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Numerical Study of Surfactant-Laden Drop-Drop Interactions

Published online by Cambridge University Press:  20 August 2015

Jian-Jun Xu ,
Zhilin Li ,
John Lowengrub  and
Hongkai Zhao
Jian-Jun Xu*
Affiliation:
School of Mathematical and Computational Sciences, Xiangtan University, Xiangtan, Hunan 411105, China
Zhilin Li*
Affiliation:
Department of Mathematics, North Carolina State University, Raleigh, NC, 27695, USA
John Lowengrub*
Affiliation:
Department of Mathematics, University of California Irvine, Irvine, CA, 92697, USA
Hongkai Zhao*
Affiliation:
Department of Mathematics, University of California Irvine, Irvine, CA, 92697, USA
*
Corresponding author.Email:jjxu21@xtu.edu.cn
Email:zhilin@math.ncsu.edu
Email:lowengrb@math.uci.edu
Email:zhao@math.uci.edu
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Abstract

In this paper, we numerically investigate the effects of surfactant on drop-drop interactions in a 2D shear flow using a coupled level-set and immersed interface approach proposed in (Xu et al., J. Comput. Phys., 212 (2006), 590-616). We find that surfactant plays a critical and nontrivial role in drop-drop interactions. In particular, we find that the minimum distance between the drops is a non-monotone function of the surfactant coverage and Capillary number. This non-monotonic behavior, which does not occur for clean drops, is found to be due to the presence of Marangoni forces along the drop interfaces. This suggests that there are non-monotonic conditions for coalescence of surfactant-laden drops, as observed in recent experiments of Leal and co-workers. Although our study is two-dimensional, we believe that drop-drop interactions in three-dimensional flows should be qualitatively similar as the Maragoni forces in the near contact region in 3D should have a similar effect.

Keywords

47.55.D-47.55.dk47.55.N-Surfactantsdropscoalescencesurface tensionMarangoni effectlevel-set methodimmersed interface methodStokes flow

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Type
Research Article
Information
Communications in Computational Physics , Volume 10 , Issue 2 , August 2011 , pp. 453 - 473
Copyright
Copyright © Global Science Press Limited 2011

