Skip to main content
Log in

A new method for determining hydrodynamic effects on the collision of two spheres

  • Articles
  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

Abstract

A sphere falling in a fluid may collide with another sphere falling more slowly if, when the spheres are far apart vertically, the horizontal distance between their centers is less than or equal to a critical radius. Accurate prediction of aerosol particle coagulation requires a good understanding of this process. Previously reported optical techniques for measuring hydrodynamic effects on this phenomenon have inherent difficulties detecting grazing collisions and hence in determining the critical radius. In this work, a novel detection technique is demonstrated and it is shown that the critical radius may be determined from the sound generated by the collision of two spheres in a viscous liquid. The technique is shown to provide a more precise and decisive indication of when hard spheres collide.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. C. E. Abbott, Experimental cloud droplet collection efficiencies,J. Geophys. Res. 79:3098 (1974).

    Google Scholar 

  2. M. K. Alam, The effect of van der Waals and viscous forces on aerosol coagulation,Aerosol Sci. Technol. 6:41 (1987).

    Google Scholar 

  3. G. K. Batchelor, Sedimentation in a dilute polydisperse system of interacting spheres. Part 1. General theory,J. Fluid Mech. 119:379 (1982).

    Google Scholar 

  4. K. V. Beard and H. R. Pruppacher, An experimental test of the theoretically calculated collision efficiency of cloud drops,J. Geophys. Res. 73:6407 (1968).

    Google Scholar 

  5. H. Brenner, The slow motion of a sphere through a viscous fluid towards a plane surface,Chem. Eng. Sci. 16:242 (1961).

    Google Scholar 

  6. R. Clift, J. R. Grace, and M. E. Weber,Bubbles, Drops and Particles (Academic Press, 1978), Chapter 9.

  7. R. H. Davis, The rate of coagulation of a dilute polydisperse system of sedimenting spheres,J. Fluid Mech. 145:179 (1984).

    Google Scholar 

  8. R. H. Davis, J. M. Serayssol, and E. J. Hinch, The elastohydrodynamic collision of two spheres,J. Fluid Mech. 163:479 (1986).

    Google Scholar 

  9. M. H. Davis, Collision of small cloud droplets: Gas kinetic effects,J. Atmos. Sci. 29:911 (1972).

    Google Scholar 

  10. I. H. Dunbar and J. Fermandjian, Comparison of sodium aerosol codes, Nuclear Safety ad Technology, Containment Loading and Response Safety Working Group Fast Reactor Coordinating Committee, Commission of the European Communities, ECI-878-B-7221-82UK (1984).

  11. N. Fuchs, On the theory of overhead precipitation of warm clouds,Dokl. Akad. Nauk SSSR 81:1043 (1951).

    Google Scholar 

  12. A. L. Graham, L. A. Mondy, J. D. Miller, N. J. Wagner, and W. A. Cook, Numerical simulations of eccentricity and end effects in falling ball rheometry,J. Rheol. 33:1107 (1989).

    Google Scholar 

  13. J. Happel and H. Brenner,Low Reynolds Number Hydrodynamics (Murtinus Nijhoff, Boston, 1973), Chapter 7.

    Google Scholar 

  14. L. M. Hocking, The effect of slip on the motion of a sphere close to a wall and of two adjacent spheres,J. Eng. Math. 7:207 (1973).

    Google Scholar 

  15. V. G. Horguani, Determination of the trapping coefficient of cloud particles of comparable size by a model experiment,Izv. Atm. Oceanic Phys. 1:208 (1965).

    Google Scholar 

  16. P. R. Jonas, The collision efficiency of small drops,Q. J. R. Meteorol. Soc. 98:681 (1972).

    Google Scholar 

  17. J. D. Klett and M. H. Davis, Theoretical collision efficiencies of cloud droplets at small Reynolds numbers,J. Atmos. Sci. 30:107 (1973).

