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The intrinsic conductivity and viscosity for ellipses in a two dimensional fluid can be obtained from the triaxial ellipsoid results in three dimensions by taking appropriate limits and averaging. The average required to obtain virial coefficients in d=2 involves rotationally averaging around the z axis of the ellipsoid and letting a3 (the semi-axis length in the z direction) approach infinity. We then need to evaluate A1, A2, and A3 from Appendix B in the limit a3 → ∞. From (71) and (72) along with the limit a3 → ∞ we find,
and the Ai sum rule in d=2 becomes,
The rotational average in d=2 for the polarization is then,
since the 2nd-rank polarization tensor is diagonal for orthorhombic symmetry and higher. The intrinsic conductivity then equals,
which agrees with Garboczi et al. .
The intrinsic viscosity in d=2 is found in the same way as the shear modulus from an averaging of the elastic stiffness tensor G having rectangular symmetry and is given by,
After inserting the appropriate combinations of A1, A2, and A3 into Haber and Brenner's formalism given in Appendix C and Ref. , the rotationally averaged intrinsic viscosity becomes
where the Qi's are defined in Ref. . This reduces to
which is exactly equal to the intrinsic conductivity given in eqn. (117).