In this paper, we have proposed a three-dimensional immersed boundary method for interfacial flows with insoluble surfactant. We enforce an equi-arclength parametrization on the interface by adding two artificial tangential velocity to the Lagrangian marker velocity so that the surface Lagrangian mesh can be uniformly distributed in numerical computations without any re-meshing. With this mesh-control technique, the
computa-0 2 4 6 0.55
0.6 0.65 0.7
x
z
(a)
0 2 4 6
x (b) t=20 t=30 t=40 t=50
Figure 13: The sectional views of interface with fixed y=π/2 at different times. (a) and (b) correspond to Figs. 11 and 12, respectively.
tions are stable up to long time simulations. Meanwhile, under this new parametrization, the surfactant concentration equation is modified accordingly. And we develop a conser-vative scheme to solve the modified surfactant equation so that the total surfactant mass is conserved numerically. We have performed a series of numerical tests to validate our present scheme. In the future, we plan to extend our current method to handle the case of compact interfaces such as a droplet, and to the soluble surfactant case as well.
Acknowledgments
The work of M.-C. Lai was supported in part by Ministry of Science and Technology of Taiwan under research grant MOST-104-2115-M-009-014-MY3 and NCTS. We also like to thank Dr. Te-sheng Lin for his helpful discussions.
References
[1] S. Adami, X.Y. Hu, N.A. Adams, A conservative SPH method for surfactant dynamics, J.
Comput. Phys., 229 (2010) 1909-1926.
[2] D.M. Ambrose, M. Siegel, S. Tlupova, A small-scale decomposition for 3D boundary integral computations with surface tension, J. Comput. Phys., 247 (2013), 168-191.
[3] J.W. Barrett, H. Garcke, R. N ¨urnberg, Stable finite element approximations of two-phase flow with soluble surfactant, J. Comput. Phys., 297 (2015), 530-564.
[4] I.B. Bazhlekov, P.D. Anderson, H.E.H. Meijer, Numerical investigation of the effect of insol-uble surfactants on drop deformation and breakup in simple shear flow, J. Colloid Interface Sci., 298 (2006), 369-394.
[5] C. Canuto, M.Y. Hussaini, A. Quarteroni, T.A. Zang, Spectral Methods in Fluid Dynamics, Springer-Verlag, 1988.
[6] H.D. Ceniceros, The effects of surfactants on the formation and evolution of capillary waves, Phys. Fluids, 15 (2003), 245-256.
[7] K.-Y. Chen, M.-C. Lai, A conservative scheme for solving coupled surface-bulk convection-diffusion equations with an application to interfacial flows with soluble surfactant, J. Com-put. Phys., 257 (2014), 1-18.
[8] R.V. Craster, O.K. Matar, Dynamics and stability of thin liquid films, Rev. Mod. Phys., 81 (2009), 1131-1198.
[9] W.C. de Jesus, A.M. Roma, M.R. Pivello, M.M. Villar, A. da Silveira-Netob, A 3D front-tracking approach for simulation of a two-phase fluid with insoluble surfactant, J. Comput.
Phys., 281 (2015), 403-420.
[10] M.P. do Carmo, Differential Geometry of Curves and Surfaces, Pearson, 2009.
[11] M.A. Drumright-Clarke, Y. Renardy, The effect of insoluble surfactant at dilute concentration on drop breakup under shear with inertia, Phys. Fluids, 16 (2004), 14-21.
[12] C.D. Eggleton, T.-M. Tsai, K.J. Stebe, Tip streaming from a drop in the presence of surfactants, Phys. Rev. Lett., 87 (2001), 048302.
[13] S. Frijters, F. G ¨unther, J. Harting, Effects of nanoparticles and surfactant on droplets in shear flow, Soft Matter, 8 (2012), 6542-6556.
[14] H. Fujioka, A continuum model of interfacial surfactant transport for particle methods, J.
Comput. Phys., 234 (2013), 280-294.
[15] S. Ganesan, L. Tobiska, A coupled arbitrary Lagrangian-Eulerian and Lagrangian method for computation of free surface flows with insoluble surfactants, J. Comput. Phys., 208 (2009), 2859-2873.
[16] S. Ganesan, L. Tobiska, Arbitrary Lagrangian-Eulerian finite-element method for computa-tion of two-phase flows with soluble surfactant, J. Comput. Phys., 231 (2012), 3685-3702.
[17] S. Ganesan, Simulations of impinging droplets with surfactant-dependent dynamic contact angle, J. Comput. Phys., 301 (2015), 178-200.
[18] J.L. Guermond, P. Minev, J. Shen, An overview of projection methods for incompressible flows, Comput. Methods Appl. Mech. Engrg., 195 (2006), 6011-6045.
[19] M. Hameed, M. Siegel, Y.-N. Young, J. Li, M.R. Booty, D.T. Papageorgiou, Influence of insol-uble surfactant on the deformation and breakup of a bubble or thread in a viscous fluid, J.
Fluid Mech., 594 (2008), 307340.
[20] F.H. Harlow, J.E. Welsh, Numerical calculation of time-dependent viscous incompressible flow of fluid with a free surface, Phys. Fluids., 8 (1965), 2181-2189.
[21] A.J. James, J. Lowengrub, A surfactant-conserving volume-of-fluid method for interfacial flows with insoluble surfactants, J. Comput. Phys., 201 (2004), 685-722.
[22] S. Khatri, A.-K. Tornberg, A numerical method for two phase flows with insoluble surfac-tants, Comput. Fluids, 49 (2011), 150-165.