    Google Scholar 

  18. G. L. Lin and S. C. Lee, Collision efficiency of water drops in the atmosphere,J. Atmos. Sci. 32:1412 (1975).

    Google Scholar 

  19. S. K. Loyalka, R. C. Warder, W. Meyer, and T. S. Storvick, Experimental measurements of aerosol gravitational collision efficiency,Trans. Am. Nucl. Soc. 38:401 (1987).

    Google Scholar 

  20. A. D. Maude, End effects in a falling-sphere viscometer,Br. J. Appl. Phys. 12:293 (1961).

    Google Scholar 

  21. H. T. Ochs, R. R. Czys, and K. V. Beard, Laboratory measurements of coalescence efficiencies for small precipitation drops,J. Atmos. Sci. 43:225 (1986).

    Google Scholar 

  22. M. E. O'Neill and S. R. Majumdar, Asymmetrical slow viscous fluid motions caused by the translation or rotation of two spheres. Part I: The determination of exact solutions for any values of the ratio of radii and separation parameters,Z. Angew. Math. Phys. 21:164 (1970).

    Google Scholar 

  23. M. E. O'Neill and S. R. Majumdar, Asymmetrical slow viscous fluid motions caused by the translation or rotation of two spheres. Part II: Asymptotic forms of the solutions when the minimum clearance between the spheres approaches zero,Z. Angew. Math. Phys. 21:180 (1970).

    Google Scholar 

  24. G. A. Pertmer and S. K. Loyalka, Gravitational collision efficiency of post hypothetical core disruptive accident liquid-metal fast breeder reactor aerosols: Spherical particles,Nucl. Technol. 47:70 (1980).

    Google Scholar 

  25. J. R. Pruppacher and J. D. Klett,Microphysics of Clouds and Precipitation (D. Reidel, Dordrecht, Holland, 1980).

    Google Scholar 

  26. R. J. Schlamp, S. N. Grover, H. R. Pruppacher, and A. E. Hamielec, A numerical investigation of the effect of electric charges and vertical external electric fields on the collision efficiency of cloud drops,J. Atmos. Sci. 33:1747 (1976).

    Google Scholar 

  27. J. M. Serayssol and R. H. Davis, The influence of surface interactions on the elastohydrodynamic collision of two spheres,J. Colloid Interface Sci. 114:54 (1986).

    Google Scholar 

  28. R. M. Schotland, The collision efficiency of cloud drops of equal size,J. Meteorol. 14:381 (1957).

    Google Scholar 

  29. E. H. Steinberger, H. R. Pruppacher, and M. Neiburger, On the hydrodynamics of pairs of spheres falling along their line of centres in a viscous medium,J. Fluid Mech. 34:809 (1968).

    Google Scholar 

  30. M. Stimson and G. B. Jeffery, The motion of two spheres in a viscous fluid,Proc. R. Soc. A 111:110 (1926).

    Google Scholar 

  31. J. W. Telford, N. S. Thorndike, and E. G. Bowen, The coalescence between small water drops,Q. J. R. Meteorol. Soc. 81:241 (1955).

    Google Scholar 

  32. P. D. Thorne and D. J. Foden, Generation of underwater sound by colliding spheres,J. Acoust. Soc. Am. 84:2144 (1988).

    Google Scholar 

  33. P. D. Thorne, Private communication (1989).

  34. C. S. Wen and G. K. Batchelor, The rate of coagulation in a dilute suspension of small particles,Sci. Sinica A 28:172 (1985).

    Google Scholar 

  35. J. D. Woods and B. J. Mason, The wake capture of water drops in air,Q. J. R. Meteorol. Soc. 91:35 (1965).

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gelbard, F., Mondy, L.A. & Ohrt, S.E. A new method for determining hydrodynamic effects on the collision of two spheres. J Stat Phys 62, 945–960 (1991). https://doi.org/10.1007/BF01128170

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01128170

Key words

Navigation