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An increase of the viscosity of suspensions and the shear modulus of solids is generally observed upon adding rigid particles to the medium. The introduction of rigid inclusions perturbs the stress field of the sheared pure medium since locally the field lines cannot penetrate the hard inclusions. There is evidently a qualitative analogy with the electrical conduction problem in a suspension of highly conducting particles where the electric field lines are similarly screened from the interiors of the conducting particles. Many authors have commented on the mathematical resemblance between electrical polarization and linearized flow theory calculations [7,98,99], which follow as a consequence of this physical analogy. In the following we develop an approximate relation between the electrical conductivity and suspension viscosity problems.
Einstein , as part of his investigation of the molecular nature of matter, first calculated the incremental increase of the viscosity η of a dilute hard sphere suspension:
where ηo is the pure solution viscosity. (Notably, his original calculation did not give the correct 5/2 coefficient  in eqn. (13) and was later corrected in light of experimental observations by Bancelin .) The leading virial coefficient is the 'intrinsic viscosity' [η], defined by
Experiments on nearly spherical particles in low concentration suspensions [φ < O(1%)] commonly yield a value of [η] ≈ 2.7, which is slightly higher  than the Einstein calculation [η] = 2.5. This small deviation is often ascribed to small particle asphericity or particle clustering  and, at any rate, the revised Einstein result (13) is a good approximation.
Rayleigh , Goodier , and Hill and Power  have pointed out a fundamental analogy between the hydrodynamics of suspensions and the elastostatics of incompressible solids with rigid inclusions, which implies that Einstein's virial expansion for the viscosity of hard sphere suspensions also describes the shear modulus G virial expansion [17,19],
for an incompressible elastic continuum of modulus Go containing stiff spherical inclusions at low concentration. The 'intrinsic shear modulus' [G] is defined by the limit,
For compressible spherical particles [G] depends on the Poisson ratio ν  ,
where ν → 1/2 in the incompressible limit. Derivation of eqn. (17) assumes that the matrix material always remains in contact with the inclusion ('sticks') under deformation. A tabulation of experiments from a variety of sources indicates that the shear modulus and suspension viscosity have a common concentration dependence ,
for nearly spherical, rigid inclusions in an incompressible elastic matrix and a Newtonian fluid, respectively. This behavior is consistent with the simple incompressibility assumption (ν = 1/2) and the viscoelastic and elastostatic analogy of Rayleigh . It is emphasized that (18) is observed to hold regardless of the concentration of the suspended matter! We note that the simple relation between intrinsic viscosity and intrinsic shear modulus is limited to spherical particles and an incompressible suspending medium (see Appendix C).
Experience also indicates that the addition of 'softer' materials to liquids and solids does not generally increase the viscosity and shear modulus. This physical situation is analogous to the addition of insulating material to a conducting medium , since the inclusions are 'permeable' to the shear-induced stress field lines in the suspending fluid or solid medium. In the extreme case, where the particle inclusions are highly deformable and the matrix is incompressible, [G] becomes :
so that the solid becomes softer with an increasing volume fraction of soft inclusions. The magnitude of [G] for holes is comparable to [σ]o for an insulator in a conducting matrix [see eqn. (1)].
The introduction of liquid drops into another viscous fluid or a solid introduces some important additional features. In this case, momentum can propagate into the interior of the droplet and induce internal circulation within the droplet, so that the dissipation is altered from the hard sphere case. In many physical circumstances surface tension or internal pressure tends to make the drop resist deformation, however. Taylor  showed that the intrinsic viscosity of idealized indeformable liquid drops of viscosity ηdrop equals,
Despite the fundamental importance of [η] in determination of molecular shape , there are few analytical calculations of [η] corresponding to non-spherical objects. Onsager  long ago calculated asymptotic results for long hard prolate ellipsoids, and these results were later generalized by Saito  to analytical estimates for arbitrary aspect ratios. Kirkwood and Riseman  and Debye and Bueche  estimated [η] for random coil polymer chains, but these calculations involved uncontrolled approximations. Rallison  and Haber and Brenner  recently obtained exact results for triaxial ellipsoids. The formalism required to treat the triaxial ellipsoid case is quite sophisticated and treatment of these more general shapes is necessarily complicated. The reason for the limited progress in calculating [η], relative to [σ]∞, is simple: Solution of the steady-state Navier-Stokes equation on the exterior of the hard particles is a significantly more difficult technical problem than the corresponding solution of the Laplace equation.
