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3D phase-field simulations of interfacial dynamics in Newtonian and viscoelastic fluids. (English) Zbl 1355.76037

Summary: This work presents a three-dimensional finite-element algorithm, based on the phase-field model, for computing interfacial flows of Newtonian and complex fluids. A 3D adaptive meshing scheme produces fine grid covering the interface and coarse mesh in the bulk. It is key to accurate resolution of the interface at manageable computational costs. The coupled Navier-Stokes and Cahn-Hilliard equations, plus the constitutive equation for non-Newtonian fluids, are solved using second-order implicit time stepping. Within each time step, Newton iteration is used to handle the nonlinearity, and the linear algebraic system is solved by preconditioned Krylov methods. The phase-field model, with a physically diffuse interface, affords the method several advantages in computing interfacial dynamics. One is the ease in simulating topological changes such as interfacial rupture and coalescence. Another is the capability of computing contact line motion without invoking ad hoc slip conditions. As validation of the 3D numerical scheme, we have computed drop deformation in an elongational flow, relaxation of a deformed drop to the spherical shape, and drop spreading on a partially wetting substrate. The results are compared with numerical and experimental results in the literature as well as our own axisymmetric computations where appropriate. Excellent agreement is achieved provided that the 3D interface is adequately resolved by using a sufficiently thin diffuse interface and refined grid. Since our model involves several coupled partial differential equations and we use a fully implicit scheme, the matrix inversion requires a large memory. This puts a limit on the scale of problems that can be simulated in 3D, especially for viscoelastic fluids.

MSC:

76M10 Finite element methods applied to problems in fluid mechanics
76A05 Non-Newtonian fluids
76A10 Viscoelastic fluids
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
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[1] Joanicot, M.; Ajdari, A., Droplet control for microfluidics, Science, 309, 887-888 (2005)
[2] Sethian, J. A.; Smereka, P., Level set methods for fluid interfaces, Ann. Rev. Fluid Mech., 35, 341-372 (2003) · Zbl 1041.76057
[3] Xie, X. Y.; Musson, L. C.; Pasquali, M., An isochoric domain deformation method for computing steady free surface flows with conserved volumes, J. Comput. Phys., 226, 398-413 (2007) · Zbl 1310.76102
[4] Aggarwal, N.; Sarkar, K., Deformation and breakup of a viscoelastic drop in a Newtonian matrix under steady shear, J. Fluid Mech., 584, 1-21 (2007) · Zbl 1118.76016
[5] Yue, P.; Feng, J. J.; Bertelo, C. A.; Hu, H. H., An arbitrary Lagrangian-Eulerian method for simulating bubble growth in polymer foaming, J. Comput. Phys., 226, 2229-2249 (2007) · Zbl 1127.82045
[6] Yue, P.; Feng, J. J.; Liu, C.; Shen, J., Diffuse-interface simulations of drop coalescence and retraction in viscoelastic fluids, J. Non-Newtonian Fluid Mech., 129, 163-176 (2005) · Zbl 1195.76120
[7] Yue, P.; Feng, J. J.; Liu, C.; Shen, J., A diffuse-interface method for simulating two-phase flows of complex fluids, J. Fluid Mech., 515, 293-317 (2004) · Zbl 1130.76437
[8] Feng, J. J.; Liu, C.; Shen, J.; Yue, P., An energetic variational formulation with phase field methods for interfacial dynamics of complex fluids: advantages and challenges, (Calderer, M.-C. T.; Terentjev, E. M., Modeling of Soft Matter (2005), Springer: Springer New York), 1-26 · Zbl 1181.76019
[9] Lin, P.; Liu, C., Simulations of singularity dynamics in liquid crystal flows: a C-0 finite element approach, J. Comput. Phys., 215, 348-362 (2006) · Zbl 1101.82039
