Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-25T17:25:45.057Z Has data issue: false hasContentIssue false
  • English
  • Français

The hydrodynamic interaction of two small freely-moving spheres in a linear flow field

Published online by Cambridge University Press:  29 March 2006

G. K. Batchelor  and
J. T. Green
G. K. Batchelor
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge
J. T. Green
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge
Get access Rights & Permissions [Opens in a new window]

Abstract

Two rigid spheres of radii a and b are immersed in infinite fluid whose velocity at infinity is a linear function of position. No external force or couple acts on the spheres, and the effect of inertia forces on the motion of the fluid and the spheres is neglected. The purpose of the paper is to provide a systematic and explicit description of those aspects of the interaction between the two spheres that are relevant in a calculation of the mean stress in a suspension of spherical particles subjected to bulk deformation. The most relevant aspects are the relative velocity of the two sphere centres (V) and the force dipole strengths of the two spheres (S′ij, S″ij), as functions of the vector r separating the two centres.

It is shown that V, S′ij and S″ij depend linearly on the rate of strain at infinity and can be represented in terms of several scalar parameters which are functions of r/a and b/a alone. These scalar functions provide a framework for the expression of the many results previously obtained for particular linear ambient flows or for particular values of r/a or of b/a. Some new results are established for the asymptotic forms of the functions both for r/(a + b) [Gt ] 1 and for values of r − (a + b) small compared with a and b. A reasonably complete numerical description of the interaction of two rigid spheres of equal size is assembled, the main deficiency being accurate values of the scalar functions describing the force dipole strength of a sphere in the intermediate range of sphere separations.

In the case of steady simple shearing motion at infinity, some of the trajectories of one sphere centre relative to another are closed, a fact which has consequences for the rheological problem. These closed forms are described analytically, and also numerically in the case b/a = 1.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 56 , Issue 2 , 28 November 1972 , pp. 375 - 400
Copyright
© 1972 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.
Batchelor, G. K. 1972 Sedimentation in a dilute suspension of spheres. J. Fluid Mech. 52, 245.Google Scholar
Batchelor, G. K. & Green, J. T. 1972 The determination of the bulk stress in a suspension of spherical particles to order c2. J. Fluid Mech. 56, 401.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. 1969 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.CrossRefGoogle Scholar
Cox, R. G., Zia, I. Y. Z. & Mason, S. G. 1968 Particle motions in sheared suspensions. XXV. Streamlines around cylinders and spheres. J. Colloid Interface Sci. 27, 7.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
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. 1967a Slow viscous motion of a sphere parallel to a plane wall. I. Motion through a quiescent fluid. Chem. Engng Sci. 22, 637.Google Scholar
Goldman, A. J., Cox, R. G. & Brenner, H. 1967b Slow viscous motion of a sphere parallel to a plane wall. II. Couette flow. Chem. Engng Sci. 22, 653.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
Goren, S. L. & O'Neill, M. E. 1971 On the hydrodynamic resistance to a particle of a dilute suspension when in the neighbourhood of a large obstacle. Chem. Engng Sci. 26, 325.Google Scholar
Green, J. T. 1971 Properties of suspensions of rigid spheres. Dissertation for Ph.D., University of Cambridge.
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 shear field. J. Fluid Mech. 43, 35.Google Scholar
O'Neill, M. E. 1968 A sphere in contact with a plane wall in a slow linear shear flow. Chem. Engng Sci. 23, 1293.Google Scholar
O'Neill, M. E. & Majumdar, S. R. 1970 Asymmetrical slow viscous motions caused by the translation or rotation of two spheres. Part II. Zeit. 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 111, 110.Google Scholar
Wakiya, S. 1971 Slow motion in shear flow of a doublet of two spheres in contact. J. Phys. Soc. Japan 31, 1581 (and a correction, 33, 278).Google Scholar
Wakiya, S., Darabaner, C. L. & Mason, S. G. 1967 Particle motions in sheared suspensions. XXI. Interactions of rigid spheres (Theoretical). Rheol. Acta 6, 264.Google Scholar