A diffuse-interface method for two-phase flows with soluble surfactants

Journal of Computational Physics

Volumn 230, Issue 2, 2011, Pages 375-393

  Erik Teigen, Knut ^a   Song, Peng ^b   Lowengrub, John ^b   Voigt, Axel ^c  

^a NORWEGIAN UNIVERSITY OF SCIENCE AND TECHNOLOGY   (Norway)

^b UNIVERSITY OF CALIFORNIA   (United States)

^c DRESDEN UNIVERSITY OF TECHNOLOGY   (Germany)

Author keywords

Adaptive grid; Adsorption; Bulk phase; Complex geometry; Desorption; Diffuse interface; Finite difference; Interfacial dynamics; Multigrid; Multiphase flows; Phase field; Soluble surfactant; Surface phase; Surfactant

Indexed keywords

EQUATIONS OF STATE; FINITE DIFFERENCE METHOD; FINITE ELEMENT METHOD; NAVIER STOKES EQUATIONS; NONLINEAR EQUATIONS; PHASE INTERFACES; SHEAR FLOW; TWO PHASE FLOW;

ADAPTIVE GRIDS; BULK PHASE; COMPLEX GEOMETRIES; DIFFUSE INTERFACE; FINITE DIFFERENCE; INTERFACIAL DYNAMICS; MULTIGRIDS; PHASE FIELDS; SOLUBLE SURFACTANTS; SURFACE PHASIS;

SURFACE ACTIVE AGENTS;

EID: 78149408227     PISSN: 00219991     EISSN: 10902716     Source Type: Journal    
DOI: 10.1016/j.jcp.2010.09.020     Document Type: Article

References (83)

