On the use of interaction tensors to describe and predict rod interactions in rod suspensions

Abstract

Recently, Ferec et al. (2009a) proposed a model for nondilute rod-like suspensions, where particle interactions are taking into account via a micromechanical approach. The derived governing equation used the well-known second- and fourth-order orientation tensors (a2 and a4) and novel second- and fourth-order interaction tensors (b2 and b4). To completely close the model, it is necessary to express a4, b2, and b4 in terms of a2. This paper gives the general framework to elaborate these new relations. Firstly, approximations for b2 are developed based on linear combinations of a2 and a4. Moreover, a new closure approximation is also derived for b4, based on the orthotropic fitted closure approach. Unknown parameters are determined by a least-square fitting technique with assumed exact solutions constructed from the probability distribution function (PDF). As numerical solutions for the PDF are difficult to obtain given the nonlinearity of the problem, a combination of steady state solutions is used to generate PDF designed to cover uniformly the entire domain of possible orientations. All these proposed approximations are tested against the particle-based simulations in a variety of flow fields. Improvements of the different approximations are observed, and the couple iORW-CO4P3 gives efficient results.