Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-25T13:35:46.708Z Has data issue: false hasContentIssue false
  • English
  • Français

On Diffuse Interface Modeling and Simulation of Surfactants in Two-Phase Fluid Flow

Published online by Cambridge University Press:  03 June 2015

Stefan Engblom ,
Minh Do-Quang ,
Gustav Amberg  and
Anna-Karin Tornberg
Stefan Engblom*
Affiliation:
Division of Scientific Computing, Department of Information Technology, Uppsala University, SE-751 05 Uppsala, Sweden
Minh Do-Quang*
Affiliation:
Linné Flow Centre, Department of Mechanics, School of Engineering Science, Royal Institute of Technology, S-100 44 Stockholm, Sweden
Gustav Amberg*
Affiliation:
Linné Flow Centre, Department of Mechanics, School of Engineering Science, Royal Institute of Technology, S-100 44 Stockholm, Sweden
Anna-Karin Tornberg*
Affiliation:
Linné Flow Centre, Department of Mechanics, School of Engineering Science, Royal Institute of Technology, S-100 44 Stockholm, Sweden
*
Corresponding author.Email:[email protected]
Email:[email protected]
Email:[email protected]
Email:[email protected]
Get access

Abstract

An existing phase-fieldmodel of two immiscible fluids with a single soluble surfactant present is discussed in detail. We analyze the well-posedness of the model and provide strong evidence that it is mathematically ill-posed for a large set of physically relevant parameters. As a consequence, critical modifications to the model are suggested that substantially increase the domain of validity. Carefully designed numerical simulations offer informative demonstrations as to the sharpness of our theoretical results and the qualities of the physical model. A fully coupled hydrodynamic test-case demonstrates the potential to capture also non-trivial effects on the overall flow.

Keywords

76T3065M6065Z05Phase-field modelCahn-Hilliard equationsurface active agentGinzburg-Landau free energywell-posedness
Type
Research Article
Information
Communications in Computational Physics , Volume 14 , Issue 4 , October 2013 , pp. 879 - 915
Copyright
Copyright © Global Science Press Limited 2013

