An Analytical Approach to Model the Effect of Evaporation on Oscillation Amplitude of Liquid Drops in Gaseous Environment

  • Gautham Varma Raja KochanattuEmail author
  • Gianpietro Elvio Cossali
  • Simona Tonini
Conference paper
Part of the Fluid Mechanics and Its Applications book series (FMIA, volume 121)


The combined effect of evaporation and oscillation of liquid drops in gaseous stagnant environment is analytically modelled. Mechanical energy and mass balances are used to derive the time evolution of drop size and amplitude of oscillation. Two approaches, based on different assumptions about the kinetic energy distribution inside the drop, are used to evaluate the energy loss due to evaporation. Conditions for oscillation damping by evaporation are derived. Application of the model to the case of water, acetone and n-dodecane drops evaporating in hot air shows a non neglectful decrease of drop lifetime, with respect to non-oscillating drops.


  1. 1.
    Sazhin, S.S.: Droplets and Sprays. Springer, London (2014)CrossRefGoogle Scholar
  2. 2.
    Sazhin, S.S.: Modelling fuel droplet heating and evaporation: recent results and unsolved problems. Fuel 196, 69–101 (2017)CrossRefGoogle Scholar
  3. 3.
    Fuchs, N.A.: Vaporisation and droplet growth in gaseous media. Pergamon Press, London (1959)Google Scholar
  4. 4.
    Abramzon, B., Sirignano, W.A.: Droplet vaporization model for spray combustion calculations. Int. J. Heat Mass Transfer 32(9), 1605–1618 (1989)CrossRefGoogle Scholar
  5. 5.
    Aggarwal, S.K., Mongia, H.C.: Multicomponent and high-pressure effects on droplet vaporization. J. Eng. Gas Turbines Power 24, 248–255 (2002)CrossRefGoogle Scholar
  6. 6.
    Tonini, S., Cossali, G.E.: A novel vaporisation model for a single-component drop in high temperature air streams. Int. J. Therm. Sci. 75, 194–203 (2014)CrossRefGoogle Scholar
  7. 7.
    Sobac, B., Talbot, P., Haut, B., Rednikov, A., Colinet, P.: A comprehensive analysis of the evaporation of a liquid spherical drop. J. Colloid Interface Sci. 438, 306–317 (2015)CrossRefGoogle Scholar
  8. 8.
    Cossali, G.E., Tonini, S.: An analytical model of heat and mass transfer from liquid drops with temperature dependence of gas thermo-physical properties. Int. J. Heat Mass Transfer 138, 1166–1177 (2019)CrossRefGoogle Scholar
  9. 9.
    Tong, A.Y., Sirignano, W.A.: Multicomponent transient droplet vaporization with internal circulation: integral equation formulation. Numerical Heat Transfer 10, 253–278 (1986)CrossRefGoogle Scholar
  10. 10.
    Zeng, Y., Lee, Y.C.F.: Multicomponent-fuel film-vaporization model for multidimensional computations. J. Propulsion Power 16, 964–973 (2000)CrossRefGoogle Scholar
  11. 11.
    Al Qubeissi, M., Sazhin, S.S., Crua, C., Turner, J., Heikal, M.R.: Modelling of biodiesel fuel droplet heating and evaporation: effects of fuel composition. Fuel 154, 308–318 (2015)CrossRefGoogle Scholar
  12. 12.
    Li, J., Zhang, J.: A theoretical study of the spheroidal droplet evaporation in forced convection. Phys. Lett. A 378(47), 3537–3543 (2014)CrossRefGoogle Scholar
  13. 13.
    Tonini, S., Cossali, G.E.: One-dimensional analytical approach to modelling evaporation and heating of deformed drops. Int. J. Heat Mass Transfer 97, 301–307 (2016)CrossRefGoogle Scholar
  14. 14.
    Zubkov, V.S., Cossali, G.E., Tonini, S., Rybdylova, O., Crua, C., Heikal, M., Sazhin, S.S.: Mathematical modelling of heating and evaporation of a spheroidal droplet. Int. J. Heat Mass Transfer 108, 2181–2190 (2017)CrossRefGoogle Scholar
  15. 15.
    Sirignano, W.A.: Fluid Dynamics and Transport of Droplets and Sprays. 2nd edn, Cambridge University Press (2010)Google Scholar
  16. 16.
    Gusev, I.G., Krutitskii, P.A., Sazhin, S.S., Elwardany, A.E.: New solutions to the species diffusion equation inside droplets in the presence of the moving boundary. Int. J. Heat Mass Transfer 55(7), 2014–2021 (2012)CrossRefGoogle Scholar
  17. 17.
