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Dynamics of Liquid Droplets

  • A. Mashayek
  • N. Ashgriz
Chapter

Abstract

In this chapter the basic physics and methods of calculation of the effective drag forces acting on drops in isolated-drop and multidrop configurations relevant to sprays are provided. The effect of various physical phenomena such as drop deformation, nonuniformity of the incoming flow, drop–drop interactions, drop–gas interactions, and evaporation on the drag coefficient on the drop, with special focus on the underlying physics, is highlighted.

Keywords

Drag coefficient Drag of deformed drops Droplet motion Evaporating droplets Flow past a droplet Interacting drops Interacting drops 

References

  1. 1.
    Batchelor, G. K., An Introduction to Fluid Dynamics, Cambridge University Press, pp. 331–343, 1967.Google Scholar
  2. 2.
    Oseen, C. W., Hydrodynamic, Akademische Verlag, Leipzig, 1927.Google Scholar
  3. 3.
    Voloshuk, V. M. and Sedunow, J. S., The Processes of Coagulation in Dispersed Systems, Nauka, Moscow, 1976.Google Scholar
  4. 4.
    Van Dyke, M., An Album of Fluid Motion, The Parabolic Press, Stanford, (1982).Google Scholar
  5. 5.
    Kundu, P. K. and Cohen, I. M., Fluid Mechanics, Fourth edition, Academic Press, San Diego, 2008.Google Scholar
  6. 6.
    Kelbaliyev, G. and Ceylan, K., Development of new empirical equations for estimation of drag coefficient, shape deformation, and rising velocity of gas bubbles or liquid drops. Chem. Eng. Commun. 194, 1623–1637, 2007.CrossRefGoogle Scholar
  7. 7.
    Hadamard, J. S., Mouvement Permanent Lent d’une Sphere Liquid et Visqueuse dans une Liquid Visquese, C.R. Acad. Sci. 152, 1735–1738, 1911.zbMATHGoogle Scholar
  8. 8.
    Taylor, T. and Acrivos, A., On the deformation and drag of a falling drop at low Reynolds numbers, J. Fluid Mech., 18, 466–476, 1964.zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Happer, J. and Moore, D. W., The motion of a spherical liquid drop at high Reynolds number, J. Fluid Mech., 32, part 2, 367–391, 1968.CrossRefGoogle Scholar
  10. 10.
    Rivkind, V. Y. and Ryskin, G. M. Flow structure in motion of a spherical drop in a fluid medium at intermediate Reynolds numbers. Fluid Dynamics (English translation of: Izv. Akad. Nauk SSSR Mekh. Zhidk. Gaza), 11, 5–12, 1976.CrossRefGoogle Scholar
  11. 11.
    Oliver, D. L. R. and Chung, J. N. Flow about a fluid sphere at low to moderate Reynolds numbers. J. Fluid Mech., 177, 1–18, 1987.zbMATHCrossRefGoogle Scholar
  12. 12.
    Feng, Z. G. and Michaelides, E. E., Drag coefficients of viscous spheres at intermediate and high Reynolds numbers, J. Fluids Eng. 123(Issue), 2001.Google Scholar
  13. 13.
    Schiller, L. and Naumann, A. Über die grundlegenden berechungen bei der schwerkraftaufbereitung. Vereines Deutscher Ingenieure 7, 318, 1933.Google Scholar
  14. 14.
    Putnam, A. Integratable form of droplet drag coefficient. ARS J. 1961.Google Scholar
  15. 15.
    Clift, R. and Gauvin, W. H. The motion of particles in turbulent gas streams. Proc. Chemeca, 1970.Google Scholar
  16. 16.
    Schaaf, S. A. and Chambre, P. L. High speed aerodynamics and jet propulsion VIII, Princeton University Press, Princeton, 1958.Google Scholar
