Advertisement

Transport in Porous Media

, Volume 122, Issue 2, pp 253–277 | Cite as

A Two-Phase SPH Model for Dynamic Contact Angles Including Fluid–Solid Interactions at the Contact Line

  • P. Kunz
  • S. M. Hassanizadeh
  • U. Nieken
Article
  • 383 Downloads

Abstract

The description of wetting phenomena on the continuum scale is a challenging problem, since intermolecular interactions, like van der Waals forces between liquid and solid, alter the flow field at the contact line. Recently, these effects were included in the smoothed particle hydrodynamics method by introducing a contact line force (CLF) on the continuum scale. This physically based contact line force model is employed here to simulate two-phase flow in a wide range of wetting dynamics parametrized by capillary number. In particular, dynamic contact angles at various capillary number values are calculated by CLF model and compared to measured values. We find that there is significant disagreement between simulated and measured results, specially at low wetting speeds. It is indeed expected that most of the driving force is dissipated to overcome strong liquid–solid interactions, which are not adequately accounted for in the existing CLF model. Therefore, we have extended that model to account for stick-slip (SSL) behavior of the contact line caused by solid–fluid interactions. The new SSL model results in dynamic contact angle values that are in good agreement with experimental data for the full range of wetting dynamics.

Keywords

Dynamic contact angle Contact line force Two-phase flow Stick-slip behavior Smoothed particle hydrodynamics (SPH) 

List of symbols

A

Arbitrary field variable

B

Transition function for continuous surface force at solid boundary

c

Color function

\(Ca_\mathrm{Tp}\)

Model parameter (capillary number at transition point)

d

Distance between walls/tape in simulation domain, L

\(\mathbf {d}\)

Distance to solid wall, L

E

Energy, L\(^2\)MT\(^{-2}\)

\(\mathbf {f}_\mathrm{wn}\)

Force per unit area acting on fluid–fluid interface, M L\(^{-1}\) T\(^{-2}\)

\(\mathbf {f}_\mathrm{wns}\)

Force per unit line acting at contact line, M T\(^{-2}\)

\(\mathbf {F}^\mathrm{VOL}\)

Force per unit volume, M L\(^{-2}\) T\(^{-2}\)

\(\mathbf {g}\)

Gravitational field strength, L T\(^{-2}\)

h

Smoothing length of kernel function, L

K

PI controller parameter

l

Domain length, L

\(\mathbf {L}\)

Kernel correction matrix

m

Mass, M

\(\hat{\mathbf {n}}\)

Unit normal vector

p

Pressure, M L\(^{-1}\) T\(^{-2}\)

\(\mathbf {r}\)

Distance, L

S

Shepard correction for kernel function

t

Time, T

\(\mathbf {v}\)

Velocity, L T\(^{-1}\)

V

Volume, L\(^3\)

W

Kernel function, L\(^{-3}\)

\(\mathbf {x}\)

Position, L

\(\alpha \)

Model parameter (fractional viscosity increase at the contact line)

\(\beta \)

Dimensionless relation of wall distances

\(\delta _\mathrm{wn}\)

Dirac delta distribution and volume reformulation at fluid–fluid interface, L\(^{-1}\)

\(\delta _\mathrm{wns}\)

Dirac delta distribution and volume reformulation at contact line, L\(^{-2}\)

\(\Delta x\)

Initial particle spacing, L

\(\zeta \)

Multiplicative factor for volume reformulation at contact line

\(\theta \)

Contact angle

\(\kappa \)

Curvature, L\(^{-1}\)

\(\mu \)

Dynamic viscosity, M L\(^{-1}\) T\(^{-1}\)

\(\hat{\mathbf {\nu }}\)

Unit vector

\(\rho \)

Density, M L\(^{-3}\)

\(\sigma \)

Surface tension, M T\(^{-2}\)

Notes

Acknowledgements

This work was funded by the German Research Foundation (DFG) within the framework of the International Research Training Group “NUPUS, Non-linearities and upscaling in porous media” (IRTG 1398) and the research unit “Multiskalen-Analyse komplexer Dreiphasensysteme” (FOR 2397). The second author would like to thank European Research Council (ERC) for the support received under the ERC Advanced Grant Agreement No. 341225. The constructive comments of three anonymous reviewers helped to improve the manuscript significantly.

