North Carolina State University.Google Scholar
[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), 985–1000.Google Scholar
[27] Levy, R. & Shearer, M. (2006) The motion of a thin liquid film driven by surfactant and gravity. SIAM J. Appl. Math. 66 (5), 1588–1609.CrossRefGoogle Scholar
[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), 679–708.CrossRefGoogle Scholar
[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), 385–410.Google Scholar
[30] Oron, A., Davis, S. H. & Bankoff, S. G. (1997) Long-scale evolution of thin liquid films. Rev. Mod. Phys. 69 (3), 931.Google Scholar
[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), 1323–1333.Google Scholar
[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.Google Scholar
[33] Peterson, E. R. (2010) Flow of thin liquid films with surfactant: Analysis, numerics, and experiment.Google Scholar
[34] Peterson, E. R. & Shearer, M. (2011) Radial spreading of a surfactant on a thin liquid film. Appl. Math. Res. Express 2011 (1), 1–22.Google Scholar
[35] Reis, P., Holmberg, K., Watzke, H., Leser, M. E. & Miller, R. (2009) Lipases at interfaces: A review. Adv. Colloid Interface Sci. 147–148, 237–250.Google Scholar
[36] Renardy, M. (1996) A singularly perturbed problem related to surfactant spreading on thin films. Nonlinear Anal.: Theory, Methods Appl. 27 (3) (1996), 287–296.Google Scholar
[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, 304–313.Google Scholar
[38] Strickland, S. L. Surfactant dynamics: Spreading and wave induced dynamics of a monolayer.Google Scholar
[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.CrossRefGoogle Scholar
[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, 63–79.Google Scholar
[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), 2301–2306.Google Scholar
[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, 1496–1499.CrossRefGoogle ScholarPubMed
[43] Troian, S. M., Herbolzheimer, E. & Safran, S. A. (1990) Model for the fingering instability of spreading surfactant drops. Phys. Rev. Lett. 65, 333–336.Google Scholar
[44] Warner, M., Craster, R. & Matar, O. (2004) Fingering phenomena associated with insoluble surfactant spreading on thin liquid films. J. Fluid Mech. 510, 169–200.Google Scholar
[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.Google Scholar