Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-27T13:37:26.065Z Has data issue: false hasContentIssue false
  • English
  • Français

A method for the calculation of the effective transport properties of suspensions of interacting particles

Published online by Cambridge University Press:  19 April 2006

R. W. O'brien
R. W. O'brien
Affiliation:
Department of Theoretical and Applied Mechanics, University of New South Wales, New South Wales 2033, Australia
Get access Rights & Permissions [Opens in a new window]

Abstract

The early attempts at calculating effective transport properties of suspensions of interacting spherical particles resulted in non-absolutely convergent expressions. In this paper we provide a physical interpretation for these convergence difficulties and we present a new method of determining the effective transport properties which clarifies difficulties in existing methods.

This method, which is described for simplicity in the context of the thermal conduction problem, is based on an expression that gives the temperature gradient ∇T at a point x in the matrix in terms of integrals over the surrounding particles and an integral over a large surface Γ which encloses x and which we term the ‘macroscopic boundary’. Without the integral over Γ, this expression for ∇T would be non-absolutely convergent, for the contribution to ∇T(x) from a distant particle is proportional to 1/r3, where r is the distance of the particle from x. On comparing the expression for ∇T with the formula used by Rayleigh (1892) in his investigation of the effective conductivity of a cubic array of spheres, we find that Rayleigh's convergence difficulties arose simply from an incorrect assessment of the macroscopic boundary integral.

By combining the expression for ∇T(x) with a formula for the dipole strength of a sphere placed in an ambient temperature field, we obtain a convergent expression relating the dipole strength of a sphere to integrals over the surrounding particles. An expression for the effective conductivity of a random suspension of spheres correct to O(ϕ2) is obtained simply by averaging this expression for the thermal dipole strength. By a similar procedure we obtain expressions for the effective viscosity and effective elastic moduli correct to O(ϕ2). Most of these results have been obtained by earlier workers using a ‘renormalization’ procedure due to Batchelor; the method presented here has the advantage that the renormalization quantity arises naturally from the macroscopic boundary integral referred to earlier, so there is no uncertainty about its choice.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 91 , Issue 1 , 9 March 1979 , pp. 17 - 39
Copyright
© 1979 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. 1970 The stress system in a suspension of force-free particles. J. Fluid Mech. 41, 545.Google Scholar
Batchelor, G. K. 1972 Sedimentation in a dilute dispersion of spheres. J. Fluid Mech. 52, 245.Google Scholar
Batchelor, G. K. 1974 Transport properties of two-phase materials with random structure. Ann. Rev. Fluid Mech. 6, 227.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
Bertaux, M. G., Bienfait, G. & Jolivet, J. 1975 Etudes des propriétés thermiques des milieux granulaires. Ann. Geophys. 31, 191.Google Scholar
Chen, H. S. & Acrivos, A. 1978 On the effective elastic moduli of composite materials containing spherical inclusions at non-dilute concentrations. Int. J. Solids Struc. (to appear)Google Scholar
Eshelby, J. D. 1957 The determination of the elastic field of an ellipsoidal inclusion and related problems. Proc. Roy. Soc. A 241, 376.Google Scholar
Jeffrey, D. J. 1973 Conduction through a random suspension of spheres. Proc. Roy. Soc. A 335, 355.Google Scholar
Jeffrey, D. J. 1974 Group expansions for the bulk properties of a statistically homogeneous, random suspension. Proc. Roy. Soc. A 338, 503.Google Scholar
Jeffrey, D. J. 1977 The physical significance of non-convergent integrals in expressions for effective transport properties. S.M. Study no. 12. Waterloo Press.
Korringa, J. 1973 Theory of elastic constants of heterogeneous media. J. Math. Phys. 14, 509.Google Scholar
Landau, L. D. & Lifshitz, E. M. 1959 Theory of Elasticity. Pergamon.
Levine, H. 1966 Effective conductivity of regular composite materials. J. Inst. Math. Appl. 2, 12.Google Scholar
McKenzie, D. R. & McPhedran, R. C. 1978 The conductivity of lattices of spheres. I. The simple cubic lattice. Proc. Roy. Soc. A 359, 45.Google Scholar
Meredith, R. E. & Tobias, C. W. 1960 Resistance to potential flow through a cubical array of spheres. J. Appl. Phys. 31, 1270.Google Scholar
O'Brien, R. W. 1977 Properties of suspensions of interacting particles. Ph.D. dissertation, Cambridge University.
Panofsky, W. K. H. & Phillips, M. 1902 Classical Electricity and Magnetism, 2nd edn. Addison-Wesley.
Protter, M. H. & Weinberger, H. F. 1967 Maximum Principles in Differential Equations. Prentice-Hall.
Rayleigh, Lord 1892 On the influence of obstacles arranged in rectangular order on the properties of the medium. Phil. Mag. 34, 481.Google Scholar
Willis, J. R. & Acton, J. R. 1976 The overall elastic moduli of a dilute suspension of spheres. Quart. J. Mech. Appl. Math. 29, 163.Google Scholar
Zuzovsky, M. & Brenner, H. 1977 Effective conductivities of composite materials composed of cubic arrangements of spherical particles embedded in an isotropic matrix. J. Appl. Math. Phys. 28, 979.Google Scholar