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Diffuse-interface modelling of droplet impact

Published online by Cambridge University Press:  22 May 2007

V. V. KHATAVKAR ,
P. D. ANDERSON ,
P. C. DUINEVELD  and
H. E. H. MEIJER
V. V. KHATAVKAR
Affiliation:
Materials Technology, Dutch Polymer Institute, Eindhoven University of Technology, PO box 513, 5600 MB Eindhoven, The Netherlandsp.d.anderson@tue.nl
P. D. ANDERSON
Affiliation:
Materials Technology, Dutch Polymer Institute, Eindhoven University of Technology, PO box 513, 5600 MB Eindhoven, The Netherlandsp.d.anderson@tue.nl
P. C. DUINEVELD
Affiliation:
Philips Research, Prof. Holstlaan, 4, 5656 AA, Eindhoven, The Netherlands
H. E. H. MEIJER
Affiliation:
Materials Technology, Dutch Polymer Institute, Eindhoven University of Technology, PO box 513, 5600 MB Eindhoven, The Netherlandsp.d.anderson@tue.nl
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Abstract

The impact of micron-size drops on a smooth, flat, chemically homogeneous solid surface is studied using a diffuse-interface model (DIM). The model is based on the Cahn–Hilliard theory that couples thermodynamics with hydrodynamics, and is extended to include non-90° contact angles. The (axisymmetric) equations are numerically solved using a combination of finite- and spectral-element methods. The influence of various process and material parameters such as impact velocity, droplet diameter, viscosity, surface tension and wettability on the impact behaviour of drops is investigated. Relevant dimensionless parameters are defined and, depending on the values of the Reynolds number, the Weber number and the contact angle, which for the cases considered here range from 1.3 to 130, 0.43 to 150 and 45° to 135°, respectively, the model predicts the spreading of a droplet with or without recoil or even rebound of the droplet, totally or partially, from the solid surface. The wettability significantly affects the impact behaviour and this is particularly demonstrated with an impact at Re = 130 and We = 1.5, where for θ < 60° the droplet oscillates a few times before attaining equilibrium while for θ ≥ 60° partial rebound of the droplet occurs, i.e. the droplet breaks into two unequal sized drops. The size of the part that remains in contact with the solid surface progressively decreases with increasing θ until at a value θ ≈ 120° a transition to total rebound happens. When the droplet rebounds totally, it has a top-heavy shape.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 581 , 25 June 2007 , pp. 97 - 127
Copyright
Copyright © Cambridge University Press 2007

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References

REFERENCES

Anderson, D. M., McFadden, G. B. & Wheeler, A. A. 1998 Diffuse-interface methods in fluid mechanics. Annu. Rev. Fluid Mech. 30, 139–165.CrossRefGoogle Scholar
Anderson, P. D., Keestra, B. J. & Hulsen, M. A. 2006 On the streamfunction-vorticity formulation in sliding bi-period frames: Application to bulk behavior for polymer blends. J. Comput. Phys. 212, 268–287.CrossRefGoogle Scholar
Asai, A., Shioya, M., Hirasawa, S. & Okazaki, T. 1993 Impact of an ink drop on paper. J. Imag. Sci. Tech. 37, 205–207.Google Scholar
