Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-28T15:11:47.150Z Has data issue: false hasContentIssue false
  • English
  • Français

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:[email protected]
Email:[email protected]
Email:[email protected]
Email:[email protected]
Get access

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

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] 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