Oscillation characteristics of low Weber number impinging micro-droplets


Oscillation characteristics of micro-droplets, when in partial contact with a dry and homogeneous substrate, are investigated using a volume of fluid (VOF) numerical method. Water is used as a fluid in both numerical and experimental studies. The velocity vectors are plotted along the phase boundary line, i.e. along the droplet interface, to show how the contact angle impacts the droplet shape during the entire oscillation process. It has been predicted that when the surface/liquid combination is of larger contact angle, the water droplet tends to spread partially as the contact velocity dynamics dominate over inertia, thereby restricting the change in shape, i.e. resulting in lesser mode of oscillations. However, all droplets that are considered here show a damped harmonic motion with the amplitude gradually decreasing to zero. Particularly, at a lower Weber number impact, it is predicted that both the height and spreading dynamics exhibit a unique decaying function for each droplet size considered.

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  1. 1.

    de Gennes, P.G.: Wetting: statics and dynamics. Rev. Mod. Phys. 57(3), 827–863 (1985)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Xie, J., Wong, T.N., Duan, F.: Modelling on the dynamics of droplet impingement and bubble boiling in spray cooling. Int. J. Therm. Sci. 104, 469–479 (2016)

    Article  Google Scholar 

  3. 3.

    Yonemoto, Y., Kunugi, T.: Analytical consideration of liquid droplet impingement on solid surfaces. Sci. Rep. 7(1), 2362 (2017)

    Article  Google Scholar 

  4. 4.

    Serras-Pereira, J., Aleiferis, P.G., Walmsley, H.L., Davies, T.J., Cracknell, R.F.: Heat flux characteristics of spray wall impingement with ethanol, butanol, iso-octane, gasoline and E10 fuels. Int. J. Heat Fluid Flow 44, 662–683 (2013)

    Article  Google Scholar 

  5. 5.

    Yarin, A.L.: Drop impact dynamics: splashing, spreading, receding, bouncing..., Annu. Rev. Fluid Mech. 38(1), 159–192 (2006)

    MathSciNet  MATH  Article  Google Scholar 

  6. 6.

    Bayer, I.S., Megaridis, C.M.: Contact angle dynamics in droplets impacting on flat surfaces with different wetting characteristics. J. Fluid Mech. 558, 415–449 (2006)

    MATH  Article  Google Scholar 

  7. 7.

    Gatne, K.P., Jog, M.A., Manglik, R.M.: Surfactant-induced modification of low Weber number droplet impact dynamics. Langmuir 25(14), 8122–8130 (2009)

    Article  Google Scholar 

  8. 8.

    Rioboo, R., Marengo, M., Tropea, C.: Time evolution of liquid drop impact onto solid, dry surfaces (in English). Exp. Fluids 33(1), 112–124 (2002)

    Article  Google Scholar 

  9. 9.

    Manglik, R.M., Jog, M.A., Gande, S.K., Ravi, V.: Damped harmonic system modeling of post-impact drop-spread dynamics on a hydrophobic surface. Phys. Fluids 25(8), 082112 (2013)

    Article  Google Scholar 

  10. 10.

    Clanet, C., Béguin, C., Dric, Eacute, Richard, D., Quéré, D.: Maximal deformation of an impacting drop. J. Fluid Mech. 517, 199–208 (2004)

    MATH  Article  Google Scholar 

  11. 11.

    Mao, T., Kuhn, D.C.S., Tran, H.: Spread and rebound of liquid droplets upon impact on flat surfaces. AIChE J. 43(9), 2169–2179 (1997)

    Article  Google Scholar 

  12. 12.

    Hamlett, C.A.E., et al.: Transitions of water-drop impact behaviour on hydrophobic and hydrophilic particles. Eur. J. Soil Sci. 64(3), 324–333 (2013)

    Article  Google Scholar 

  13. 13.

    Bertola, V., Wang, M.: Dynamic contact angle of dilute polymer solution drops impacting on a hydrophobic surface. Colloids Surf. A Physicochem. Eng. Asp. 481, 600–608 (2015)

    Article  Google Scholar 

  14. 14.

    Bolleddula, D.A., Berchielli, A., Aliseda, A.: Impact of a heterogeneous liquid droplet on a dry surface: application to the pharmaceutical industry. Adv. Colloid Interface Sci. 159(2), 144–159 (2010)

    Article  Google Scholar 

  15. 15.

    Petit, J., et al.: A dimensional analysis approach for modelling the size of droplets formed by bi-fluid atomisation. J. Food Eng. 149, 237–247 (2015)

    Article  Google Scholar 

  16. 16.

    Wu, J., Huang, J.J., Yan, W.W.: Lattice Boltzmann investigation of droplets impact behaviors onto a solid substrate. Colloids Surf. A Physicochem. Eng. Asp. 484, 318–328 (2015)

    Article  Google Scholar 

  17. 17.