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

[1]Ariel, G., Diamant, H. and Andelman, D., Kinetics of surfactant adsorption at fluid-fluid interfaces: Surfactant mixtures, Langmuir, 15(10) (1999), 3574–3581.CrossRefGoogle Scholar
[2]Barrett, J. W., Blowey, J. F. and Garcke, H., Finite element approximation of the Cahn-Hilliard equation with degenerate mobility, SIAM J. Num. Anal., 37(1) (1999), 286–318.Google Scholar
[3]Bazhlekov, I. B., Anderson, P. D. and Meijer, H. E. H., Numerical investigation of the effect of insoluble surfactants on drop deformation and breakup in simple shear flow, J. Coll. Interface Sci., 298(1) (2006), 369–394.Google Scholar
[4]Bertozzi, A. L., Ju, N. and Lu, H.-W., A biharmonic-modified forward time stepping method for fourth order nonlinear diffusion equations, Discrete Contin. Dyn. Syst., 29(4) (2011), 1367–1391.CrossRefGoogle Scholar
[5]Booty, M. R. and Siegel, M., A hybrid numerical method for interfacial fluid flow with soluble surfactant, J. Comput. Phys., 229(10) (2010), 3864–3883.CrossRefGoogle Scholar
[6]Copetti, M. I. M. and Elliott, C. M., Numerical analysis of the Cahn-Hilliard equation with a logarithmic free energy, Numer. Math., 63(1) (1992), 39–65.CrossRefGoogle Scholar
[7]Cueto-Felgueroso, L. and Peraire, J., A time-adaptive finite volume method for the Cahn-Hilliard and Kuramoto-Sivashinsky equations. J. Comput. Phys., 227(24) (2008), 9985–10017.Google Scholar
[8]Davis, T. A., Algorithm 832: UMFPACK V4.3-an unsymmetric-pattern multifrontal method, ACM Trans. Math. Software, 30(2) (2004), 196–199.CrossRefGoogle Scholar
[9]Diamant, H. and Andelman, D., Kinetics of surfactant adsorption at fluid-fluid interfaces, J. Phys. Chem., 100(32) (1996), 13732–13742.CrossRefGoogle Scholar
[10]Diamant, H., Ariel, G. and Andelman, D., Kinetics of surfactant adsorption: the free energy approach. Coll. Surf. A: Physicochem. Eng. Aspects, 183–185 (2001), 259-276.Google Scholar
[11]Ding, H., Spelt, P. D. M. and Shu, C., Diffuse interface model for incompressible two-phase flows with large density ratios, J. Comput. Phys., 226(2) (2007), 2078–2095.CrossRefGoogle Scholar
[12]Do-Quang, M., Villanueva, W., Singer-Loginova, I. and Amberg, G., Parallel adaptive computation of some time-dependent materials-related microstructural problems, Bull. Pol. Ac. Tech., 55(2) (2007), 229–237.Google Scholar
[13]Eggleton, C. D. and Stebe, K. J., An adsorption-desorption-controlled surfactant on a deforming droplet, J. Coll. Interface Sci., 208(1) (1998), 68–80.Google Scholar
[14]Elliott, C. M. and Garcke, H., Diffusional phase transitions in multicomponent systems with a concentration dependent mobility matrix, Phys. D, 109(3-4) (1997), 242–256.Google Scholar
[15]Eyre, D. J., An unconditionally stable one-step scheme for gradient systems, http: //www.math.utah.edu/∼eyre/research/methods/stable.ps, 1998.Google Scholar
[16]Eyre, D. J., Unconditionally gradient stable time marching the Cahn-Hilliard equation, In Computational and Mathematical Models of Microstructure Evolution, volume 529, pages 39–46, Warrendale, PA, 1998. Materials Research Society.Google Scholar
[17]Greenbaum, A., Iterative Methods for Solving Linear Systems, Number 17 in Frontiers in Applied Mathematics. SIAM, Philadelphia, PA, 1997.Google Scholar
[18]Guermond, J.-L. and Quartapelle, L., A projection FEM for variable density incompressible flows, J. Comput. Phys., 165(1) (2000), 167–188.CrossRefGoogle Scholar
[19]Gustafsson, B., Kreiss, H.-O. and Oliger, J., Time Dependent Problems and Difference Methods, Pure and Applied Mathematics, John Wiley & Sons, New York, 1995.Google Scholar
[20]Gustafsson, K. and Soöderlind, G., Control strategies for the iterative solution of nonlinear equations in ODE solvers, SIAM J. Sci. Comput., 18(1) (1997), 23–40.Google Scholar
[21]Hairer, E. and Wanner, G., Solving Ordinary Differential Equations II, Stiff and Differential-Algebraic Problems, Springer, Berlin, 2nd edition, 1996.Google Scholar
[22]Halpern, D., Jensen, O. E. and Grotberg, J. B., A theoretical study of surfactant and liquid delivery into the lung, J. Appl. Physiol., 85(1) (1998), 333–352.CrossRefGoogle ScholarPubMed
[23]Hansen, R. S., The theory of diffusion controlled absorption kinetics with accompanying evaporation, J. Phys. Chem., 64(5) (1960), 637–641.CrossRefGoogle Scholar
[24]Hesthaven, J. S. and Kirby, R. M., Filtering in Legendre spectral methods, Math. Comput., 77(263) (2008), 1425–1452.CrossRefGoogle Scholar
[25]Hesthaven, J. S., Gottlieb, S. and Gottlieb, D., Spectral Methods for Time-Dependent Problems, Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, Cambridge, 2007.Google Scholar
[26]Hills, B. A., An alternative view of the role(s) of surfactant and the alveolar model, J. Appl. Physiol., 87(5) (1999), 1567–1583.Google Scholar
[27]Jacqmin, D., Calculation of two-phase Navier-Stokes flows using phase-field modeling, J. Comput. Phys., 155(1) (1999), 96–127.Google Scholar
[28]James, A. J. and Lowengrub, J., A surfactant-conserving volume-of-fluid method for interfacial flows with insoluble surfactant, J. Comput. Phys., 201(2) (2004), 685–722.Google Scholar