Attinger, D. & Poulikakos, D. 2001 Melting and resolidification of a substrate caused by molten microdroplet impact. Trans. ASME: J. Heat Transfer. 123, 1110–1122.CrossRefGoogle Scholar
Attinger, D., Zhao, Z. & Poulikakos, D. 2000 An experimental study of molten microdroplet surface deposition and solidification: Transient behavior and wetting angle dynamics. Trans. ASME: J. Heat Transfer. 122, 544–556.CrossRefGoogle Scholar
Aziz, S. D. & Chandra, S. 2000 Impact, recoil and splashing of molten metal droplets. Intl J. Heat Mass Transfer. 43, 2841–2857.CrossRefGoogle Scholar
Barosan, I., Anderson, P. D. & Meijer, H. E. H. 2006 Application of mortar elements to diffuse-interface methods. Computers Fluids 35 (10), 1384–1399.CrossRefGoogle Scholar
Bergeron, V., Bonn, D., Martin, J. Y. & Vovelle, L. 2000 Controlling droplet deposition with polymer additives. Nature. 405, 772–775.CrossRefGoogle ScholarPubMed
Beveridge, G. S. G. & Schechter, R. S. 1970 Optimization: Theory and Practice. McGraw-Hill.Google Scholar
Bhola, R. & Chandra, S. 1999 Parameters controlling solidification of molten wax droplets falling on a solid surface. J. Mater. Sci. 34, 4883–4894.CrossRefGoogle Scholar
Blake, T. D. 1993 Dynamic Contact Angles and Wetting Kinetics. Surfactant Science Series, vol. 49 Marcel Dekker.Google Scholar
Boettinger, W. J., Warren, J. A., Beckermann, C. & Karma, A. 2002 Phase-field simulation of solidification. Annu. Rev. Mater. Res. 32, 163–194.CrossRefGoogle Scholar
Briant, A. J., Wagner, A. J. & Yeomans, J. M. 2004 Lattice Boltzmann simulations of contact line motion. I. Liquid-gas systems. Phys. Rev. E 69, 031602.Google ScholarPubMed
Briant, A. J. & Yeomans, J. M. 2004 Lattice Boltzmann simulations of contact line motion. II. Binary fluids. Phys. Rev. E 69, 031603.Google ScholarPubMed
Bussmann, M., Mostaghimi, J. & Chandra, S. 1999 On a three-dimensional volume tracking model of droplet impact. Phys. Fluids. 11, 1406–1417.CrossRefGoogle Scholar
Cahn, J. W. 1964 Phase separation by spinodal decomposition in isotropic systems. J. Chem. Phys. 42 (1), 93–99.CrossRefGoogle Scholar
Cahn, J. W. 1977 Critical point wetting. J. Chem. Phys. 66 (8), 3667–3672.CrossRefGoogle Scholar
Cahn, J. W. & Hilliard, J. E. 1958 Free energy of a nonuniform system. I. Interfacial energy. J. Chem. Phys. 28 (2), 258–267.CrossRefGoogle Scholar
Chandra, S. & Avedisian, C. T. 1991 On the collision of a droplet with a solid surface. Proc. R. Soc. Lond. A 432, 13–41.Google Scholar
Chella, R. & Viñals, J. , J. 1996 Mixing of a two-phase fluid by cavity flow. Phys. Rev. E 53 (4), 3832–3840.Google ScholarPubMed
Chen, H.-Y., Jasnow, D. & Viñals, J. 2000 Interface and contact line motion in a two phase fluid under shear flow. Phys. Rev. Lett. 85 (8), 1686–1689.CrossRefGoogle Scholar
Crooks, R. C. & Boger, D. V. 2001 Influence of fluid elasticity on drop impacting on dry surfaces. J. Rheol. 44, 973–996.CrossRefGoogle Scholar
Crooks, R. C., Cooper-White, J. J. & Boger, D. V. 2001 The role of dynamic surface tension and elasticity in the dynamics of drop impact. Chem. Engng Sci. 56, 5575–5592.CrossRefGoogle Scholar
Crooks, R. C., Cooper-White, J. J. & Boger, D. V. 2002 A drop impact study of worm-like viscoelastic surfactant solutions. Colloids Surf. A 210, 105–123.Google Scholar