Recently, Hubbard and Douglas  have observed an interesting relation between hydrodynamic and electrostatic problems that suggests a route for developing a direct approximate relation between [η] and [σ]. They observed that the angular average of the Green's function for the steady-state free space Navier-Stokes equation equals the Green's function of the free space Laplacian . From this observation and the physical angular averaging associated with the Brownian particle diffusion process, they deduced that the scalar translational friction coefficient of arbitrarily shaped rigid paricles approximately equals,
where C is the electrostatic capacitance. (C is the Newtonian capacity as opposed to the logarithmic capacity discussed in Sec. 5 . The units of C are chosen so that a sphere of radius R has a capacitance C = R.) The capacitance C governs the far field decay of the solution of Laplace's equation where the solution equals 1 on the boundary and approaches zero at great distances from the boundary [26,80]. Eqn. (22), which is consistent (within about 1% accuracy) with exactly known values of fT, serves as an explicit connection between hydrodynamic and electrostatic problems. Direct comparisons of the average stress and electrostatic (or thermal) 'dipole coefficients' [110,111,112] in the calculation of [η] and [σ]∞ , respectively, suggests that [η] is simply proportional to [σ]∞ within angular averaging. In other words, it seems reasonable to preangularly average the steady-state Navier-Stokes Green's function so that the hydrodynamic problem reduces to the solution of the Laplace equation on the exterior of the particle as in the former calculations relating translational friction and capacity . This procedure seems reasonable for a dilute particle suspension of randomly-oriented particles since [η] is then an invariant under suspension rotations. In this paper we are interested in checking the numerical accuracy of this relation. The existence of small numerical discrepancies in exact analytical results, described below, show that this relation is not exact, but rather a very good approximation for objects having diverse shapes.
The constant of proportionality between [η] and [σ]∞ can be fixed by exact calculations for sphere suspensions [5,113] in d spatial dimensions,
We choose the sphere case to determine the proportionality constant since the preaveraging argument for the Oseen tensor leads to exact results for spheres. Of course, this is a rather trivial case and other shapes must be considered to check the conjectured relation (23) .
Further motivation of the approximation, eqn. (23), derives from calculations by Kanwal , which show an exact relation between the rotational friction coefficient fR and αe for a certain class of bodies,
corresponding to the rotation of a body of revolution, having an otherwise arbitrary profile, about its axis of symmetry. αe(T) is the polarizability component normal to the axis of symmetry. Exact fR(T) results for a variety of complex-shaped particles can be directly obtained from eqn. (24) and from tabulations of αe(T) .
Riseman and Kirkwood  have noted that a proportionality relation should exist between the rotational friction coefficient and [η] and this observation is consistent with the approximation (23) and (24). The rotational friction coefficient becomes difficult to measure and to calculate for non-symmetric objects and for flexible objects so we do not pursue this connection further.
Brenner  has developed the necessary mathematical machinery for calculating [η] for rigid axisymmetric particles. It is useful to utilize this formalism to obtain some exact results that can be tested against eqn. (23). The particle shape made from two touching spheres of radius a is an interesting test case. Exact calculation, using the formalism of Brenner  and associated results for the stress dipole due previously to Wakiya , gives an exact value for the intrinsic viscosity of two touching and rigidly joined spheres,
(We note that the value of [η] given on p. 263 of  is incorrect). An exact calculation of αe (and thus implicitly [σ]∞) for touching spheres is summarized by Schiffer and Szego . The electrical polarizability components along the symmetry axis αe(L) and normal to the symmetry axis αe(T) equal,
where ζ is the Riemann zeta function, i.e., ζ(3) = 1.20206... The intrinsic conductivity [σ]∞ for touching spheres is then
Eqns. (25) and (26-27) imply that the ratio [η] / [σ]∞ for touching spheres equals,
which agrees well with the estimate from (23),
We also compare the exact result to recent 'bead' model calculations of [η] for touching spheres by de la Torre and Bloomfield . They find  = 3.493 for touching spheres which is accurate to within 1% in comparison with the exact result (25).
Exact polarizability results are also known for the disc and needle limits of an ellipsoid of revolution. For a disc of radius a the polarizability components  and [σ]∞ in number density units equal,
and from the formalism of Brenner  we can also obtain an exact calculation of the intrinsic viscosity of a disc as,
which is also given in number density units. This result is probably known but we could not find a reference to it. For a disc we then obtain the exact ratio,
which is rather close to the estimate (23).
In the opposite needle limit (x → ∞), corresponding to an extended prolate ellipsoid, the asymptotic scaling of [σ]∞ with x can be deduced analytically,
where x is the ratio of the semi-major axis length to the semi-minor axis length. Onsager  calculated the corresponding asymptotic prolate ellipsoid result for [η] as,
which is consistent with more general calculations given later by Saito . The exact limiting ratio for [η] / [σ]∞ for a 'needle' (x → ∞) then equals,
which is identical to the ratio obtained for a disc. The ratio [η] / [σ]∞ is thus found to be nearly invariant for a significant range of particle shapes, as expected from eqn. (23). In Fig. 4a we plot log [η] vs. log[σ]∞ for a wide range of aspect ratios for prolate ellipsoids of revolution, while Fig. 4b shows a similar graph for oblate ellipsoids of revolution, where the abscissa is now the inverse of the aspect ratio. The straight line is a fit which gives an average value of 0.8 for the intrinsic viscosity-conductivity ratio. Table 5 gives the numerical data shown in Fig. 4, where the [η] results are taken from the original tabulation of Scheraga .
