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The rate of collisions due to Brownian or gravitational motion of small drops

Published online by Cambridge University Press:  26 April 2006

Xiaoguang Zhang  and
Robert H. Davis
Xiaoguang Zhang
Affiliation:
Department of Chemical Engineering, University of Colorado, Boulder, CO 80309-0424, USA
Robert H. Davis
Affiliation:
Department of Chemical Engineering, University of Colorado, Boulder, CO 80309-0424, USA
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Abstract

A dilute dispersion containing drops of one fluid dispersed in a second, immiscible fluid is considered. The drops are sufficiently small that inertia is negligible and that they remain spherical. Two drops of different size are in relative motion due to either Brownian diffusion or gravitational sedimentation. When the drops become close, they interact with each other owing to hydrodynamic disturbances and van der Waals attractions, and, under favourable conditions, they will collide with each other and coalesce. The rate at which two drops collide is predicted by solving the diffusion equation for Brownian coalescence, and by using a trajectory analysis to follow the relative motion of pairs of drops for gravity-induced coalescence.

The emphasis of our analysis is on the effects of drop interactions on their collision rate, and these are described by the collision efficiency. Since the hydrodynamic resistance to the drop relative motion reduces with a decreasing ratio of the viscosities of the drop fluid and the surrounding fluid, the collision efficiency increases with decreasing viscosity ratio. A qualitative difference in the collision behaviour of viscous drops from that of rigid spheres is demonstrated; finite collision rates for drops are predicted even in the absence of attractive forces, provided that drop deformation is negligible, whereas rigid particles with smooth surfaces will not come into contact in a fluid continuum unless an attractive force is present which is able to overcome the lubrication forces resisting the relative motion. Hydrodynamic interactions between two spherical drops are accounted for exactly by determining the two-sphere relative mobility functions from previous solutions for two drops moving along and normal to their line of centres. These solutions are based on the method of reflections for widely separated drops, lubrication theory for drops in near-contact, and bispherical coordinates for general separations. The hydrodynamic interactions have a greater effect on reducing the rate of gravity collisions than the rate of Brownian collisions.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 230 , September 1991 , pp. 479 - 504
Copyright
© 1991 Cambridge University Press

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References

Batchelor, G. K. 1982 Sedimentation in a dilute polydisperse system of interacting spheres. J. Fluid Mech. 119, 379–408.Google Scholar
Davis, R. H. 1984 The rate of coagulation of a dilute polydisperse system of sedimenting spheres. J. Fluid Mech. 145, 179–199.Google Scholar
Davis, R. H., Schonberg, J. A. & Rallison, J. M. 1989 The lubrication force between two iscous drops. Phys. Fluids Al, 77–81.Google Scholar
Derjaguin, B. & Landau, L. 1941 Theory of the stability of strongly charged lyophobic sols and of the adhesion of strongly charged particles in solution of electrolytes. Acta. Physicochim. 14, 633.Google Scholar
Fuentes, Y. O., Kim, S. & Jeffrey, D. J. 1988 Mobility functions for two unequal viscous drops in Stokes flow. Part 1. Axisymmetric motions. Phys. Fluids 31, 2445–2455.Google Scholar
Fuentes, Y. O., Kim, S. & Jeffrey, D. J. 1989 Mobility functions for two unequal viscous drops in Stokes flow. Part 2. Nonaxisymmetric motions. Phys. Fluids Al, 61–76.Google Scholar
Haber, S., Hetsroni, G. & Solan, A. 1973 On the low Reynolds number motion of two droplets. Intl J. Multiphase Flow 1, 57–71.Google Scholar
Hamaker, H. C. 1937 The London-van der Waals attraction between spherical particles. Physica 4, 1058.Google Scholar
Hetsroni, G. & Haber, S. 1978 Low Reynolds number motion of two drops submerged in an unbounded arbitrary velocity field. Intl J. Multiphase Flow 4, 1–17.Google Scholar
Ho, N. F. H. & Higuchi, W. I. 1968 Preferential aggregation and coalescence in heterodispersed system. J. Pharm. Sci. 57, 436–442.Google Scholar
Kim, S & Zukoski, C. F. 1990 A model of growth by hetero-coagulation in seeded colloidal dispersions. J. Colloid Interface Sci. 139, 198–212.Google Scholar
Lamb, S. H. 1945 Hydrodynamics, p. 600. Dover.
Melik, D H. 1984 Flocculation in quiescent media. Ph.D. thesis, University of Michigan.
Melik, D. H. & Fogler, H. S. 1984 Gravity-induced flocculation. J. Colloid Interface Sci. 101, 72–83.Google Scholar
Reed, L. D. & Morrison, F. A. 1974 The slow motion of two touching fluid spheres along their line of centers. Intl J. Multiphase Flow 1, 573–583.Google Scholar
Rogers, J. R. & Davis, R. H. 1990 The influence of van der Waals attractions on cloud droplet growth by coalescence. J. Atmos. Sci. 47, 1057–1080.Google Scholar
Rushton, E. & Davies, G. A. 1973 The slow unsteady settling of two fluid spheres along their line of centers. Appl. Sci. Res. 28, 37–61.Google Scholar
Schenkel, J. H. & Kitchner, J. A. 1960 A test of the Derjaguin-Verwey-Overbeek theory with a colloidal suspension. Trans. Faraday Soc. 56, 161.Google Scholar
Smoluchowski, M. von 1917 Versuch einer mathematishen Theorie dar koagulationskinetik kollider LoUsungen. Z. Phys. Chem. 92, 129–168.Google Scholar
Spielman, L. A. 1970 Viscous interactions in Brownian coagulation. J. Colloid Interface Sci. 33, 562–571.Google Scholar
valiolis, I. A. & List, E. J. 1984 Collision efficiencies of diffusing spherical particles: hydrodynamic, van der Waals and electrostatic forces. Adv. Colloid Interface Sci. 20, 1–20.Google Scholar
verwey, E. J. W. & Overbeek, J. Th. G. 1948 Theory of the Stability of Lyophobic Colloids. Elsevier.
Wen, C.-S. & Batchelor, G. K. 1985 The rate of coagulation in a dilute suspension of small particles. Scientia Sinica A28, 172–184.Google Scholar
Yiantsios, S. G. & Davis, R. H. 1991 Close approach and deformation of two viscous drops due to gravity and van der Waals forces. J. Colloid Interface Sci. (in press).Google Scholar
Zhang, X. 1992 The collision rate of a dilute polydisperse system of spherical drops. Ph.D. thesis, University of Colorado.
Zinchenko, A. Z. 1978 Calculation of hydrodynamic interactions between drops at low Reynolds number. Prikl. Mat. Mech. 42, 955–959.Google Scholar
Zinchenko, A. Z. 1980 The slow asymmetric motion of two drops in a viscous medium. Prikl. Mat. Mech. 44, 30–37.Google Scholar
Zinchenko, A. Z. 1982 Calculations of the effectiveness of gravitational coagulation of drops with allowance for internal circulation. Prikl. Mat. Mech. 46, 58–65.Google Scholar