The stress field of slender particles oriented by a non-Newtonian extensional flow

Abstract

An analysis is presented of the creeping motion around a flow-oriented slender particle in a material medium subject to a uniaxial extension in the far field. A general quasi-steady rheological model is adopted, of a kind representing isotropic (Noll) fluids subject to time-independent velocity gradients, or isotropic solids subject to time-independent strain fields. The analysis is based on the premise of a shear-dominated motion in the near field, which is joined asymptotically to the extension-dominated motion in the far field. For axisymmetric particles, and to the order of terms in slenderness considered here, the far-field perturbation due to the particle can be represented as a distributed coaxial line force in a transversely isotropic medium whose strength is governed by the structure of the near-field rheology.On the basis of the results for a single particle, a formula is derived for the stress contribution due to the presence of oriented slender fibres in dilute suspension in a non-Newtonian fluid. For certain simple rheological models exhibiting a strong shear-thinning behaviour, the particle contribution to tensile stress is greatly diminished relative to the Newtonian case, as was predicted by an earlier rudimentary treatment (Goddard 1975). The present analysis is thought to be highly promising for applications to general composite materials.