  17. 17.
    Crowe, C. T., Babcock, W., Willoughby, P. G., and Carlson, R. L., Measurement of particle drag coefficient in flow regimes encountered by particles in a rocket nozzle, United Technology Report 2296-FR, 1969.Google Scholar
  18. 18.
    Hermsen, R. W. Review of particle drag models. Subcommittee 12th Meeting Minutes, CPIA Publication, 1979.Google Scholar
  19. 19.
    Wadhwa, A. R., Magi, V., and Abraham, J. Transient deformation and drag of decelerating drops in axisymmetric flows. Phys. Fluids, 19, 113301, 2007.CrossRefGoogle Scholar
  20. 20.
    Mashayek, A. and Ashgriz, N., Model or deformation of drops and liquid jets in gaseous crossflows. AIAA J. 47(2) (2009).Google Scholar
  21. 21.
    Desantes, J. M., Margot, X., Pastor, J. M., Chavez, M., and Pinzello, A. CFD-Phenomenological diesel spray analysis under evaporative conditions. Energy Fuels 23, 3919–3929, 2009.CrossRefGoogle Scholar
  22. 22.
    Achenbach, E. Distribution of local pressure and skin friction around a circular cylinder in crossflow up to Re = 5×106. Fluid Mechanics 34, 625–639, 1968.Google Scholar
  23. 23.
    O’Rourke, P. J. and Amsden, A. A. The TAB Method for Numerical Calculation of Spray Droplet Breakup, SAE Paper 872089.Google Scholar
  24. 24.
    Liu, A. B., Mather, D., and Reitz, R. D. Modeling the effects of drop drag and breakup on fuel sprays, SAE Paper 930072.Google Scholar
  25. 25.
    Clair, B. L. and Hamielec, A. Viscous flow through particle assemblages at intermediate Reynolds numbers. IεEC Fundam. 7, 308–315, 1968.Google Scholar
  26. 26.
    Poo J. Y. and Ashgriz N. Variation of drag coefficients in an interacting drop stream, Exp. Fluids, 11, 1–8, 1991.CrossRefGoogle Scholar
  27. 27.
    Cybulski, A., Dalen, M. V., Verkerk, J., and Berg, P. V. D. Gas-particle heat-transfer coefficients in packed beds. Chem. Eng. Sci. 30, 1015, 1975.CrossRefGoogle Scholar
  28. 28.
    Difelice, R. The voidage function for fluid-particle interaction systems. Int. J. Multiph. Flow 20, 153–159, 1994.CrossRefGoogle Scholar
  29. 29.
    Dwyer, H., Nirschl, H., Kerschl, P., and Denk, V. Heat, mass and momentum transfer about arbitrary groups of particles. Twenty-Fifth Symposium on Combustion, Irvine, pp. 389–395, 1994.Google Scholar
  30. 30.
    Ergun, S. 1952 Fluid flow through packed columns. Chem. Eng. Prog. 48(2), 89–94, 1952.Google Scholar
  31. 31.
    Gibilaro, L. G., Felice, R. I. D., and Waldram, S. P. Generalized friction factor and drag coefficient correlations for fluid-particle interactions. Chem. Eng. Sci. 40, 1817–1823, 1985.CrossRefGoogle Scholar
  32. 32.
    Hill, R. J. Koch, D. L., and Ladd, A. J. C. The first effect of fluid inertia on flows in ordered and random arrays of spheres. J. Fluid Mech. 448, 213–248, 2001a.zbMATHMathSciNetGoogle Scholar
  33. 33.
    Hill, R. J. Koch, D. L., and Ladd, A. J. C. Moderate-Reynolds-number flows in ordered and random arrays of spheres. J. Fluid Mech. 448, 243–278, 2001b.zbMATHMathSciNetGoogle Scholar
  34. 34.
    Ishii, M. and Zuber, N. Drag coefficient and relative velocity in bubbly, droplet or particulate flows. AIChE J. 25(5), 843–855, 1979.CrossRefGoogle Scholar