vanDam, D. B. Dam, D. B. & Clerc, C. L. 2004 Experimental study of the impact of an ink-jet printed droplet on a solid substrate. Phys. Fluids. 16, 3403–3414.Google Scholar
Davis, H. T. & Scriven, L. E. 1982 Stress and structure in fluid interfaces. Adv. Chem. Phys. 49, 357–454.CrossRefGoogle Scholar
Duineveld, P. 2003 The stability of ink-jet printed lines of liquid with zero receding contact angle on a homogeneous substrate. J. Fluid Mech. 477, 175–200.CrossRefGoogle Scholar
DussanV., E. B. V., E. B. & Davis, S. H. 1974 On the motion of a fluid-fluid interface along a surface. J. Fluid Mech. 65, 71–95.CrossRefGoogle Scholar
Fedorchenko, A. I., Wang, A.-B. & Wang, Y.-H. 2005 Effect of capillary and viscous forces on spreading of a liquid drop impinging on a solid surface. Phys. Fluids. 17, 0931041–0931048.CrossRefGoogle Scholar
Fournier, A., Bunge, H. P., Hollerbach, R. & Villote, J. P. 2004 Application of the spectral-element method to the axisymmetric Navier-Stokes equation. Geophys. J. Intl. 156, 682–700.CrossRefGoogle Scholar
Francois, M. & Shyy, W. 2003 Computations of drop dynamics with the immersed boundary method, Part 2: Drop impact and heat transfer. Numer. Heat Transfer B 44, 119–143.CrossRefGoogle Scholar
Fukai, J., Shiiba, Y., Yamamoto, T., Miyatake, O., Poulikakos, D., Megaridis, C. M. & Zhao, Z. 1995 Wetting effects on the spreading of a liquid droplet colliding with a flat surface: Experiment and modeling. Phys. Fluids. 7, 236–247.CrossRefGoogle Scholar
Fukai, J., Zhao, Z., Poulikakos, D., Megaridis, C. M. & Miyatake, O. 1993 Modeling of the deformation of a liquid droplet impinging upon a flat surface. Phys. Fluids. 5, 2588–2599.CrossRefGoogle Scholar
deGans, B.-J. Gans, B.-J., Duineveld, P. & Schubert, U. S. 2004 Inkjet printing of polymers: State of the art and future developments. Adv. Mater. 16, 203–213.Google Scholar
deGennes, P. G. Gennes, P. G. 1985 Wetting: statics and dynamics. Rev. Mod. Phys. 57, 827–862.Google Scholar
Gerritsma, M. I. & Phillips, T. N. 2000 Spectral element methods for axisymmetric Stokes problem. J. Comput. Phys. 164, 81–103.CrossRefGoogle Scholar
Gunjal, P. R., Ranade, V. V. & Chaudhari, R. V. 2005 Dynamics of drop impact on solid surface: Experiments and VOF simulations. AIChE J. 51, 59–78.CrossRefGoogle Scholar
Gunton, J. D., Miguel, M. S. & Sahni, P. S. 1983 The Dynamics of First-order Phase Transitions. Phase Transitions and Critical Phenomena, vol. 8. Academic.CrossRefGoogle Scholar
Hocking, L. M. 1977 A moving fluid interface. Part 2. The removal of force singularity by a slip flow. J. Fluid Mech. 79, 209–229.CrossRefGoogle Scholar
Huh, C. & Scriven, L. E. 1971 Hydrodynamic model of steady movement of a solid/liquid/fluid contact line. J. Colloid Interface Sci. 35, 85–100.CrossRefGoogle Scholar
Jacqmin, D. 1999 Calculation of two-phase Navier-Stokes flows using phase-field modeling. J. Comput. Phys. 155, 96–127.CrossRefGoogle Scholar
Jacqmin, D. 2000 Contact-line dynamics of a diffuse fluid interface. J. Fluid Mech. 402, 57–88.CrossRefGoogle Scholar
Jasnow, D. & Viñals, J. 1996 Coarse-grained description of thermo-capillary flow. Phys. Fluids 8 (3), 660–669.CrossRefGoogle Scholar
Joseph, D. D. & Renardy, Y. Y. 1993 Fundamentals of Two-fluid Dynamics. Springer.Google Scholar
