Acta Mechanica Sinica

, Volume 35, Issue 1, pp 32–44 | Cite as

Quasi-static simulation of droplet morphologies using a smoothed particle hydrodynamics multiphase model

  • Xiangwei Dong
  • Jianlin LiuEmail author
  • Sai Liu
  • Zengliang Li
Research Paper


Numerical simulation of the morphology of a droplet deposited on a solid surface requires an efficient description of the three-phase contact line. In this study, a simple method of implementing the contact angle is proposed, combined with a robust smoothed particle hydrodynamics multiphase algorithm (Zhang 2015). The first step of the method is the creation of the virtual liquid–gas interface across the solid surface by means of dummy particles, thus the calculated surface tension near the triple point serves to automatically modulate the dynamic contact line towards the equilibrium state. We simulate the evolution process of initially square liquid lumps on flat and curved surfaces. The predictions of droplet profiles are in good agreement with the analytical solutions provided that the macroscopic contact angle is accurately implemented. Compared to the normal correction method, the present method is straightforward without the need to manually alter the normal vectors. This study presents a robust algorithm capable of capturing the physics of the static wetting. It may hold great potentials in bio-inspired superhydrophobic surfaces, oil displacement, microfluidics, ore floatation, etc.


Smoothed particle hydrodynamics Virtual interface method Multiphase flow Macroscopic contact angle Droplet morphology Curved surfaces 



The work was supported by the National Natural Science Foundation of China (Grants 11672335 and 11611530541), China Postdoctoral Science Foundation (Grant 2017M622307), Shandong Natural Science Foundation (Grant ZR201709210320), Fundamental Research Funds for the Central Universities (Grant 18CX02153A), and the Endeavour Australia Cheung Kong Research Fellowship Scholarship from the Australian government.