[10] Lin, P.; Liu, C.; Zhang, H., An energy law preserving C-0 finite element scheme for simulating the kinematic effects in liquid crystal dynamics, J. Comput. Phys., 227, 1411-1427 (2007) · Zbl 1133.65077
[11] Yue, P.; Feng, J. J.; Liu, C.; Shen, J., Interfacial force and Marangoni flow on a nematic drop retracting in an isotropic fluid, J. Colloid Interf. Sci., 290, 281-288 (2005)
[12] Zhou, C.; Yue, P.; Feng, J. J., Formation of simple and compound drops in microfluidic devices, Phys. Fluid, 18, 092105 (2006)
[13] Pismen, L. M., Nonlocal diffuse interface theory of thin films and the moving contact line, Phys. Rev. E, 64, 021603 (2001)
[14] P. Yue, C. Zhou, J.J. Feng, Sharp interface limit of the Cahn-Hilliard model for moving contact lines, J. Fluid Mech., in press.; P. Yue, C. Zhou, J.J. Feng, Sharp interface limit of the Cahn-Hilliard model for moving contact lines, J. Fluid Mech., in press. · Zbl 1189.76074
[15] Lowengrub, J.; Truskinovsky, L., Quasi-incompressible Cahn-Hilliard fluids and topological transitions, Proc. Roy. Soc. Lond. A, 454, 2617-2654 (1998) · Zbl 0927.76007
[16] Jacqmin, D., Calculation of two-phase Navier-Stokes flows using phase-field modelling, J. Comput. Phys., 155, 96-127 (1999) · Zbl 0966.76060
[17] Keestra, B. J.; van Puyvelde, P. C.J.; Anderson, P. D.; Meijer, H. E.H., Diffuse interface modeling of the morphology and rheology of immiscible polymer blends, Phys. Fluid, 15, 2567-2575 (2003) · Zbl 1186.76273
[18] Badalassi, V. E.; Ceniceros, H. D.; Banerjee, S., Computation of multiphase systems with phase field model, J. Comput. Phys., 190, 371-397 (2003) · Zbl 1076.76517
[19] Caginalp, G.; Chen, X., Convergence of the phase field model to its sharp interface limits, Eur. J. Appl. Math., 9, 417-445 (1998) · Zbl 0930.35024
[20] Yue, P.; Zhou, C.; Feng, J. J.; Ollivier-Gooch, C. F.; Hu, H. H., Phase-field simulations of interfacial dynamics in viscoelastic fluids using finite elements with adaptive meshing, J. Comput. Phys., 219, 47-67 (2006) · Zbl 1137.76318
[21] Yue, P.; Zhou, C.; Feng, J. J., A computational study of the coalescence between a drop and an interface in Newtonian and viscoelastic fluids, Phys. Fluid, 18, 102102 (2006)
[22] Zhou, C.; Yue, P.; Feng, J. J., Simulation of neutrophil deformation and transport in capillaries using simple and compound drop models, Ann. Biomed. Eng., 35, 766-780 (2007)
[23] Zhou, C.; Yue, P.; Feng, J. J., The rise of Newtonian drops in a nematic liquid crystal, J. Fluid Mech., 593, 385-404 (2007) · Zbl 1151.76374
[24] Zhou, C.; Yue, P.; Feng, J. J., Dynamic simulation of droplet interaction and self-assembly in a nematic liquid crystal, Langmuir, 24, 3099-3110 (2008)
[25] Jacqmin, D., Onset of wetting failure in liquid-liquid systems, J. Fluid Mech., 517, 209-228 (2004) · Zbl 1131.76316
[26] Joseph, D. D., Lubricated pipelining, Powder Technol., 94, 211-215 (1997)
[27] Yue, P.; Zhou, C.; Dooley, J.; Feng, J. J., Elastic encapsulation in bicomponent stratified flow of viscoelastic fluids, J. Rheol., 52, 1027-1042 (2008)
[28] Anderson, D. M.; McFadden, G. B.; Wheeler, A. A., Diffuse-interface methods in fluid mechanics, Ann. Rev. Fluid Mech., 30, 139-165 (1998) · Zbl 1398.76051
[29] Zhou, C.; Yue, P.; Feng, J. J.; Liu, C.; Shen, J., Heart-shaped bubbles rising in anisotropic liquids, Phys. Fluid, 19, 041703 (2007) · Zbl 1146.76583
[30] C. Zhou, P. Yue, J.J. Feng, Dynamic simulation of capillary breakup of nematic fibers: molecular orientation and interfacial rupture, J. Comput. Theor. Nanosci., in press.; C. Zhou, P. Yue, J.J. Feng, Dynamic simulation of capillary breakup of nematic fibers: molecular orientation and interfacial rupture, J. Comput. Theor. Nanosci., in press.