Kawase, T., Shimoda, T., Newsome, C., Sirringhaus, H. & Friend, R. H. 2003 Inkjet printing of polymer thin film transistor. Thin Solid Films 438–439, 279–287.CrossRefGoogle Scholar
Keestra, B. J., vanPuyvelde, P. C. J. Puyvelde, P. C. J., Anderson, P. D. & Meijer, H. E. H. 2003 Diffuse interface modeling of the morphology and rheology of immiscible polymer blends. Phys. Fluids. 15, 2567–2575.CrossRefGoogle Scholar
Khatavkar, V. V., Anderson, P. D. & Meijer, H. E. H. 2006 On scaling of diffuse-interface models. Chem. Engng Sci. 61 (8), 2364–2378.CrossRefGoogle Scholar
Khatavkar, V. V., Anderson, P. D. & Meijer, H. E. H. 2007 Capillary spreading of droplet in a partially wetting regime using a diffuse-interface model. J. Fluid Mech. 572, 367–387.CrossRefGoogle Scholar
Kim, H.-Y., Park, S.-Y. & Min, K. 2003 Imaging high-speed impact of mircodrop on solid surface. Rev. Sci. Instrum. 74, 4930–4937.CrossRefGoogle Scholar
Lee, H.-G., Lowengrub, J. S. & Goodman, J. 2002a Modeling pinchoff and reconnection in a Hele-Shaw cell. I. The models and their calibration. Phys. Fluids 14 (2), 492–513.CrossRefGoogle Scholar
Lee, H.-G., Lowengrub, J. S. & Goodman, J. 2002b Modeling pinchoff and reconnection in a Hele-Shaw cell. II. Analysis and simulation in the non-linear regime. Phys. Fluids 14 (2), 514–545.CrossRefGoogle Scholar
Lowengrub, J. & Truskinovsky, L. 1998 Quasi-incompressible Cahn Hilliard fluids. Proc. R. Soc. Lond. A 454, 2617–2654.CrossRefGoogle Scholar
Mao, T., Kuhn, D. C. S. & Tran, H. 1997 Spread and rebound of liquid droplets upon impact on flat surfaces. AIChE J. 43, 2169–2179.CrossRefGoogle Scholar
Naumann, E. & He, D. 2001 Nonlinear diffusion and phase separation. Chem. Engng Sci. 56, 1999–2018.CrossRefGoogle Scholar
Okamoto, T., Suzuki, T. & Yamamoto, N. 2000 Microarray fabrication with covalent attachment of DNA using bubble jet technology. Nat. Biotechnol. 18, 438–441.CrossRefGoogle ScholarPubMed
Park, H., Carr, W. W., Zhu, J. & Morris, J. F. 2003 Single drop impaction on a solid surface. AIChE J. 49, 2461–2471.CrossRefGoogle Scholar
Pasandideh-Fard, M., Qiao, Y. M., Chandra, S. & Mostaghimi, J. 1996 Capillary effects during droplet impact on a solid surface. Phys. Fluids. 8, 650–659.CrossRefGoogle Scholar
Pasandideh-Fard, M., Qiao, Y. M., Chandra, S. & Mostaghimi, J. 2002 On a three-dimensional model of droplet impact and solidification. Intl J. Heat Mass Transfer. 45, 2229–2242.CrossRefGoogle Scholar
Patera, A. T. 1984 A spectral element method for fluid dynamics. J. Comput. Phys. 54, 468–488.CrossRefGoogle Scholar
Range, K. & Feuillebois, F. 1998 Influence of surface roughness on liquid drop impact. J. Colloid Interface Sci. 203, 16–30.CrossRefGoogle Scholar
Rein, M. 1993 Phenomena of liquid drop impact on solid and liquid surfaces. Fluid Dyn. Res. 12, 61–93.CrossRefGoogle Scholar
Renardy, M., Renardy, Y. & Li, J. 2001 Numerical simulation of moving contact line problems using a volume-of-fluid method. J. Comput. Phys. 171, 243–263.CrossRefGoogle Scholar
Renardy, Y., Popinet, S., Duchemin, L., Renardy, M., Zaleski, S., Josserand, C., Drumright-Clarke, M. A., Richard, D., Clanet, C. & Quere, D. 2003 Pyramidal and toroidal water drops after impact on a solid surface. J. Fluid Mech. 484, 69–83.CrossRefGoogle Scholar