  1. 1.
    De Gans, B.J., Duineveld, P.C., Schubert, U.S.: Inkjet printing of polymers: state of the art and future developments. Adv. Mater. 16, 203–213 (2004)CrossRefGoogle Scholar
  2. 2.
    Goldszal, A., Jacques, B.: Wet agglomeration of powders: from physics toward process optimization. Powder Technol. 117, 221–231 (2001)CrossRefGoogle Scholar
  3. 3.
    Squires, T.M., Quake, S.R.: Microfluidics: fluid physics at the nanoliter scale. Rev. Mod. Phys. 77, 977–1026 (2005)CrossRefGoogle Scholar
  4. 4.
    Sun, M., Luo, C., Xu, L., et al.: Artificial lotus leaf by nanocasting. Langmuir ACS J. Surf. Colloids. 21, 8978–8981 (2005)CrossRefGoogle Scholar
  5. 5.
    Pate, J.S., Atkins, C.A., White, S.T., et al.: Nitrogen nutrition and xylem transport of nitrogen in ureide-producing grain legumes. Plant Physiol. 65, 961–965 (1980)CrossRefGoogle Scholar
  6. 6.
    Gao, X., Jiang, L.: Biophysics: water-repellent legs of water striders. Nature 432, 36 (2004)CrossRefGoogle Scholar
  7. 7.
    Nguyen, V.T., Park, W.G.: A volume-of-fluid (VOF) interface-sharpening method for two-phase incompressible flows. Comput. Fluids 152, 104–119 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Liu, G.R., Liu, M.B.: Smoothed Particle Hydrodynamics: A Meshfree Particle Method. World Scientific, Singapore (2003)CrossRefzbMATHGoogle Scholar
  9. 9.
    Zhang, Z.L., Ma, T., Liu, M.B., et al.: Numerical study on high velocity impact welding using a modified SPH method. Int. J. Comput. Methods (2018). zbMATHGoogle Scholar
  10. 10.
    Xu, A., Zhao, T.S., An, L., et al.: A three-dimensional pseudo-potential-based lattice Boltzmann model for multiphase flows with large density ratio and variable surface tension. Int. J. Heat Fluid Flow 56, 261–271 (2015)CrossRefGoogle Scholar
  11. 11.
    Xu, A., Shyy, W., Zhao, T.: Lattice Boltzmann modeling of transport phenomena in fuel cells and flow batteries. Acta Mech. Sin. 33, 555–574 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Gingold, R.A., Monaghan, J.J.: Smoothed particle hydrodynamics: theory and application to non-spherical stars. Mon. Not. R. Astron. Soc. 181, 375–389 (1977)CrossRefzbMATHGoogle Scholar
  13. 13.
    Monaghan, J.J.: Simulating free surface flows with SPH. J. Comput. Phys. 110, 399–406 (1994)CrossRefzbMATHGoogle Scholar
  14. 14.
    Shao, J.R., Yang, Y., Gong, H.F., et al.: Numerical simulation of water entry with improved SPH method. Int. J. Comput. Methods (2018). zbMATHGoogle Scholar
  15. 15.
    Randles, P.W., Libersky, L.D.: Smoothed particle hydrodynamics: some recent improvements and applications. Comput. Methods Appl. Mech. Eng. 139, 375–408 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Hu, Y.X., Adams, N.A.: A multi-phase SPH method for macroscopic and mesoscopic flows. J. Comput. Phys. 213, 844–861 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Ming, F.R., Sun, P.N., Zhang, A.M.: Numerical investigation of rising bubbles bursting at a free surface through a multiphase SPH model. Meccanica 52, 2665–2684 (2017)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Zhang, A.M., Sun, P.N., Ming, F.R.: An SPH modeling of bubble rising and coalescing in three dimensions. Comput. Methods Appl. Mech. Eng. 294, 189–209 (2015)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Kunz, P., Zarikos, I.M., Karadimitriou, N.K., et al.: Study of multi-phase flow in porous media: comparison of SPH simulations with micro-model experiments. Transp. Porous Media 114, 581–600 (2016)CrossRefGoogle Scholar
  20. 20.
    Zisi, I., Messahel, R., Boudlal, A., et al.: Validation of robust SPH schemes for fully compressible multiphase flows. Int. J. Multiphysics 9, 225–234 (2016)CrossRefGoogle Scholar
  21. 21.
    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, 5011–5021 (2010)CrossRefzbMATHGoogle Scholar
  22. 22.
    Szewc, K., Pozorski, J., Minier, J.P.: Simulations of single bubbles rising through viscous liquids using smoothed particle hydrodynamics. Int. J. Multiph. Flow 50, 98–105 (2013)CrossRefGoogle Scholar
  23. 23.
    Shao, J.R., Li, S.M., Liu, M.B.: Numerical simulation of violent impinging jet flows with improved SPH method. Int. J. Comput. Methods 13, 1641001 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Nugent, S., Posch, H.A.: Liquid drops and surface tension with smoothed particle applied mechanics. Phys. Rev. E 62, 4968–4975 (2000)CrossRefGoogle Scholar
  25. 25.
    Tartakovsky, A., Meakin, P.: Modeling of surface tension and contact angles with smoothed particle hydrodynamics. Phys. Rev. E 72, 026301 (2005)CrossRefGoogle Scholar
  26. 26.
    Brackbill, J.U., Kothe, D.B., Zemach, C.: A continuum method for modeling surface tension. J. Comput. Phys. 100, 335–354 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Morris, J.P.: Simulating surface tension with smoothed particle hydrodynamics. Int. J. Numer. Methods Fluids 33, 333–353 (2000)CrossRefzbMATHGoogle Scholar
  28. 28.
    Liu, M.B., Liu, G.R.: Meshfree particle simulation of micro channel flows with surface tension. Comput. Mech. 35, 332–341 (2005)CrossRefzbMATHGoogle Scholar
  29. 29.
    Hu, Y.X., Adams, N.A.: A multi-phase SPH method for macroscopic and mesoscopic flows. J. Comput. Phys. 213, 844–861 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Das, A.K., Das, P.K.: Equilibrium shape and contact angle of sessile drops of different volumes—computation by SPH and its further improvement by DI. Chem. Eng. Sci. 65, 4027–4037 (2010)CrossRefGoogle Scholar
  31. 31.
    Breinlinger, T., Polfer, P., Hashibon, A., et al.: Surface tension and wetting effects with smoothed particle hydrodynamics. J. Comput. Phys. 243, 14–27 (2013)CrossRefzbMATHGoogle Scholar
  32. 32.
    Yeganehdoust, F., Yaghoubi, M., Emdad, H., et al.: Numerical study of multiphase droplet dynamics and contact angles by smoothed particle hydrodynamics. Appl. Math. Model 40, 8493–8512 (2016)MathSciNetCrossRefGoogle Scholar
  33. 33.
    de Gennes, P.G., Brochard-Wyart, F., Quere, D., et al.: Capillarity and Wetting Phenomena: Drops, Bubbles, Pearls, Waves. Springer (2004)Google Scholar
  34. 34.
    Bormashenko, E.: Young, Boruvka-Neumann, Wenzel and Cassie-Baxter equations as the transversality conditions for the variational problem of wetting. Colloids Surf. A 345, 163–165 (2009)CrossRefGoogle Scholar
  35. 35.
    Dilts, G.A.: Moving least-squares particle hydrodynamics II: conservation and boundaries. Int. J. Numer. Methods Eng. 48, 1503–1524 (2000)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Chinese Society of Theoretical and Applied Mechanics; Institute of Mechanics, Chinese Academy of Sciences and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Xiangwei Dong
    • 1
    • 2
  • Jianlin Liu
    • 1
    Email author
  • Sai Liu
    • 1
  • Zengliang Li
    • 2
  1. 1.Department of Engineering Mechanics, College of Pipeline and Civil EngineeringChina University of Petroleum (East China)QingdaoChina
  2. 2.College of Mechanical and Electronic EngineeringChina University of Petroleum (East China)QingdaoChina

Personalised recommendations