The hydrodynamic stress in a suspension of rods

TLDR

The theory builds on a diagrammatic multiple‑scattering expansion of the averaged Green’s function from prior work.

Abstract

A theory is presented to describe the momentum transport properties of suspensions containing randomly placed, slender fibers. The theory is based on a diagrammatic representation of the multiple scattering expansion for the averaged Green’s function as developed in the authors’ previous work on the heat and mass transfer properties of fiber dispersions [Phys. Fluids A 1, 3 (1989)]. The ‘‘best one-body approximation’’ is used to calculate the wavenumber-dependent, ensemble-averaged stress for both aligned and isotropically oriented fiber dispersions. Both the dilute and semidilute concentration regimes are considered. The effective viscosity is calculated as a limit unit of the previously obtained wavenumber-dependent properties. In the semidilute concentration regime the scaling form originally suggested by Batchelor [J. Fluid Mech. 46, 813 (1971)] is recovered for both orientation distributions and its relation to short range ‘‘screening’’ is discussed. Corrections to this result in a ‘‘semidilute expansion’’ for small volume fraction are calculated and the dependence of these corrections on orientation distribution and particle shape is demonstrated.