[29]Kay, D., Styles, V. and Welford, R., Finite element approximation of a Cahn-Hilliard-Navier-Stokes system, Interfaces Free Bound., 10 (1) (2008), 15–43.Google Scholar
[30]Khatri, S., A Numerical Method for Two Phase Flows with Insoluble and Soluble Surfactants, PhD thesis, Courant Institute of Mathematical Sciences, New York University, 2009.Google Scholar
[31]Khatri, S. and Tornberg, A.-K., A numerical method for two phase flows with insoluble surfactants, Comput. Fluids, 49(1) (2011), 150–165.Google Scholar
[32]Lagzi, I., Soh, S., Wesson, P. J., Browne, K. P. and Grzybowski, B. A., Maze solving by chemotactic droplets, J. Am. Chem. Soc., 132(4) (2010), 1198–1199.Google Scholar
[33]Lai, M.-C., Tseng, Y.-H. and Huang, H., An immersed boundary method for interfacial flows with insoluble surfactant, J. Comput. Phys., 227(15) (2008), 7279–7293.Google Scholar
[34]Lamura, A., Gonnella, G. and Yeomans, J. M., A lattice Boltzmann model of ternary fluid mixtures, Euro Phys. Lett. EPL, 45(3) (1999), 314–320.CrossRefGoogle Scholar
[35]Laradji, M., Guo, H., Grant, M. and Zuckermann, M. J., The effect of surfactants on the dynamics of phase separation, J. Phys. Condens. Matter, 4(32) (1992), 6715–6728.CrossRefGoogle Scholar
[36]Liu, H. and Zhang, Y., Phase-field modeling droplet dynamics with soluble surfactants, J. Comput. Phys., 229(24) (2010), 9166–9187.Google Scholar
[37]de Mello, E. V. L. and da Silveira Filho, O. T., Numerical study of the Cahn-Hilliard equation in one, two and three dimensions, Phys. A, 347 (2005), 429–443.Google Scholar
[38]Mohrbach, H., Kinetics of nonionic surfactant adsorption at a fluid-fluid interface from a micellar solution, J. Chem. Phys., 123(12) (2005), 1–4.CrossRefGoogle Scholar
[39]Muradoglu, M. and Tryggvason, G., A front-tracking method for computation of interfacial flows with soluble surfactants, J. Comput. Phys., 227(4) (2008), 2238–2262.Google Scholar
[40] National Institute of Standards and Technology, Digital Library of Mathematical Functions, Release date 2010-05-07. http://dlmf.nist.gov.Google Scholar
[41]Novick-Cohen, A., Chapter 4: The Cah.n-Hilliard equation, In Dafermos, C. M. and Pokornyý, M., editors, Handbook of Differential Equations: Evolutionary Equations, volume IV, pages 201–228. North-Holland, Amsterdam, 2008.Google Scholar
[42]Pawar, Y. and Stebe, K. J., Marangoni effects on drop deformation in an extensional flow: the role of surfactant physical chemistry. I. insoluble surfactants, Phys. Fluids, 8(7) (1996), 1738–1751.Google Scholar
[43]Shen, J. and Yang, X., Numerical approximations of Allen-Cahn and Cahn-Hilliard equations, Discrete Contin. Dyn. Syst., 28(4) (2010), 1669–1691.Google Scholar
[44]van der Sman, R. G. M. and der Graaf, S. van, Diffuse interface model of surfactant adsorption onto flat and droplet interfaces, Rheol. Acta, 46(1) (2006), 3–11.Google Scholar
[45]Söderlind, G., Digital filters in adaptive time-stepping, ACM Trans. Math. Software, 29(1) (2003), 1–26.CrossRefGoogle Scholar
[46]Söderlind, G. and Wang, L., Adaptive time-stepping and computational stability, J. Comput. Appl. Math., 185(2) (2006), 225–243.Google Scholar
[47]Stone, H. A., A simple derivation of the time-dependent convective-diffusion equation for surfactant transport along a deforming interface, Phys. Fluids A, 2(1) (1990), 111–112.Google Scholar
[48]Stuart, A. M. and Humphries, A. R., Model problems in numerical stability theory for initial value problems, SIAM Rev., 36(2) (1994), 226–257.Google Scholar
[49]Stuart, A. M. and Humphries, A. R., Dynamic Systems and Numerical Analysis, Number 2 in Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, Cambridge, 1998.Google Scholar
[50]Teigen, K. E., Song, P., Lowengrub, J. and Voigt, A., A diffuse-interface method for two-phase flows with soluble surfactants, J. Comput. Phys., 230(2) (2011), 375–393.CrossRefGoogle ScholarPubMed
[51]Teramoto, T. and Yonezawa, F., Droplet growth dynamics in a water/oil/surfactant system, J. Coll. Interface Sci., 235(2) (2001), 329–333.Google Scholar
[52]Theissen, O. and Gompper, G., Lattice-Boltzmann study of spontaneous emulsification, Eur. Phys. J. B, 11(1) (1999), 91–100.CrossRefGoogle Scholar
[53]Ward, A. F. H. and Tordai, L., Time-dependence of boundary tensions of solutions I: The role of diffusion in time-effects, J. Chem. Phys., 14(7) (1946), 453–461.CrossRefGoogle Scholar
[54]Witelski, T. P., Bernoff, A. J. and Bertozzi, A. L., Blowup and dissipation in a critical-case unstable thin film equation, European J. Appl. Math., 15(2) (2004), 223–256.Google Scholar
[55]Xu, J.-J., Li, Z., Lowengrub, J. and Zhao, H., A level-set method for interfacial flows with surfactant, J. Comput. Phys., 212(2) (2006), 590–616.CrossRefGoogle Scholar
[56]Ye, X., The Legendre collocation method for the Cahn-Hilliard equation, J. Comput. Appl. Math., 150(1) (2003), 87–108.CrossRefGoogle Scholar
[57]Zhang, J., Eckmann, D. M. and Ayyaswamy, P. S., A front tracking method for a deformable intravascular bubble in a tube with soluble surfactant transport, J. Comput. Phys., 214(1) (2006), 366–396.CrossRefGoogle Scholar
[58]Zhang, S. and Wang, M., A nonconforming finite element method for the Cahn-Hilliard equation, J. Comput. Phys., 229(19) (2010), 7361–7372.Google Scholar