    Sazhin, S.S., Krutitskii, P.A., Gusev, I.G., Heikal, M.R.: Transient heating of an evaporating droplet. Int. J. Heat Mass Transfer 53(13), 2826–2836 (2010)CrossRefGoogle Scholar
  18. 18.
    Sazhin, S.S., Krutitskii, P.A., Gusev, I.G., Heikal, M.R.: Transient heating of an evaporating droplet with presumed time evolution of its radius. Int. J. Heat Mass Transfer 54(5), 1278–1288 (2011)CrossRefGoogle Scholar
  19. 19.
    Tonini, S., Cossali, G.E.: Modelling liquid drop heating and evaporation: the effect of drop shrinking. Comp. Therm. Sci 10(3), 273–283 (2018)CrossRefGoogle Scholar
  20. 20.
    Rayleigh, L.: On the capillary phenomena of jets. Proc. R. Soc. Lond. A 29, 71–97 (1879)CrossRefGoogle Scholar
  21. 21.
    Lamb, H.: On the oscillations of a viscous spheroid. Proc. Lond. Math. Soc. 13, 51–66 (1881)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Lamb, H.: Hydrodynamics, 6th edn. Cambridge University Press, United Kingdom (1932)zbMATHGoogle Scholar
  23. 23.
    Chandrasekhar, S.: The oscillations of a viscous liquid globe. Proc. London Math. Soc. 9, 141–149 (1959)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Miller, C., Scriven, L.: The oscillations of a fluid droplet immersed in another fluid. J. Fluid Mech. 32, 417–435 (1968)CrossRefGoogle Scholar
  25. 25.
    Prosperetti, A.: Free oscillations of drops and bubbles: the initial-value problem. J. Fluid Mech. 100(2), 333–347 (1980)CrossRefGoogle Scholar
  26. 26.
    Tsamopoulos, J.A., Brown, R.A.: Nonlinear oscillations of inviscid drops and bubbles. J. Fluid Mech. 127, 519–537 (1983)CrossRefGoogle Scholar
  27. 27.
    Trinh, E.H., Zwern, A., Wang, T.G.: An experimental study of small-amplitude drop oscillations in immiscible liquid systems. J. Fluid Mech. 115, 453–474 (1982)CrossRefGoogle Scholar
  28. 28.
    Trinh, E.H., Wang, T.G.: Large-amplitude free and driven drop-shape oscillations: experimental observations. J. Fluid Mech. 122, 315–338 (1982)CrossRefGoogle Scholar
  29. 29.
    Becker, E., Hiller, W., Kowalewski, T.: Experimental and theoretical investigation of large-amplitude oscillations of liquid droplets. J. Fluid Mech. 231, 189–210 (1981)CrossRefGoogle Scholar
  30. 30.
    Becker, E., Hiller, W., Kowalewski, T.: Nonlinear dynamics of viscous droplets. J. Fluid Mech. 258, 191–216 (1994)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Brenn, G., Teichtmeister, S.J.: Linear shape oscillations and polymeric time scales of viscoelastic drops. J. Fluid Mech. 733, 504–527 (2013)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Trinh, E.H., Thiessen, D.B., Holt, R.G.: Driven and freely decaying nonlinear shape oscillations of drops and bubbles immersed in a liquid: experimental results. J. Fluid Mech. 364, 253–272 (1998)CrossRefGoogle Scholar
  33. 33.
    Wang, T.G., Anilkumar, A.V., Lee, C.P.: Oscillations of liquid drops: results from USML-1 experiments in Space. J. Fluid Mech. 308, 1–14 (1996)CrossRefGoogle Scholar
  34. 34.
    Azuma, H., Yoshihara, S.: Three-dimensional large-amplitude drop oscillations: experiments and theoretical analysis. J. Fluid Mech. 393, 309–332 (1999)CrossRefGoogle Scholar
  35. 35.
    Al Zaitone, B.: Oblate spheroidal drop evaporation in an acoustic levitator. Int. J. Heat Mass Transfer 126, 164–172 (2018)CrossRefGoogle Scholar
  36. 36.
    Mashayek, F.: Dynamics of evaporating drops, Part I: formulation and evaporation model. Int. J. Heat Mass Transf. 44(8), 1517–1526 (2001)CrossRefGoogle Scholar
  37. 37.
    Mashayek, F.: Dynamics of evaporating drops, Part II: free oscillations. Int. J. Heat Mass Transf. 44(8), 1527–1541 (2001)CrossRefGoogle Scholar
  38. 38.
    Tonini, S., Cossali, G.E.: An evaporation model for oscillating spheroidal drops. Int. Comm. Heat Mass Transf. 51, 18–24 (2014)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  • Gautham Varma Raja Kochanattu
    • 1
    Email author
  • Gianpietro Elvio Cossali
    • 1
  • Simona Tonini
    • 1
  1. 1.Department of Engineering and Applied SciencesUniversity of BergamoBergamoItaly

Personalised recommendations