References

  1. Adami, S., Hu, X.Y., Adams, N.A.: A new surface-tension formulation for multi-phase sph using a reproducing divergence approximation. J. Comput. Phys. 229(13), 5011–5021 (2010).  https://doi.org/10.1016/j.jcp.2010.03.022 CrossRefGoogle Scholar
  2. Angeli, P., Gavriilidis, A.: Taylor flow in microchannels. In: Li, D. (ed.) Encyclopedia of Microfluidics and Nanofluidics, pp. 1971–1976. Springer, Boston (2008).  https://doi.org/10.1007/978-0-387-48998-8_1526 CrossRefGoogle Scholar
  3. Aslannejad, H., Hassanizadeh, S.M., Raoof, A., de Winter, D., Tomozeiu, N., van Genuchten, M.: Characterizing the hydraulic properties of a porous coating of paper using fib-sem tomography and 3d pore-scale modeling. Chem. Eng. Sci. (2016).  https://doi.org/10.1016/j.ces.2016.11.021 Google Scholar
  4. Blake, T.D.: The contact angle and two-phase flow. Ph.D. thesis, University of Bristol (1969)Google Scholar
  5. Blake, T.D.: Dynamic contact angles and wetting kinetics. In: Berg, J.C. (ed.) Wettability, Surfactant Science Series, pp. 251–310. M. Dekker, New York (1993)Google Scholar
  6. Blake, T.D.: Forced wetting of a reactive surface. In: Interfaces, Wettability, Surface Forces and Applications: Special Issue in honour of the 65th Birthday of John Ralston, vol. 179–182, pp. 22–28 (2012). https://doi.org/10.1016/j.cis.2012.06.002
  7. Blake, T.D., Shikhmurzaev, Y.D.: Dynamic wetting by liquids of different viscosity. J. Colloid Interface Sci. 253(1), 196–202 (2002).  https://doi.org/10.1006/jcis.2002.8513 CrossRefGoogle Scholar
  8. Bonet, J., Lok, T.S.L.: Variational and momentum preservation aspects of smooth particle hydrodynamic formulations. Comput. Methods Appl. Mech. Eng. 180(1–2), 97–115 (1999).  https://doi.org/10.1016/S0045-7825(99)00051-1 CrossRefGoogle Scholar
  9. Brackbill, J.U., Kothe, D.B., Zemach, C.: A continuum method for modeling surface tension. J. Comput. Phys. 100(2), 335–354 (1992).  https://doi.org/10.1016/0021-9991(92)90240-Y CrossRefGoogle Scholar
  10. Brochard, F.: Motions of droplets on solid surfaces induced by chemical or thermal gradients. Langmuir 5(2), 432–438 (1989).  https://doi.org/10.1021/la00086a025 CrossRefGoogle Scholar
  11. Chen, Y., Kulenovic, R., Mertz, R.: Numerical study on the formation of taylor bubbles in capillary tubes. Nano Micro Mini Channels Comput. Heat Transf. 48(2), 234–242 (2009).  https://doi.org/10.1016/j.ijthermalsci.2008.01.004 Google Scholar
  12. Cummins, S.J., Rudman, M.: An SPH projection method. J. Comput. Phys. 152(2), 584–607 (1999).  https://doi.org/10.1006/jcph.1999.6246 CrossRefGoogle Scholar
  13. de Gennes, P.-G.: Wetting: statics and dynamics. Rev. Mod. Phys. 57(3), 827–863 (1985).  https://doi.org/10.1103/RevModPhys.57.827 CrossRefGoogle Scholar
  14. de Gennes, P.-G., Brochard-Wyart, F., Quéré, D.: Capillarity and Wetting Phenomena: Drops, Bubbles, Pearls, Waves. Springer Science & Business Media, Berlin (2003)Google Scholar
  15. Dussan, E.B.: On the spreading of liquids on solid surfaces: static and dynamic contact lines. Annu. Rev. Fluid Mech. 11(1), 371–400 (1979)CrossRefGoogle Scholar
  16. Extrand, C.: Water contact angles and hysteresis of polyamide surfaces. J. Colloid Interface Sci. 248(1), 136–142 (2002)CrossRefGoogle Scholar
  17. Ferrari, A., Jimenez-Martinez, J., Le Borgne, T., Méheust, Y., Lunati, I.: Challenges in modeling unstable two-phase flow experiments in porous micromodels. Water Resour. Res. 51(3), 1381–1400 (2015).  https://doi.org/10.1002/2014WR016384 CrossRefGoogle Scholar