Roisman, I. V., Rioboo, R. & Tropea, C. 2002 Normal impact of a liquid drop on a dry surface: model for spreading and receding. Proc. R. Soc. Lond. A 458, 1411–1430.CrossRefGoogle Scholar
Rowlinson, J. S. & Widom, B. 1989 Molecular Theory of Capillarity. Clarendon.Google Scholar
Schiaffino, S. & Sonin, A. 1997 Molten droplet deposition and solification at low Weber number. Phys. Fluids. 9, 3172–3187.CrossRefGoogle Scholar
Segal, A. 1995 SEPRAN manual. Leidschendam, The Netherlands.Google Scholar
Sekerka, R. 2004 Morphology: from sharp interface to phase field models. J. Crystal Growth. 264, 530–540.CrossRefGoogle Scholar
Seppecher, P. 1996 Moving contact lines in the Cahn–Hilliard theory. Intl J. Engng Sci. 34 (9), 977–992.CrossRefGoogle Scholar
Sikalo, S., Wilhelm, H.-D., Roisman, I. V., Jakirlic, S. & Tropea, C. 2005 Dynamic contact angle of spreading droplets: Experiments and simulations. Phys. Fluids. 17, 0621031–06210313.CrossRefGoogle Scholar
Sirringhaus, H., Kawase, T., Friend, R. H., Shimoda, T., Inbasekaran, M., Wu, W. & Woo, E. P. 2000 High-resolution inkjet printing of all-polymer transistor circuits. Science. 290, 2123–2126.CrossRefGoogle ScholarPubMed
Timmermans, L. J. P., vande Vosse, F. N. de Vosse, F. N. & Minev, P. D. 1994 Taylor-Galerkin-based spectral element methods for convection-diffusion problems. Intl J. Numer. Meth. Fluids. 18, 853–870.CrossRefGoogle Scholar
Verschueren, M. 1999 A diffuse-interface model for structure development in flow. PhD thesis, Eindhoven University of Technology, The Netherlands (downloadable from http://www.mate.tue.nl/mate/pdfs/56.pdf).Google Scholar
Verschueren, M., vande Vosse, F. N. de Vosse, F. N. & Meijer, H. E. H. 2000 Diffuse-interface modelling of thermocapillary flow instabilities in a Hele-Shaw cell. J. Fluid Mech. 434, 153–166.CrossRefGoogle Scholar
vander Waals, J. D. der Waals, J. D. 1893 The thermodynamic theory of capillarity under the hypothesis of a continuous variation of density. Verhandel. Konink. Akad. Weten. Amsterdam 1 (English translation by J. S. Rowlinson in J. Statist. Phys. 20 (1979), 197–244).Google Scholar
Worthington, A. M. 1876 On the forms assumed by drops of liquids falling vertically on a horizontal plate. Proc. R. Soc. Lond. A 25, 261–271.Google Scholar
Yarin, A. L., Yazicioglu, A. G., Megaridis, C. M., Rossi, M. P. & Gogotsi, Y. 2005 Theoretical and experimental investigation of aqueous liquids contained in carbon nanotubes. J. Appl. Phys. 97, 1243091.CrossRefGoogle Scholar
Yue, P., Feng, J. J., Liu, C. & Shen, J. 2004 A diffuse-interface method for simulating two-phase flows of complex fluids. J. Fluid Mech. 515, 293–317.CrossRefGoogle Scholar
Zhang, X. & Basaran, O. A. 1997 Dynamic Surface Tension Effects in Impact of a Drop with a Solid Surface. J. Colloid Interface Sci. 187, 166–178.CrossRefGoogle ScholarPubMed
Zhao, Z., Poulikakos, D. & Fukai, J. 1996 a Heat transfer and fluid dynamics during the collision of a liquid droplet on a substrate-I. Modeling. Intl J. Heat Mass Transfer. 39, 2771–2789.CrossRefGoogle Scholar
Zhao, Z., Poulikakos, D. & Fukai, J. 1996 b Heat transfer and fluid dynamics during the collision of a liquid droplet on a substrate-II. Experiments. Intl J. Heat Mass Transfer. 39, 2791–2802.CrossRefGoogle Scholar