  35. 35.
    Kim, I., Elghobashi, S., and Sirignano, W. Three-dimensional flow over two spheres placed side by side. J. Fluid Mech. 246, 465–488, 1993.zbMATHCrossRefGoogle Scholar
  36. 36.
    Mulholland, J., Srivastava, R., and Wendt, J. Influence of droplet spacing on drag coefficient in nonevaporating, monodisperse streams. AIAA J. 26(10), 1231–1237, 1988.CrossRefGoogle Scholar
  37. 37.
    Poo, J. and Ashgriz, N. Variation of drag coefficients in an interacting drop stream. Exp. Fluids 11, 1–8, 1991.CrossRefGoogle Scholar
  38. 38.
    Racmachandran, R., Kleinstreuer, C., and Wang, T.-Y. Forced convection heat transfer of interacting spheres. Numer. Heat Transf. 15, 471–487, 1989.CrossRefGoogle Scholar
  39. 39.
    Tal, R., Lee, D., and Siriganano, W. Hydrodynamics and heat transfer in sphere assemblages cylindrical cell models. Int. J. Heat Mass Transf. 26(9), 1265–1273, 1983.zbMATHCrossRefGoogle Scholar
  40. 40.
    Tal, R., Lee, D., and Siriganano, W. Heat and mass momentum transfer around a pair of spheres in viscous flow. Int. J. Heat Mass Transf. 27(11), 1953–1962, 1984.zbMATHCrossRefGoogle Scholar
  41. 41.
    Tal, R. and Sirignano, W. Cylindrical cell model for hydrodynamics of particles assemblages at intermediate Reynolds numbers. AIChE J. 28(2), 233–237, 1982.CrossRefGoogle Scholar
  42. 42.
    Zhu, C., Liang, S.-C., and Fan, L.-S. Particle wake effects on the drag force of an interactive particle. Int. J. Multiph. Flow 20(1), 117–129, 1994.zbMATHCrossRefGoogle Scholar
  43. 43.
    Sadhal, S.S., Ayyaswamy, P.S., and Chung, J.N. Transport Phenomena with drops and bubbles, 1997.Google Scholar
  44. 44.
    Giles D. K., Energy conversion and distribution in pressure atomization, Trans. ASAE, 31(6), 1668–1673, 1988.Google Scholar
  45. 45.
    Rhee J.B., Young W., and Bode L. E., Transport of spray droplets from flat-fan nozzles, ASAE paper No. 90–1001, 1990.Google Scholar
  46. 46.
    Rusche, H. and Issa, R. I., The Effects of voidage on the drag force on particles, droplets and bubbles in dispersed two-phase flow, Proceedings of the 2nd Japanese-European Two-Phase Flow Group Meeting, Tsukuba, Japan, 2000.Google Scholar
  47. 47.
    Rudinger, G., Fundamentals of Gas-Particle Flow, Handbook of Powder Technology, Vol. 2, Elsevier Scientific Publishing Co., Amesterdam, 1980.Google Scholar
  48. 48.
    Eisenklam, P., Arunachlaman, S. A., and Weston, J. A., Evaporation rates and drag resistances of burning drops, 11th International Symposium on Combustion, Pittsburgh, pp. 715–728, 1967.Google Scholar
  49. 49.
    Barnea, E. and Mizrahi, J. A. Generalized approach to the fluid-dynamics of particulate systems, part 1: General correlation for fluidization and sedimentation. Chem. Eng. 5, 171–189, 1973.CrossRefGoogle Scholar
  50. 50.
    Taned, S. Experimental investigation of the wakes behind cylinders and plates at low Reynolds numbers. Phys. Soc. Jpn. 11, 302–307, 1956.CrossRefGoogle Scholar

Copyright information

© Springer US 2011

Authors and Affiliations

  • A. Mashayek
    • 1
  • N. Ashgriz
  1. 1.Department of PhysicsUniversity of TorontoTorontoCanada

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