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Resistance functions for spherical particles, droplets and bubbles in cylindrical tubes

Published online by Cambridge University Press:  26 April 2006

J. J. L. Higdon  and
G. P. Muldowney
J. J. L. Higdon
Affiliation:
Department of Chemical Engineering, University of Illinois, Urbana, IL 61801, USA
G. P. Muldowney
Affiliation:
Department of Chemical Engineering, University of Illinois, Urbana, IL 61801, USA Present address: Mobil Research and Development Co, Billingsport Rd, Paulsboro, NJ 08066, USA.
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Abstract

Numerical computations are performed to evaluate the resistance functions for low Reynolds number flow past spherical particles, droplets and bubbles in cylindrical domains. Spheres of arbitrary radius a and radial position b move with arbitrary velocity U within a cylinder of radius R. The undisturbed fluid may be at rest, or subject to a pressure-driven flow with maximum velocity U0. The spectral boundary element method is employed to compute the resistance force for torque-free bodies in three cases: rigid solids, fluid droplets with viscosity ratio λ = 1, and bubbles with viscosity ratio λ = 0. A lubrication theory is developed to predict the limiting resistance of bodies near contact with the cylinder walls. Compact algebraic expressions are developed which accurately represent the numerical data over the entire range of particle positions 0 < b/(R − a) < 1 for all particle sizes in the range 0 < a/R < 0.9. The resistance functions are consistent with known analytical results and are presented in a form suitable for further studies of particle migration in cylindrical vessels.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 298 , 10 September 1995 , pp. 193 - 210
Copyright
© 1995 Cambridge University Press

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References

Bungay, P. M. & Brenner, H. 1973 The motion of a closely fitting sphere in a fluid-filled tube. Intl. J. Multiphase Flow 1, 25–56.Google Scholar
Coutanceau, M. & Thizon, P. 1981 Wall effect on the bubble behaviour in highly viscous liquids. J. Fluid Mech. 107, 339–373.Google Scholar
Cox, R. G. 1974 The motion of suspended particles almost in contact. Intl J. Multiphase Flow 1, 343–371.Google Scholar
Davis, R. H., Schonberg, J. A. & Rallison, J. M. 1989 The lubrication force between two viscous drops. Phys. Fluids A 1, 77–81.Google Scholar
Falade, A. & Brenner, H. 1985 Stokes wall effects for particles moving near cylindrical boundaries. J. Fluid Mech. 154, 145–162.Google Scholar
Happel, J. & Brenner, H. 1965 Low Reynolds Number Hydrodynamics. Prentice-Hall.
Hasimoto, H. 1976 Slow motion of a small sphere in a cylindrical domain. J. Phys. Soc. Japan 41 2143–2144 (and erratum, J. Phys. Soc. Japan 42, 1047).Google Scholar
Hetsroni, G., Haber, S. & Wacholder, E. 1970 The flow fields in and around a droplet moving axially within a tube. J. Fluid Mech. 41, 689–705.Google Scholar
Hirschfeld, B. R., Brenner, H. & Falade, A. 1984 First and second order wall effects upon the slow viscous asymmetric motion of an arbitrarily shaped, positioned and oriented particle within a circular cylinder. Physiochem. Hydrodyn. 5, 99–133.Google Scholar
Jeffrey, D. J. & Onishi, Y. 1984 Calculations of the resistance and mobility functions for two unequal rigid spheres in low Reynolds number flow. J. Fluid Mech. 139, 261–290.Google Scholar
Kim, S. & Karrila, S. J. 1991 Microhydrodynamics: Principles and Selected Applications. Butterworth-Heinemann.
Liron, N. & Shahar, R. 1978 Stokes flow due to a Stokeslet in a pipe. J. Fluid Mech. 86, 727–744.Google Scholar
Muldowney, G. P. & Higdon, J. J. L. 1995 A spectral boundary element approach to three-dimensional Stokes flow. J. Fluid Mech. 298, 167–192.Google Scholar
Pozrikidis, C. 1992 Boundary Integral and Singularity Methods for Linearized Viscous Flow. Cambridge University Press.
Tozeren, H. 1982 Torque on eccentric spheres flowing in tubes. J. Appl. Mech. 49, 279–283.Google Scholar
Tozeren, H. 1983 Drag on eccentrically positioned spheres translating and rotating in tubes. J. Fluid Mech. 129, 77–90.Google Scholar
Zheng, G.-H., Powell, R. L. & Stroeve, P. 1992 Torque and frictional force on a slowly rotating sphere arbitrarily positioned in a circular cylinder. Ind. Engng Chem. Res. 31, 1190–1194.Google Scholar