The Fast Exact Closure for Jeffery's Equation with Diffusion

TL;DR

This paper introduces the Fast Exact Closure (FEC), a novel method for solving Jeffery's equation with diffusion that avoids approximation of higher order moments, providing exact solutions in non-interacting cases and improved accuracy for dense fiber suspensions.

Contribution

The FEC method uses conversion tensors to derive a pair of ODEs, eliminating the need for closure approximations and enhancing computational efficiency and accuracy.

Findings

FEC provides exact solutions when fibers do not interact.

Numerical results show improved accuracy over existing closures.

FEC achieves comparable or better computational speeds.

Abstract

Jeffery's equation with diffusion is widely used to predict the motion of concentrated fiber suspensions in flows with low Reynold's numbers. Unfortunately, the evaluation of the fiber orientation distribution can require excessive computation, which is often avoided by solving the related second order moment tensor equation. This approach requires a `closure' that approximates the distribution function's fourth order moment tensor from its second order moment tensor. This paper presents the Fast Exact Closure (FEC) which uses conversion tensors to obtain a pair of related ordinary differential equations; avoiding approximations of the higher order moment tensors altogether. The FEC is exact in that when there are no fiber interactions, it exactly solves Jeffery's equation. Numerical examples for dense fiber suspensions are provided with both a Folgar-Tucker (1984) diffusion term and the recent anisotropic rotary diffusion term proposed by Phelps and Tucker (2009). Computations demonstrate that the FEC exhibits improved accuracy with computational speeds equivalent to or better than existing closure approximations.