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References

[1] Adami, S., Hu, X. and Adams, N., A conservative SPH method for surfactant dynamics, J. Comput. Phys., 229 (2010), 1909–1926.Google Scholar
[2] 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. Int. Sci., 298 (2006), 369.CrossRefGoogle ScholarPubMed
[3] Chesters, A., The modelling of coalescence processes in fluid liquid dispersions–a review of currrent understanding, Trans. Inst. Chem. Eng., 69 (1991), 259.Google Scholar
[4] Chesters, A. K. and Bazhlekov, I. B., Effect of insoluble surfactants on drainage and rupture of a film between drops interacting under a constant force, J. Coll. Int. Sci., 230 (2000), 229.CrossRefGoogle Scholar
[5] Cristini, V., Blawzdziewicz, J. and Loewenberg, M., Near-contact motion of surfactant-covered shperical drops, J. Fluid. Mech., 11 (1998), 251.Google Scholar
[6] Cristini, V., Blawzdziewicz, J. and Loewenberg, M., An adaptive mesh algorithm for evolving surfaces: simulations of drop breakup and coalescence, J. Comput. Phys., 168 (2001), 445–463. CrossRefGoogle Scholar
[7] Dai, B. and Leal, L. G., Mechanism of surfactant effects on drop coalescence, Phys. Fluids., 19 (2008), 023102.Google Scholar
[8] Drumwright-Clarke, M. A. and Renardy, Y., The effect of insoluble surfactant at dilute concentration on drop breakup under shear with inertia, Phys. Fluids., 16 (2004), 14–21.Google Scholar
[9] Eggleton, C. D., Tsai, T.-M. and Stebe, K. J., Tip streaming from a drop in the presence of surfactants, Phys. Rev. Lett., 87 (2001), 048302.Google Scholar
[10] Ganesan, S. and Tobiska, L., A coupled arbitrary Lagrangian-Eulerian and Lagrangian method for computation of free surface flows with insoluble surfactants, J. Comput. Phys., 228 (2009), 2859–2873.Google Scholar
[11] Hameed, M., Siegel, M., Young, Y.-N., Li, J., Booty, M. R. and Papageorgiou, D. T., Influence of insoluble surfactant on the deformation and breakup of a bubble or thread in a viscous fluid, J. Fluid. Mech., 594 (2008), 307–340.Google Scholar
[12] Hu, Y. T., Pine, D. J. and Leal, L. G., Drop deformation, breakup, and coalsecence with com-patibilizer, Phys. Fluids., 12 (2000), 484–489.Google Scholar
[13] James, A. J. and Lowengrub, J., A surfactant-conserving volume-of-fluid method for interfacial flows with insoluble surfactant, J. Comput. Phys., 201 (2004), 685–722.Google Scholar
[14] Jiang, G.-S. and Peng, D., Weighted ENO schemes for Hamilton-jacobi equations, SIAM J. Sci. Comput., 21 (2000), 2126–2143.Google Scholar
[15] Ha, J. W., Yoon, Y. and Leal, L. G., The effect of compatibilizer on the coalescence of two drops in flow, Phys. Fluids., 15 (2003), 849–867.Google Scholar
[16] Khatri, S. M., A Numerical Method for Two-Phase Flows with Insoluble and Soluble Surfactant, PhD thesis, Courant Institute, NYU, 2009.Google Scholar
[17] Lai, M.-C., Yu-Hau Tseng and Huaxiong Huang, An immersed boundary method for interfacial flows with insoluble surfactant, J. Comput. Phys., 227 (2008), 7279–7293.CrossRefGoogle Scholar
[18] Landau, L. D. and Lifshitz, E. M., Fluid Mechanics, Pergamon press, 1958.Google Scholar
[19] Leal, L. G., Flow induced coalecsence of drops in a viscous fluid, Phys. Fluids., 16 (2004), 1833–1851.Google Scholar
[20] Lee, J. and Pozrikidis, C., Effect of surfactants on the deformation of drops and bubbles in Navier-Stokes flow, Comput. Fluids., 35 (2006), 43–60.Google Scholar
[21] Le Veque, R. and Li, Z., The immersed interface method for elliptic equations with discontinuous coefficients and singular sources, SIAM J. Numer. Anal., 31 (1994), 1019–1044.Google Scholar
[22] Le Veque, R. and Li, Z., Immersed interface methods for Stokes flow with elastic boundaries or surface tension, SIAM. J. Sci. Comput., 18 (1997), 709–735.Google Scholar
[23] Li, X. and Pozrikidis, C., Effect of surfactants on drop deformation and on the rheology of dilute emulsion in Stokes flow, J. Fluid. Mech., 385 (1999), 79–99.Google Scholar
[24] Loewenberg, M. and Hinch, E. J., Collision of two deformable drops in shear flow, J. Fluid. Mech., 388 (1997), 299–315.Google Scholar
