On the isotropic and anisotropic viscosity of suspensions containing particles of diverse shapes and orientations

The classical problem of effective viscosity of a Newtonian fluid containing rigid particles is discussed. For spherical particles, it is shown that the usual Einstein's formula μ/μ0=1+2.5ϕ represents an incorrect formulation of the non-interaction approximation (NIA): it violates a rigorous lower bound for the effective viscosity. The correct formulation yields the effective viscosity in the form μ/μ0=1-2.5ϕ-1 that agrees with the bounds and remains accurate at substantial volume fractions of particles ϕ (up to 20%-30% according to various data sets). This result is extended to ellipsoidal particles, with the emphasis on mixtures of particles of diverse aspect ratios and cases of anisotropic viscosity (due to non-random orientations of particles). For mixtures of particles of diverse shapes (such as ellipsoids of diverse aspect ratios), the effective viscosity cannot generally be expressed in terms of either ϕ or any other of concentration parameter, and the very concept of concentration parameters becomes questionable. The case of thin platelets is considered in detail; in this case, the concentration parameter is identified, and it is different from the volume fraction.