    Yonemoto, Y., Kunugi, T.: Experimental investigation on variations in geometric variables of a droplet on a low-surface-energy solid. Int. J. Heat Mass Transf. 73, 810–818 (2014)

    Article  Google Scholar 

  18. 18.

    LeClear, S., LeClear, J., Abhijeet, Park, K.-C., Choi, W.: Drop impact on inclined superhydrophobic surfaces. J. Colloid Interface Sci. 461, 114–121 (2016)

    Article  Google Scholar 

  19. 19.

    Wang, M.-J., Lin, F.-H., Ong, J.Y., Lin, S.-Y.: Dynamic behaviors of droplet impact and spreading–Water on glass and paraffin. Colloids Surf. A Physicochem. Eng. Asp. 339(1–3), 224–231 (2009)

    Article  Google Scholar 

  20. 20.

    Olgac, U., Izbassarov, D., Muradoglu, M.: Direct numerical simulation of an oscillating droplet in partial contact with a substrate. Comput. Fluids 77, 152–158 (2013)

    Article  Google Scholar 

  21. 21.

    Gunjal, P.R., Ranade, V.V., Chaudhari, R.V.: Dynamics of drop impact on solid surface: experiments and VOF simulations. AIChE J. 51(1), 59–78 (2005)

    Article  Google Scholar 

  22. 22.

    Wang, M.-J., Hung, Y.-L., Lin, F.-H., Lin, S.-Y.: Dynamic behaviors of droplet impact and spreading: a universal relationship study of dimensionless wetting diameter and droplet height. Exp. Therm. Fluid Sci. 33(7), 1112–1118 (2009)

    Article  Google Scholar 

  23. 23.

    Brant Foote, G.: A numerical method for studying liquid drop behavior: simple oscillation. J. Comput. Phys. 11(4), 507–530 (1973)

    MATH  Article  Google Scholar 

  24. 24.

    Mukherjee, S., Abraham, J.: Investigations of drop impact on dry walls with a lattice-Boltzmann model. J. Colloid Interface Sci. 312(2), 341–354 (2007)

    Article  Google Scholar 

  25. 25.

    Gupta, A., Kumar, R.: Droplet impingement and breakup on a dry surface. Comput. Fluids 39(9), 1696–1703 (2010)

    MATH  Article  Google Scholar 

  26. 26.

    Ganesan, S.: Simulations of impinging droplets with surfactant-dependent dynamic contact angle. J. Comput. Phys. 301, 178–200 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  27. 27.

    Kwak, G., Lee, D.W., Kang, I.S., Yong, K.: A study on the dynamic behaviors of water droplets impacting nanostructured surfaces. AIP Adv. 1(4), 042139 (2011)

    Article  Google Scholar 

  28. 28.

    Sato, Y., Ničeno, B.: A new contact line treatment for a conservative level set method. J. Comput. Phys. 231(10), 3887–3895 (2012)

    MathSciNet  MATH  Article  Google Scholar 

  29. 29.

    Pashos, G., Kokkoris, G., Boudouvis, A.G.: A modified phase-field method for the investigation of wetting transitions of droplets on patterned surfaces. J. Comput. Phys. 283, 258–270 (2015)

    Article  Google Scholar 

  30. 30.

    Zhou, W., Loney, D., Degertekin, F.L., Rosen, D.W., Fedorov, A.G.: What controls dynamics of droplet shape evolution upon impingement on a solid surface? AIChE J. 59(8), 3071–3082 (2013)

    Article  Google Scholar 

  31. 31.

    Lim, C., Lam, Y.: Phase-field simulation of impingement and spreading of micro-sized droplet on heterogeneous surface (in English). Microfluid. Nanofluidics 17(1), 131–148 (2014)

    MathSciNet  Article  Google Scholar 

  32. 32.

    Jiang, F., Wang, Y., Xiang, J., Liu, Z.: A comprehensive computational fluid dynamics study of droplet-film impact and heat transfer. Chem. Eng. Technol. 38(9), 1565–1573 (2015)

    Article  Google Scholar 

  33. 33.

    Malgarinos, I., Nikolopoulos, N., Marengo, M., Antonini, C., Gavaises, M.: VOF simulations of the contact angle dynamics during the drop spreading: standard models and a new wetting force model. Adv. Colloid Interface Sci. 212, 1–20 (2014)

    Article  Google Scholar 

  34. 34.

    Tan, H., Torniainen, E., Markel, D.P., Browning, R.N.K.: Numerical simulation of droplet ejection of thermal inkjet printheads. Int. J. Numer. Methods Fluids 77(9), 544–570 (2015)

    Article  Google Scholar 

  35. 35.