1. Tryggvason G., Abdollahi-Alibeik J., Willmarth W.W., Hirsa A. Collision of a vortex pair with a contaminated free surface. Phys. Fluids A 1992, 4:1215-1229.
2. Matar O.K., Troian S.M. The development of transient fingering patterns during the spreading of surfactant coated films. Phys. Fluids 1999, 11:3232-3246.
3. Leal L.G. Flow induced coalescence of drops in a viscous fluid. Phys. of Fluids 2004, 16(6):1833-1851.
4. Bazhlekov I., Anderson P., Meijer H. Numerical investigation of the effect of insoluble surfactants on drop deformation and breakup in simple shear flow. J. Colloid Interface Sci. 2006, 298:369-394.
5. Hameed M., Siegel M., Young Y.-N., Li J., Booty M.R., Papageorgiou D.T. Influence of insoluble surfactant on the deformation and breakup of a bubble or thread in a viscous fluid. J. Fluid Mech. 2008, 594:307-340.
6. Bruijn R.D. Tip streaming of drops in simple shear flows. Chem. Eng. Sci. 1993, 48:277-284.
7. Stone H.A., Leal L.G. The effect of surfactants on drop deformation and breakup. J. Fluid Mech. 1990, 220:161-186.
8. Milliken W.J., Stone H.A., Leal L.G. The effect of surfactant on transient motion of Newtonian drops. Phys. Fluids A 1993, 5:69-79.
9. Li X., Pozrikidis C. The effect of surfactants on drop deformation and on the rheology of dilute emulsions in Stokes flow. J. Fluid Mech. 1997, 341:165-194.
10. Milliken W.J., Leal L.G. The influence of surfactant on the deformation and breakup of a viscous drop - the effect of surfactant solubility. J. Colloid Interface Sci. 1994, 166:275-285.
11. Pozrikidis C. Interfacial dynamics for Stokes flow. J. Comput. Phys. 2001, 169:250-301.
12. Tryggvason G., Bunner B., Esmaeeli A., Juric D., Al-Rawahi N., Tauber W., Han J., Nas S., Jan Y.-J. Front tracking method for the computation of multiphase flow. J. Comput. Phys. 2001, 169:708-759.
13. Y.J. Jan, Computational studies of bubble dynamics, Ph.D. thesis, University of Michigan, 1994.
14. Zhang J., Eckmann D., Ayyaswamy P. A front tracking method for a deformable intravascular bubble in a tube with soluble surfactant transport. J. Comput. Phys. 2006, 214:366-396.
15. Muradoglu M., Tryggvason G. A front-tracking method for computation of interfacial flows with soluble surfactants. J. Comput. Phys. 2008, 227:2238-2262.
16. Mittal R., Iaccarino G. Immersed boundary methods. Ann. Rev. Fluid Mech. 2005, 37:239-361.
17. Lai M.-C., Tseng Y.-H., Huang H. An immersed boundary method for interfacial flows with insoluble surfactant. J. Comput. Phys. 2008, 227:7270-7293.
18. Tseng Y.H., Ferziger J.H. A ghost-cell immersed boundary method for flow in complex geometry. J. Comput. Phys. 2003, 192:593-623.
19. Jin F., Stebe K.J. The effects of a diffusion controlled surfactant on a viscous drop injected into a viscous medium. Phys. Fluids 2007, 19:112103.
20. Ceniceros H. The effects of surfactants on the formation and evolution of capillary waves. Phys. Fluids 2003, 15:245-256.
21. Liao Y.-C., Franses E., Basaran O. Deformation and breakup of a stretching liquid bridge covered with an insoluble surfactant monolayer. Phys. Fluids 2006, 18:022101.
22. Xu Q., Liao Y.-C., Basaran O. Can surfactant be present at pinch-off of a liquid filament. Phys. Rev. Lett. 2007, 98:054503.
23. Booty M., Siegel M. A hybrid numerical method for interfacial fluid flow with soluble surfactant. J. Comput. Phys. 2010, 229:3864-3883.
24. Scardovelli R., Zaleski S. Direct numerical simulation of free surface and interfacial flows. Ann. Rev. Fluid Mech. 1999, 31:567-603.
25. Renardy Y.Y., Renardy M., Cristini V. A new volume-of-fluid formulation for surfactants and simulations of drop deformation under shear at a low viscosity ratio. European J. Mech. - B/Fluids 2002, 21:49-59.
26. James A.J., Lowengrub J. A surfactant-conserving volume-of-fluid method for interfacial flows with insoluble surfactant. J. Comput. Phys. 2004, 201:685-722.
27. Alke A., Bothe D. 3d numerical modelling of soluble surfactant at fluidic interfaces based on the volume-of-fluid method. FDMP 2009, 1:1-29.
28. Osher S., Sethian J. Fronts propagating with curvature-dependent speed- algorithms based on Hamilton-Jacobi formulations. J. Comput. Phys. 1988, 79:12-49.