  18. Francois, M., Shyy, W.: Computations of drop dynamics with the immersed boundary method, part 2: drop impact and heat transfer. Numer. Heat Transf. B Fundam. 44(2), 119–143 (2003).  https://doi.org/10.1080/713836348 CrossRefGoogle Scholar
  19. Frontiere, N., Raskin, C.D., Owen, J.M.: Crksph—a conservative reproducing kernel smoothed particle hydrodynamics scheme. J. Comput. Phys. 332((Supplement C)), 160–209 (2017).  https://doi.org/10.1016/j.jcp.2016.12.004 CrossRefGoogle Scholar
  20. Hao, L., Cheng, P.: Lattice boltzmann simulations of water transport in gas diffusion layer of a polymer electrolyte membrane fuel cell. J. Power Sources 195(12), 3870–3881 (2010).  https://doi.org/10.1016/j.jpowsour.2009.11.125 CrossRefGoogle Scholar
  21. Hassanizadeh, S.M., Gray, W.G.: Thermodynamic basis of capillary pressure in porous media. Water Resour. Res. 29(10), 3389–3405 (1993).  https://doi.org/10.1029/93WR01495 CrossRefGoogle Scholar
  22. Hu, X.Y., Adams, N.A.: A multi-phase SPH method for macroscopic and mesoscopic flows. J. Comput. Phys. 213(2), 844–861 (2006).  https://doi.org/10.1016/j.jcp.2005.09.001 CrossRefGoogle Scholar
  23. Hu, X.Y., Adams, N.A.: An incompressible multi-phase SPH method. J. Comput. Phys. 227(1), 264–278 (2007).  https://doi.org/10.1016/j.jcp.2007.07.013 CrossRefGoogle Scholar
  24. Huber, M., Dobesch, D., Kunz, P., Hirschler, M., Nieken, U.: Influence of orifice type and wetting properties on bubble formation at bubble column reactors. Chem. Eng. Sci. 152, 151–162 (2016a).  https://doi.org/10.1016/j.ces.2016.06.002 CrossRefGoogle Scholar
  25. Huber, M., Keller, F., Säckel, W., Hirschler, M., Kunz, P., Hassanizadeh, S.M., Nieken, U.: On the physically based modeling of surface tension and moving contact lines with dynamic contact angles on the continuum scale. J. Comput. Phys. 310, 459–477 (2016b).  https://doi.org/10.1016/j.jcp.2016.01.030 CrossRefGoogle Scholar
  26. Joekar, N.V., Hassanizadeh, S.M., Pyrak-Nolte, L.J., Berentsen, C.: Simulating drainage and imbibition experiments in a high-porosity micromodel using an unstructured pore network model. Water Resour. Res. 45(2), W02,430 (2009).  https://doi.org/10.1029/2007WR006641 Google Scholar
  27. Johnson, R.E., Dettre, R.H.: Contact angle hysteresis. III. Study of an idealized heterogeneous surface. J. Phys. Chem. 68(7), 1744–1750 (1964).  https://doi.org/10.1021/j100789a012 CrossRefGoogle Scholar
  28. Kunz, P., Hirschler, M., Huber, M., Nieken, U.: Inflow/outflow with dirichlet boundary conditions for pressure in ISPH. J. Comput. Phys. 326, 171–187 (2016a).  https://doi.org/10.1016/j.jcp.2016.08.046 CrossRefGoogle Scholar
  29. Kunz, P., Zarikos, I.M., Karadimitriou, N.K., Huber, M., Nieken, U., Hassanizadeh, S.M.: Study of multi-phase flow in porous media: comparison of SPH simulations with micro-model experiments. Transp. Porous Media 114(2), 581–600 (2016b).  https://doi.org/10.1007/s11242-015-0599-1 CrossRefGoogle Scholar
  30. Lafaurie, B., Nardone, C., Scardovelli, R., Zaleski, S., Zanetti, G.: Modelling merging and fragmentation in multiphase flows with surfer. J. Comput. Phys. 113(1), 134–147 (1994).  https://doi.org/10.1006/jcph.1994.1123 CrossRefGoogle Scholar
  31. Monaghan, J.J.: Smoothed particle hydrodynamics. Rep. Prog. Phys. 68(8), 1703 (2005). http://stacks.iop.org/0034-4885/68/i=8/a=R01