[25] Lowengrub, J., Xu, J. and Voigt, A., Surface phase separation and flow in a simple model of multicomponent drops and vesicles, Fluid. Dyn. Mater. Proc., 3 (2007), 1–19.Google Scholar
[26] Lyu, S., Jones, T. D., Bates, F. S. and Macosko, C. W., Role of block copolymers on suppression of droplet coalescence, Macormolecules, 35 (2002), 7845–7855.Google Scholar
[27] Macklin, P. and Lowengrub, J., Evolving interfaces via gradients of geometry-dependent interior poisson problems: application to tumor growth, J. Comput. Phys., 203 (2005), 191–220.CrossRefGoogle Scholar
[28] Milliken, W. J., Stone, H. A. and Leal, L. G., The effect of surfactant on transient motion of newtonian drops, Phys. Fluids. A., 5 (1993), 69–79.Google Scholar
[29] Milner, S. T. and Xi, H., How copolymers promote mixing of immiscible homopolymers, J. Rheol., 40 (2000), 663–687.Google Scholar
[30] Muradoglu, M. and Tryggvason, G., A front-tracking method for the computation of interfacial flows with soluble surfactants, J. Comput. Phys., 227 (2008), 2238–2262.Google Scholar
[31] Osher, S. and Fedkiw, R. P., Level set methods: an overview and some recent results, J. Comput. Phys., 169 (2001), 463–502.Google Scholar
[32] Osher, S. and Sethian, J. A., Fronts propagating with curvature dependent speed: algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys., 79 (1988), 12–49.Google Scholar
[33] 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 (1996), 1738–1751.Google Scholar
[34] Sethian, J. A. and Smereka, P., Level set methods for fluid interfaces, Ann. Rev. Fluid. Mech., 35 (2003), 341–372.Google Scholar
[35] Shu, C.-W., Total-variation-diminishing time discretization, SIAM J. Sci. Stat. Comput., 9 (1988), 1073–1084.Google Scholar
[36] Sussman, M., Smereka, P. and Osher, S., A level-set approach for computing solutions ot incompressible two-phase flow, J. Comput. Phys., 114 (1994), 146–159.Google Scholar
[37] Stone, H. A., A simple derivation of the time-dependent convective-diffusion equation for surfactant transport along a deforming interface, Phys. Fluids. A., 2 (1989), 111.Google Scholar
[38] Stone, H. A. and Leal, L. G., The effects of surfactants on drop deformation and breakup, J. Fluid. Mech., 220 (1990), 161–186.Google Scholar
[39] Tasoglu, S., Demirci, U. and Muradoglu, M., The effect of soluble surfactant on the transient motion of a buoyancy-driven bubble, Phys. Fluids., 20 (2008), 04085.Google Scholar
[40] Teigen, K. E., Song, P., Lowengrub, J. and Voigt, A., A diffusive-interface method for two-phase flows with soluble surfactants, J. Comput. Phys., in review.Google Scholar
[41] Wong, H., Rumschitzki, D. and Maldarelli, C., On the surfactant mass balance at a deforming fluid interface, Phys. Fluids., 8 (1996), 3203–3204.Google Scholar
[42] Xu, J. and Yuan, H., Three dimensional level-set approach for surfactant convection-diffusion equation along a moving interface, submitted.Google Scholar
[43] Xu, J., Yang, Y. and Lowengrub, J., A level-set continuum method for two-phase flows with insoluble surfactant, submitted.Google Scholar
[44] Xu, J., Li, Z., Lowengrub, J. and Zhao, H., A level set method for solving interfacial flows with surfactant, J. Comput. Phys., 212 (2006), 590–616.Google Scholar
[45] Xu, J. and Zhao, H., An Eulerian formulation for solving partial differential equations along a moving interface, J. Sci. Comput., 19 (2003), 573–594.Google Scholar
[46] Yang, X. and James, A. J., An arbitrary Lagrangian-Eulerian (ALE) method for interfacial flows with insoluble surfactants, Fluid. Dyn. Mater. Proc., 3 (2007), 65–96.Google Scholar
[47] Yoon, Y., Hsu, A. and Leal, L. G., Experimental investigation of the effects of copolymer surfactants on flow-induced coalescence of drops, Phys. Fluids., 19 (2007), 023102.Google Scholar
[48] Zhang, J., Eckmann, D. M. and Ayyaswamy, P. S., A front tracking method for a deformable in-travascular bubble in a tube with soluble surfactant transport, J. Comput. Phys., 214 (2006), 366–396.Google Scholar
[49] Zhao, H., Chan, T. F., Merriman, B. and Osher, S., A variational level set approach to multiphase motion, J. Comput. Phys., 127 (1996), 179–195.Google Scholar