Figure 4: The intrinsic viscosity [η] versus the intrinsic conductivity [σ]∞ for ellipsoids of revolution. The scales are logarithmic (base 10): a) prolate , b) oblate.
The information required to obtain [η] for triaxial ellipsoids is also known, although this information is rather inaccessible because of the complicated mathematical formalism which these calculations involve. The necessary formulas for the components of the electric polarizability  are summarized in Appendix B and a summary of the necessary results of Haber and Brenner  for [η] are provided in Appendix C. Tabulations of these virial coefficients, which should be useful in applications, are given in Table 6.
We observe from Table 5 and Table 6 that all the ellipsoid data is nearly consistent with the ratio given in eqn. (29),
so that [η] / [σ]∞ is an invariant to within a 5% accuracy. The angular averaging approximation is not as accurate for  as in previous applications to fT [27,81], but eqn. (29) is sufficiently accurate for many practical applications since measurement and numerical calculation errors are often comparable to the 5% inaccuracy indicated by (36).
Eqn. (36) also holds for the spherical 'dumbbell' at arbitrary separations. The dumbbell is defined by two identical spheres connected by a sraight wire of zero thickness and fixed length. In the calculation of the polarizability the spheres are uncharged and the wire has zero electrical resistance, while for the intrinsic viscosity calculation the wire has a vanishing hydrodynmic resistance. Brenner  has summarized the information required to calculate [η] for a dumbbell and an exact calculation of [σ] ∞ for the dumbbell is given in Appendix D.
We define the quantity r p to be the ratio of the distance between the centers of the spheres to their diameters, which completely characterizes the shape of the spherical particle dumbbell. It is then found that the value of [η] is approximately quadratic in r p. A useful approximate formula for [η], covering the range 1 < r p < 10, is given by,
which holds to about a 3% accuracy. Exact results for [η] are tabulated in Table 7 and the asymptotic variation of [η] for a dumbbell at large separation equals,
Simha  previously indicated a quadratic dependence of [η] on r p in the r p → ∞ limit, but his widely cited value for the prefactor, 3/2, is not correct. The origin of this discrepancy is not clear, but we note that Simha  ignored hydrodynamic interactions.
Schiffer and Szego  previously summarized exact results for the electric polarizability of two separated spheres without the connecting wire. The generation of a large dipole in separated spheres, however, requires the electrical connection and the calculation of [σ]∞ in the case where there is a connecting wire is given in Appendix D. A tabulation of these new results for [σ]∞, along with the dimensionless polarizability components (normalized by the particle volume), is given in Table 7 and these results are shown graphically in Fig. 5. It is hard to imagine a geometry more representative of a dipole. For large separations [σ]∞ is simply proportional to r p2,
so that we have the asymptotic result,
It is interesting that the dumbbell accords with eqn. ( 36) even in the extreme limit of infinite separation. Individual components of the polarizability are shown in Table 8.
Figure 5: Longitudinal (L) and transverse (T) components of the dimensionless (normalized by the particle volume) electric polarizability tensor αe for the spherical particle dumbbell, along with the average of these components, the intrinsic conductivity [σ]∞.
We also mention some results for the intrinsic conductivity of insulating dumbbells. From the results of Schiffer and Szego  for the effective mass M of touching spheres and (12) we have
and by finite element methods we calculate the other component
which has long defied exact analytical calculation . [The closed form estimate in (42) is based on the assumption that αm(T) is proportional to ζ(3), as in (41), in combination with accurate numerical estimates of αm(T).] We note that the known value of αm(L) is given by our finite element method to an accuracy of better than 1%, so we expect that the corresponding value for the unknown αm(T) should be correct within the same tolerance. (From previous experience, the finite element method used here is always more accurate for conducting matrix/insulating particle problems than for conducting matrix /superconducting particle problems.) We then obtain the intrinsic conductivity of the insulating doublet of spheres,
Even in this extreme limit the deviation of [σ]o from the sphere value is unimpressive. The variation of [σ]o with shape is more interesting for flat bodies (see Fig. 2). We return to a discussion of flat bodies in Sect. 6.
As a final point, we mention that exact calculation of [η] for other shapes is possible, in principle. Exact results for αe and M are known for the 'lens', 'bowl', 'spindle', and other shapes . Calculation of [η] involves similar (albeit more complicated) mathematics.