[31] Bird, R. B.; Curtiss, C. F.; Armstrong, R. C.; Hassager, O., Dynamics of polymeric liquids, Kinetic Theory, vol. 2 (1987), Wiley: Wiley New York
[32] Cahn, J. W.; Hilliard, J. E., Free energy of a nonuniform system. I. Interfacial free energy, J. Chem. Phys., 28, 258-267 (1958) · Zbl 1431.35066
[33] Jacqmin, D., Contact-line dynamics of a diffuse fluid interface, J. Fluid Mech., 402, 57-88 (2000) · Zbl 0984.76084
[34] Cahn, J. W., Critical-point wetting, J. Chem. Phys., 66, 3667-3672 (1977)
[35] Shen, J., Efficient spectral-Galerkin method. II. Direct solvers of second and fourth order equations by using Chebyshev polynomials, SIAM J. Sci. Comput., 16, 74-87 (1995) · Zbl 0840.65113
[36] Hu, H. H.; Patankar, N. A.; Zhu, M. Y., Direct numerical simulations of fluid-solid systems using the arbitrary Lagrangian-Eulerian technique, J. Comput. Phys., 169, 427-462 (2001) · Zbl 1047.76571
[37] Gan, H.; Feng, J. J.; Hu, H. H., Simulation of the sedimentation of melting solid particles, Int. J. Multiphase Flow, 29, 751-769 (2003) · Zbl 1136.76513
[38] Freitag, L. A.; Ollivier-Gooch, C. F., Tetrahedral mesh improvement using swapping and smoothing, Int. J. Numer. Methods Eng., 40, 3979-4002 (1997) · Zbl 0897.65075
[39] Ollivier-Gooch, C. F.; Boivin, C., Guaranteed-quality simplical mesh generation with cell size and grading control, Eng. Comput., 17, 269-286 (2001) · Zbl 0983.68563
[40] Boivin, C.; Ollivier-Gooch, C. F., Guaranteed-quality triangular mesh generation for domains with curved boundaries, Int. J. Numer. Methods Eng., 55, 1185-1213 (2002) · Zbl 1027.76041
[41] J.R. Shewchuk, Delaunay Refinement Mesh Generation, Ph.D. Thesis, Carnegie Mellon University, 1997.; J.R. Shewchuk, Delaunay Refinement Mesh Generation, Ph.D. Thesis, Carnegie Mellon University, 1997. · Zbl 1016.68139
[42] Watson, D. F., Computing the n-dimensional Delaunay tessellation with application to Voronoi polytopes, Comput. J., 24, 167-172 (1981)
[43] Mo, H.; Zhou, C.; Yu, W., A new method to determine interfacial tension from the retraction of ellipsoidal drops, J. Non-Newtonian Fluid Mech., 91, 221-232 (2000) · Zbl 0948.76524
[44] Son, Y.; Migler, K. B., Interfacial tension measurement between immiscible polymers: improved deformed drop retraction method, Polymer, 43, 3001-3006 (2002)
[45] Velankar, S.; Zhou, H.; Jeon, H. K.; Macosko, C. W., CFD evaluation of drop retraction methods for the measurement of interfacial tension of surfactant-laden drops, J. Colloid Interf. Sci., 272, 172-185 (2004)
[46] Ziegler, V. E.; Wolf, B. A., Interfacial tensions from drop retraction versus pendant drop data and polydispersity effects, Langmuir, 20, 8688-8692 (2004)
[47] Guido, S.; Villone, M., Measurement of interfacial tension by drop retraction analysis, J. Colloid Interf. Sci., 209, 247-250 (1999)
[48] Hooper, R. W.; de Almeida, V. F.; Macosko, C. W.; Derby, J. J., Transient polymeric drop extension and retraction in uniaxial extensional flows, J. Non-Newtonian Fluid Mech., 98, 141-168 (2001) · Zbl 1071.76004
[49] Renardy, M.; Renardy, Y.; Li, J., Numerical simulation of moving contact line problems using a volume-of-fluid method, J. Comput. Phys., 171, 243-263 (2001) · Zbl 1044.76051
[50] Mazouchi, A.; Gramlich, C. M.; Homsy, G. M., Time-dependent free surface Stokes flow with a moving contact line. I. Flow over plane surfaces, Phys. Fluid, 16, 1647-1659 (2004) · Zbl 1186.76358
[51] Villanueva, W.; Amberg, G., Some generic capillary-driven flows, Int. J. Multiphase Flow, 32, 1072-1086 (2006) · Zbl 1136.76672
[52] Khatavkar, V. V.; Anderson, P. D.; Meijer, H. E.H., Capillary spreading of a droplet in the partially wetting regime using a diffuse-interface model, J. Fluid Mech., 572, 367-387 (2007) · Zbl 1111.76016
[53] Ding, H.; Spelt, P. D.M., Inertial effects in droplet spreading: a comparison between diffuse-interface and level-set simulations, J. Fluid Mech., 576, 287-296 (2007) · Zbl 1125.76024
[54] Zosel, A., Studies of the wetting kinetics of liquid drops on solid surfaces, Colloid Polym. Sci., 271, 680-687 (1993)
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