29. Xu J.J., Zhao H. An Eulerian formulation for solving partial differential equations along a moving interface. J. Sci. Comput. 2003, 19:573-594.
30. Xu J.J., Li Z., Lowengrub J., Zhao H. A level set method for interfacial flows with surfactant. J. Comput. Phys. 2006, 212:590-616.
31. Sethian J., Smereka P. Level-set methods for fluid interfaces. Ann. Rev. Fluid Mech. 2003, 35:341-372.
32. Yang X., James A.J. An arbitrary Lagrangian-Eulerian (ALE) method for interfacial flows with insoluble surfactants. FDMP 2007, 3:65-96.
33. Young Y.-N., Booty M.R., Siegel M., Li J. Influence of surfactant solubility on the deformation and breakup of a bubble or capillary jet in a viscous fluid. Phys. Fluids 2009, 21:072105.
34. Uzgoren E., Sim J., Shyy W. Marker-based, 3d adaptive Cartesian grid method for multiphase flow around irregular geometries. Commun. Comput. Phys. 2009, 5:1-41.
35. Adami S., Hu X., Adams N. A conservative SPH method for surfactant dynamics. J. Comput. Phys. 2010, 229:1909-1926.
36. Anderson D.M., McFadden G.B., Wheeler A.A. Diffuse interface methods in fluid mechanics. Ann. Rev. Fluid Mech. 1998, 30(1):139-165.
37. Emmerich H. Advances of and by phase-field modeling in condensed-matter physics. Adv. Phys. 2008, 57:1-87.
38. Lowengrub J., Truskinovsky L. Quasi-incompressible Cahn-Hilliard fluids and topological transitions. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 1998, 454:2617-2654.
39. Jacqmin D. Calculation of two-phase Navier-Stokes flows using phase-field modeling. J. Comput. Phys. 1999, 155:96-127.
40. Lee H., Lowengrub J., Goodman J. Modeling pinchoff and reconnection in a Hele-Shaw cell i. the models and their calibration. Phys. Fluids 2002, 14:492-513.
41. Lee H., Lowengrub J., Goodman J. Modeling pinchoff and reconnection in a hele-shaw cell ii. analysis and simulation in the nonlinear regime. Phys. Fluids 2002, 14:514-545.
42. Baldalassi V., Ceniceros H., Banerjee S. Computation of multiphase systems with phase field models. J. Comput. Phys. 2004, 190:371-397.
43. Boyer F., Chupin L., Franck B. Numerical study of viscoelastic mixtures through a Cahn-Hilliard model. European J. Mech. B-Fluids 2004, 23:759-780.
44. Yue P., Feng J.J., Liu C., Shen J. A diffuse interface method for simulating two phase flows of complex fluids. J. Fluid Mech. 2004, 515:293-317.
45. Kim J. A continuous surface tension force formulation for diffuse-interface models. J. Comput. Phys. 2005, 204(2):784-804.
46. Lu H.-W., Glasner K., Bertozzi A., Kim C.-J. A diffuse interface model for electrowetting droplets in a Hele-Shaw cell. J. Fluid Mech. 2007, 590:411-435.
47. Ding H., Spelt P., Shu C. Diffuse interface model for incompressible two-phase flows with large density ratios. J. Comput. Phys. 2007, 226:2078-2095.
48. van der Sman R., van der Graaf S. Diffuse interface model of surfactant adsorption onto flat and droplet interfaces. Rheol. Acta 2006, 46:3-11.
49. Jacqmin D. Contact-line dynamics of a diffuse interface. J. Fluid Mech. 2000, 402:57-88.
50. Do-Quang M., Amberg G. The splash of a solid sphere impacting on a liquid surface: numerical simulation of the influence of wetting. Phys. Fluids 2009, 21:022102.
51. S. Aland, J. Lowengrub, A. Voigt, Two-phase flow in complex geometries: a diffuse domain approach, in preparation, 2009.
52. Rätz A., Voigt A. PDEs on surfaces-a diffuse interface approach. Commun. Math. Sci. 2006, 4:575-590.
53. Rätz A., Voigt A. A diffuse-interface approximation for surface diffusion including adatoms. Nonlinearity 2007, 20:177-192.
54. Dziuk G., Elliott C. Eulerian finite element method for parabolic PDEs on complex surfaces. Interfaces Free Bound. 2008, 10:119-138.
55. Lowengrub J., Rätz A., Voigt A. Phase-field modeling of the dynamics of multicomponent vesicles: Spinodal decomposition, coarsening, budding and fission. Phys. Rev. E 2008, 79:031926.
56. Elliott C., Stinner B. Analysis of a diffuse interface approach to an advection diffusion equation on a moving surface. Math. Models Meth. Appl. Sci. 2009, 19:787-802.
57. Kockelkoren J., Levine H., Rappel W.-J. Computational approach for modeling intra- and extracellular dynamics. Phys. Rev. E 2003, 68(3):037702. 10.1103/PhysRevE.68.037702.