  32. Monaghan, J.J.: Smoothed particle hydrodynamics and its diverse applications. Annu. Rev. Fluid Mech. 44(1), 323–346 (2011).  https://doi.org/10.1146/annurev-fluid-120710-101220 CrossRefGoogle Scholar
  33. Morris, J.P., Fox, P.J., Zhu, Y.: Modeling low reynolds number incompressible flows using SPH. J. Comput. Phys. 136(1), 214–226 (1997).  https://doi.org/10.1006/jcph.1997.5776 CrossRefGoogle Scholar
  34. Oger, G., Marrone, S., Le Touzé, D., de Leffe, M.: SPH accuracy improvement through the combination of a quasi-Lagrangian shifting transport velocity and consistent ALE formalisms. J. Comput. Phys. 313(Supplement C), 76–98 (2016).  https://doi.org/10.1016/j.jcp.2016.02.039 CrossRefGoogle Scholar
  35. Renardy, M., Renardy, Y., Li, J.: Numerical simulation of moving contact line problems using a volume-of-fluid method. J. Comput. Phys. 171(1), 243–263 (2001).  https://doi.org/10.1006/jcph.2001.6785 CrossRefGoogle Scholar
  36. Sheng, P., Zhou, M.: Immiscible-fluid displacement: contact-line dynamics and the velocity-dependent capillary pressure. Phys. Rev. A 45(8), 5694–5708 (1992)CrossRefGoogle Scholar
  37. Shikhmurzaev, Y.D.: Moving contact lines in liquid/liquid/solid systems. J. Fluid Mech. 334, 211–249 (1997).  https://doi.org/10.1017/S0022112096004569 CrossRefGoogle Scholar
  38. Sussman, M., Fatemi, E., Smereka, P., Osher, S.: An improved level set method for incompressible two-phase flows. Comput. Fluids 27(5–6), 663–680 (1998).  https://doi.org/10.1016/S0045-7930(97)00053-4 CrossRefGoogle Scholar
  39. Szewc, K., Pozorski, J., Minier, J.P.: Analysis of the incompressibility constraint in the smoothed particle hydrodynamics method. Int. J. Numer. Methods Eng. 92(4), 343–369 (2012).  https://doi.org/10.1002/nme.4339 CrossRefGoogle Scholar
  40. Tartakovsky, A., Meakin, P.: Modeling of surface tension and contact angles with smoothed particle hydrodynamics. Phys. Rev. E 72(2), 26,301 (2005).  https://doi.org/10.1103/PhysRevE.72.026301 CrossRefGoogle Scholar
  41. Thompson, P.A., Robbins, M.O.: Simulations of contact-line motion: slip and the dynamic contact angle. Phys. Rev. Lett. 63(7), 766–769 (1989)CrossRefGoogle Scholar
  42. Thompson, P.A., Robbins, M.O.: Origin of stick-slip motion in boundary lubrication. Science 250(4982), 792–794 (1990).  https://doi.org/10.1126/science.250.4982.792 CrossRefGoogle Scholar
  43. Trask, N., Maxey, M., Kim, K., Perego, M., Parks, M.L., Yang, K., Xu, J.: A scalable consistent second-order SPH solver for unsteady low Reynolds number flows. Comput. Methods Appl. Mech. Eng. 289, 155–178 (2015).  https://doi.org/10.1016/j.cma.2014.12.027 CrossRefGoogle Scholar
  44. Wang, J., Do-Quang, M., Cannon, J.J., Yue, F., Suzuki, Y., Amberg, G., Shiomi, J.: Surface structure determines dynamic wetting. Sci. Rep. 5, 8474 EP (2015)Google Scholar
  45. Washburn, E.W.: The dynamics of capillary flow. Phys. Rev. 17(3), 273–283 (1921).  https://doi.org/10.1103/PhysRev.17.273 CrossRefGoogle Scholar
  46. Wendland, H.: Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree. Adv. Comput. Math. 4(1), 389–396 (1995).  https://doi.org/10.1007/BF02123482 CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute of Chemical Process EngineeringUniversity of StuttgartStuttgartGermany
  2. 2.Department of Earth Sciences, Faculty of GeosciencesUtrecht UniversityUtrechtThe Netherlands

Personalised recommendations