    Raessi, M., Mostaghimi, J., Bussmann, M.: A volume-of-fluid interfacial flow solver with advected normals. Comput. Fluids 39(8), 1401–1410 (2010)

    MathSciNet  MATH  Article  Google Scholar 

  36. 36.

    Yokoi, K., Vadillo, D., Hinch, J., Hutchings, I.: Numerical studies of the influence of the dynamic contact angle on a droplet impacting on a dry surface. Phys. Fluids 21(7), 072102 (2009)

    MATH  Article  Google Scholar 

  37. 37.

    McDonnell, A.G., et al.: Motility induced changes in viscosity of suspensions of swimming microbes in extensional flows. Soft Matter 11(23), 4658–4668 (2015). https://doi.org/10.1039/C4SM02742F

    Article  Google Scholar 

  38. 38.

    McDonnell, A.G., Jason, N.N., Yeo, L.Y., Friend, J.R., Cheng, W., Prabhakar, R.: Extensional viscosity of copper nanowire suspensions in an aqueous polymer solution. Soft Matter 11(41), 8076–8082 (2015). https://doi.org/10.1039/C5SM01940K

    Article  Google Scholar 

  39. 39.

    Yeo, L.Y., Friend, J.R.: Surface acoustic wave microfluidics. Annu. Rev. Fluid Mech. 46, 379–406 (2014)

    MathSciNet  MATH  Article  Google Scholar 

  40. 40.

    Qi, A., Yeo, L.Y., Friend, J.R.: Interfacial destabilization and atomization driven by surface acoustic waves. Phys. Fluids 20(7), 074103 (2008)

    MATH  Article  Google Scholar 

  41. 41.

    Collins, D.J., Manor, O., Winkler, A., Schmidt, H., Friend, J.R., Yeo, L.Y.: Atomization off thin water films generated by high-frequency substrate wave vibrations. Phys. Rev. E 86(5), 056312 (2012)

    Article  Google Scholar 

  42. 42.

    Brackbill, J.U., Kothe, D.B., Zemach, C.: A continuum method for modeling surface tension. J. Comput. Phys. 100(2), 335–354 (1992)

    MathSciNet  MATH  Article  Google Scholar 

  43. 43.

    Jo, J.H., Kim, W.T.: Numerical simulation of water droplet dynamics in a right angle gas channel of a polymer electrolyte membrane fuel cell. Int. J. Hydrogen Energy 40(26), 8368–8383 (2015)

    Article  Google Scholar 

  44. 44.

    Bhattacharjee, P.K., McDonnell, A.G., Prabhakar, R., Yeo, L.Y., Friend, J.: Extensional flow of low-viscosity fluids in capillary bridges formed by pulsed surface acoustic wave jetting. New J. Phys. 13(2), 023005 (2011)

    Article  Google Scholar 

  45. 45.

    Tan, M.K., Friend, J.R., Yeo, L.Y.: Interfacial jetting phenomena induced by focused surface vibrations. Phys. Rev. Lett. 103(2), 024501 (2009)

    Article  Google Scholar 

  46. 46.

    Das, S., Morsi, Y.S., Brooks, G., Chen, J.J.J., Yang, W.: Principal characteristics of a bubble formation on a horizontal downward facing surface. Colloids Surf. A Physicochem. Eng. Asp. 411, 94–104 (2012)

    Article  Google Scholar 

  47. 47.

    Ganesan, S., Tobiska, L.: Finite element simulation of an impinging liquid droplet. In: Bertram, A., Tomas, J. (eds.) Micro-Macro-Interaction, pp. 173–185. Springer, Berlin (2008)

    Google Scholar 

  48. 48.

    Barrett, J.W., Garcke, H., Nürnberg, R.: Eliminating spurious velocities with a stable approximation of viscous incompressible two-phase Stokes flow. Comput. Methods Appl. Mech. Eng. 267, 511–530 (2013)

    MathSciNet  MATH  Article  Google Scholar 

  49. 49.

    Malgarinos, I., Nikolopoulos, N., Gavaises, M.: Coupling a local adaptive grid refinement technique with an interface sharpening scheme for the simulation of two-phase flow and free-surface flows using VOF methodology. J. Comput. Phys. 300, 732–753 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  50. 50.

    Dodge, F.T.: The spreading of liquid droplets on solid surfaces. J. Colloid Interface Sci. 121(1), 154–160 (1988)

    Article  Google Scholar 

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Das, S., Mohammed, M.I., Gibson, I. et al. Oscillation characteristics of low Weber number impinging micro-droplets. Theor. Comput. Fluid Dyn. 33, 197–213 (2019). https://doi.org/10.1007/s00162-019-00489-9

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  • VOF simulation
  • Droplet impingement
  • Oscillation characteristics
  • Low Weber number