58. Levine H., Rappel W.-J. Membrane-bound Turing patterns. Phys. Rev. E 2005, 72(6):061912. 10.1103/PhysRevE.72.061912.
59. Fenton F., Cherry E., Karma A., Rappel W.-J. Modeling wave propagation in realistic heart geometries using the phase-field method. Chaos 2005, 15:103502.
60. Li X., Lowengrub J., Rätz A., Voigt A. Solving PDEs in complex geometries: a diffuse domain approach. Commun. Math. Sci. 2009, 7:81-107.
61. Teigen K.E., Wang F., Li X., Lowengrub J., Voigt A. A diffuse-interface approach for modelling transport, diffusion and adsorption/desorption of material quantities on a deformable interface. Commun. Math. Sci. 2009, 7:1009-1037.
62. Lee H.-G., Lowengrub J., Goodman J. Modeling pinchoff and reconnection in a hele-shaw cell i: the models and their calibration. Phys. Fluids 2002, 14:492-513.
63. Lee H.-G., Lowengrub J., Goodman J. Modeling pinchoff and reconnection in a hele-shaw cell ii: analysis and simulation in the nonlinear regime. Phys. Fluids 2002, 14:514-545.
64. Stone H.A. The dynamics of drop deformation and breakup in viscous fluids. Ann. Rev. Fluid Mech. 1994, 26:65-102.
65. Sheth K.S., Pozrikidis C. Effects of inertia on the deformation of liquid drops in simple shear flow. Comput. Fluids 1995, 24(2):101-119.
66. Li J., Renardy Y.Y., Renardy M. Numerical simulation of breakup of a viscous drop in simple shear flow through a volume-of-fluid method. Phys. Fluids 2000, 12:269-282.
67. Cristini V., Blawzdziewicz J., Loewenberg M. An adaptive mesh algorithm for evolving surfaces: simulation of breakup and coalescence. J. Comput. Phys. 2001, 168:445-463.
68. Velankar S., Zhou H., Jeon H., Macosko C. CFD evaluation of drop retraction methods for the measurement of interfacial tension of surfactant-laden drops. J. Colloid Interface Sci. 2004, 272(1):172-185.
69. Eggleton C.D., Pawar Y.P., Stebe K.J. Insoluble surfactants on a drop in an extensional flow: a generalization of the stagnated surface limit to deforming interfaces. J. Fluid Mech. 1999, 385:79-99.
70. Wong H., Rumschitzki D., Maldarelli C. On the surfactant mass balance at a deforming fluid interface. Phys. Fluids 1996, 8:3203-3204.
71. Wise S., Kim J., Lowengrub J. Solving the regularized, strongly anisotropic Cahn-Hilliard equation by an adaptive nonlinear multigrid method. J. Comput. Phys. 2007, 226:414-446.
72. Shu C.W., Osher S. Efficient implementation of essentially non-oscillatory shock-capturing schemes. J. Comput. Phys. 1988, 77(2):439-471.
73. Liu S., Chan T. Weighted essentially non-oscillatory schemes. J. Comput. Phys. 1994, 115:200-212.
74. Bell J.B., Colella P., Glaz H.M. A second-order projection method for the incompressible Navier-Stokes equations. J. Comput. Phys. 1989, 85(2):257-283.
75. Kim J., Kand K., Lowengrub J. Conservative multigrid methods for ternary Cahn-Hilliard fluids. J. Comput. Phys. 2004, 193:511-543.
76. Sussman M., Almgren A.S., Bell J.B., Colella P., Howell L.H., Welcome M.L. An adaptive level set approach for incompressible two-phase flows. J. Comput. Phys. 1999, 148(1):81-124.
77. Lorensen W.E., Cline H.E. Marching cubes: a high resolution 3D surface construction algorithm. Comput. Graphics 1987, 21:163-169.
78. Prosperetti A. Motion of two superposed viscous fluids. Phys. Fluids 1981, 24:1217-1223.
79. Young N.O., Goldstein J.S., Block M.J. The motion of bubbles in a vertical temperature gradient. J. Fluid Mech. 1959, 6(03):350-356.
80. Karma A., Rappel W.-J. Phase-field method for computationally efficient modeling of solidification with arbitrary interface kinetics. Phys. Rev. E 1996, 53(4):R3017-R3020.
81. Karma A., Rappel W.-J. Quantitative phase-field modeling of dendritic growth in two and three dimensions. Phys. Rev. E 1998, 57(4):4323-4349. 10.1103/PhysRevE.57.4323.
82. Ni M.-J. Consistent projection methods for variable density incompressible Navier-Stokes equations with continuous surface forces on a rectangular collocated mesh. J. Comput. Phys. 2009, 228:6938-6956.
83. Chang C., Franses E. Adsorption dynamics of surfactants at the air/water interface: a critical review of mathematical models, data, and mechanisms. Colloids Surfaces A 1995, 100:1-45.