[23] S. Khatri, A.-K. Tornberg, An embedded boundary method for soluble surfactants with in-terface tracking for two-phase flows, J. Comput. Phys., 256 (2014), 768-790.
[24] I. Kralova, J. Sj ¨oblom, Surfactants used in food industry: a review, J. Dispers. Sci. Technol., 30 (2009), 1363-1383.
[25] M.-C. Lai, Y.-H. Tseng, H. Huang, An immersed boundary method for interfacial flows with insoluble surfactant, J. Comput. Phys., 227 (2008), 7279-7293.
[26] M.-C. Lai, Y.-H. Tseng, H. Huang, Numerical simulation of moving contact lines with sur-factant by immersed boundary method, Commun. Comput. Phys., 8 (2010), 735-757.
[27] M.-C. Lai, C.-Y. Huang, Y.-M. Huang, Simulating the axisymmetric interfacial flows with insoluble surfactant by immersed boundary method, Int. J. Numer. Anal. Model., 8 (2011) 105117.
[28] X. Li, C. Pozrikidis, The effect of surfactants on drop deformation and on the rheology of dilute emulsions in Stokes flow, J. Fluid Mech., 341 (1997), 165-194.
[29] H. Liu, Y. Zhang, Phase-field modeling droplet dynamics with soluble surfactants, J. Com-put. Phys., 229 (2010), 9166-9187.
[30] W.J. Milliken, H.A. Stone, L.G. Leal, The effect of surfactant on transient motion of Newto-nian drops, Phys. Fluids A, 5 (1993), 69-79.
[31] W.J. Milliken, L.G. Leal, The influence of surfactant on the deformation and breakup of a viscous drop: the effect of surfactant solubility, J. Colloid Interface Sci., 166 (1994), 275-285.
[32] M. Muradoglu, G. Tryggvason, A front-tracking method for computation of interfacial flows with soluble surfactants, J. Comput. Phys., 227 (2008), 2238-2262.
[33] M. Muradoglu, G. Tryggvason, Simulations of soluble surfactants in 3D multiphase flow, J.
Comput. Phys., 274 (2014), 737-757.
[34] Y. Pawar, K.J. Stebe, Marangoni effects on drop deformation in an extensional flow: The role of surfactant physical chemistry. I. Insoluble surfactants, Phys. Fluids., 8 (1996), 1738-1751.
[35] C. Pozrikidis, Interfacial dynamics for Stokes flow, J. Comput. Phys., 169 (2001), 250-301.
[36] C. Pozrikidis, A finite-element method for interfacial surfactant transport, with application to the flow-induced deformation of a viscous drop, J. Eng. Math., 49 (2004), 163-180.
[37] K.-L. Pan, Y.-H. Tseng, J.-C. Chen, K.-L. Huang, C.-H. Wang, and M.-C. Lai Controlling droplet bouncing and coalescence with surfactant, J. Fluid Mech., 799, (2016), 603-636.
[38] Y. Seol, W.-F. Hu, Y. Kim, M.-C. Lai, An immersed boundary method for simulating vesicle dynamics in three dimensions, J. Comput. Phys., 322 (2016), 125-141.
[39] D. Sinclair, R. Levy, K.E. Daniels, Simulating surfactant spreading: Impact of a physically motivated equation of state, Preprint.
[40] H.A. Stone, L.G. Leal, The effects of surfactants on drop deformation and breakup, J. Fluid Mech., 220 (1990), 161-186.
[41] H.A. Stone, 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.
[42] S.L. Strickland, M. Hin, M.R. Sayanagi, C. Gaebler, K.E. Daniels, R. Levy, Self-healing dy-namics of surfactant coatings on thin viscous films, Phys. Fluids., 26 (2014), 042109.
[43] K.E. Teigen, P. Song, J. Lowengrub, A. Voigt, A diffuse-interface method for two-phase flows with soluble surfactants, J. Comput. Phys., 203 (2011), 375-393.
[44] L.N. Trefethen, Spectral Methods in MATLAB, SIAM, Philadelphia, 2000.
[45] G. Tryggvason, R. Scardovelli, S. Zaleski, Direct numerical simulations of gas-liquid multi-phase flows, Cambridge University Press, 2011.
[46] T.M. Tsai, M.J. Miksis, The effects of surfactant on the dynamics of bubble snap-off, J. Fluid Mech., 337 (1997), 381-410.
[47] J.-J. Xu, Z. Li, J. Lowengrub, H. Zhao, A level-set method for interfacial flows with surfactant, J. Comput. Phys., 212 (2006), 590-616.
[48] J.-J. Xu, Y. Yang, J. Lowengrub, A level-set continuum method for two-phase flows with insoluble surfactant, J. Comput. Phys., 231 (2012), 5897-5909.
[49] J.-J. Xu, Y. Huang, M.-C. Lai, Z. Li, A coupled immersed interface and level set method for three-dimensional interfacial flows with insoluble surfactant, Commun. Comput. Phys., 15 (2014), 451-469.
[50] X. Yang, X. Zhang, Z. Li, G.-W. He, A smoothing technique for discrete delta functions with application to immersed boundary method in moving boundary simulations, J. Comput.
Phys., 228 (2009), 7821-7836.
[51] S. Yon, C. Pozrikidis, A Finite-volume/Boundary-element Method for Flow Past Interfaces in the Presence of Surfactants, with Application to Shear Flow Past a Viscous Drop, Comput.
Fluids, 27 (1998), 879-902.