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The hydrodynamic interaction of two unequal spheres moving under gravity through quiescent viscous fluid

Published online by Cambridge University Press:  29 March 2006

E. Wacholder  and
N. F. Sather
E. Wacholder
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge
N. F. Sather
Affiliation:
Department of Chemical Engineering, University of Washington, Seattle Present address: 31 a Ben-Jehuda Street, Haifa, Israel.
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Abstract

The hydrodynamic forces and couples that act on two spherical particles in slow motion through a quiescent fluid are determined as functions of the relative configuration of the particles from the solution of the Stokes equation for the motion of the fluid in the vicinity of the particles. General formulae that relate the translational and rotational velocities of the particles to the ratios of their radii a = a2/a1 and net densities I are obtained, as are asymptotic forms for the velocities in the limiting cases of very large and very small interparticle separation. Relative trajectories of the particles when they move solely under gravity and their own interaction are calculated for several values of I and a. A particularly interesting feature of the results is that, for certain ranges of values of I and a, trajectories of finite length and trajectories having the form of closed periodic orbits may occur.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 65 , Issue 3 , 16 September 1974 , pp. 417 - 437
Copyright
© 1974 Cambridge University Press

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References

Bartok, W. & Mason, S. G. 1957 Particle motions in sheared suspensions. V. Rigid rods and collision doublets of spheres J. Colloid Sci. 12, 243.Google Scholar
Batchelor, G. K. 1972 Sedimentation in a dilute suspension of spheres J. Fluid Mech. 52, 245.Google Scholar
Batchelor, G. K. & Green, J. T. 1972a The hydrodynamic interaction of two small freely-moving spheres in a linear flow field J. Fluid Mech. 56, 375.Google Scholar
Batchelor, G. K. & Green, J. T. 1972b The determination of the bulk stress in a suspension of spherical particles to order c2. J. Fluid Mech. 56, 401.Google Scholar
Brenner, H. 1963 The Stokes resistance of an arbitrary particle Chem. Engng Sci. 18, 1.Google Scholar
Brenner, H. 1964a The Stokes resistance of an arbitrary particle. II Chem. Engng Sci. 19, 599.Google Scholar
Brenner, H. 1964b The Stokes resistance of an arbitrary particle. III Chem. Engng Sci. 19, 631.Google Scholar
Brenner, H. & O'Neill, M. E. 1972 On the Stokes resistance of multiparticle systems in a linear shear field Chem. Engng Sci. 27, 1421.Google Scholar
Cooley, M. D. A. & O'Neill, M. E. 1968 On the slow rotation of a sphere about a diameter parallel to a nearby plane wall J. Inst. Math. Applics. 4, 163.Google Scholar
Cooley, M. D. A. & O'Neill, M. E. 1969a On the slow motion of two spheres in contact along their line of centres through a viscous fluid Proc. Camb. Phil. Soc. 66, 407.Google Scholar
Cooley, M. D. A. & O'Neill, M. E. 1969b On the slow motion generated in a viscous fluid by the approach of a sphere to a plane wall or stationary sphere Mathematika, 16, 37.Google Scholar
Curtis, A. S. G. & Hocking, L. M. 1970 Collision efficiency of equal spherical particles in a shear flow Trans. Faraday Soc. 66, 1381.Google Scholar
Darabaner, C. L. & Mason, S. G. 1967 Particle motions in sheared suspensions. XXII. Interactions of rigid spheres. (Experimental.) Rheol. Acta, 6, 273.Google Scholar
Davis, M. H. 1969 The slow translation and rotation of two unequal spheres in a viscous fluid Chem. Engng Sci. 24, 1769.Google Scholar
Davis, M. H. 1971 Two unequal spheres in a slow viscous linear flow. Nat. Center for Atmos. Res. (Boulder, Colorado) Tech. Note, NCAR-TN/STR-64.Google Scholar
Dean, W. R. & O'Neill, M. E. 1963 A slow motion of viscous liquid caused by the rotation of a solid sphere Mathematika, 10, 13.Google Scholar
Goldman, A. J., Cox, R. G. & Brenner, H. 1966 The slow motion of two identical arbitrarily oriented spheres through a viscous fluid Chem. Engng Sci. 21, 1151.Google Scholar
Goldman, A. J., Cox, R. G. & Brenner, H. 1967 Slow viscous motion of a sphere parallel to a plane wall. I. Motion through a quiescent fluid Chem. Engng Sci. 22, 637.Google Scholar
Goren, S. L. 1970 The normal force exerted by creeping flow on a small sphere touching a plane J. Fluid Mech. 41, 619.Google Scholar
Happel, J. & Brenner, H. 1965 Low Reynolds Number Hydrodynamics. Prentice-Hall.
Lin, C. J., Lee, K. J. & Sather, N. F. 1970 Slow motion of two spheres in a linear shear field J. Fluid Mech. 43, 35.Google Scholar
Nir, A. & Acrivos, A. 1973 On the creeping motion of two arbitrarily-sized touching spheres in a linear shear field J. Fluid Mech. 59, 209.Google Scholar
O'Neill, M. E. 1964a A slow motion of viscous liquid caused by a slowly moving solid sphere Mathematika, 11, 67.Google Scholar
O'Neill, M. E. 1964b Some motions of incompressible liquid generated by the movement of spheres. Ph.D. thesis, University of London.
O'Neill, M. E. 1969 On asymmmetrical slow viscous flows caused by the motion of two equal spheres almost in contact Proc. Camb. Phil. Soc. 65, 543.Google Scholar
O'Neill, M. E. & Majumdar, S. R. 1970a Asymmetrical slow viscous 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.Google Scholar
O'Neill, M. E. & Majumdar, S. R. 1970b Asymmetrical slow viscous 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.Google Scholar
O'Neill, M. E. & Stewartson, K. 1967 On the slow motion of a sphere parallel to a nearby plane wall J. Fluid Mech. 27, 705.Google Scholar
Stimson, M. & Jeffery, G. B. 1926 The motion of two spheres in a viscous fluid. Proc. Roy. Soc. A 113, 110.Google Scholar
Wakiya, S. 1957 Niigata University (Nagaoka, Japan) Coll. Engng Res. Rep. no. 6.
Wakiya, S. 1967 Slow motions of a viscous fluid around two spheres J. Phys. Soc. Japan, 22, 1101.Google Scholar
Zia, I. Y. Z., Cox, R. G. & Mason, S. G. 1967 Ordered aggregates of particles in shear flow. Proc. Roy. Soc. A 300, 421.Google Scholar
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