Exact tensor closures for the three dimensional Jeffery's equation

TL;DR

This paper introduces an exact, non-approximate formula for the fourth-moment tensor in 3D Jeffery's equation, along with a fast method for solving the equation without closures or elliptic integrals.

Contribution

It provides the first published explicit formulas for the fourth-moment tensor in 3D Jeffery's equation and introduces a fast, exact solution method called FEC.

Findings

Explicit formulas involve elliptic integrals.

Formulas are valid for isotropic fiber orientations.

FEC method avoids elliptic integrals and approximations.

Abstract

This paper presents an exact formula for calculating the fourth-moment tensor from the second-moment tensor for the three dimensional Jeffery's equation. Although this approach falls within the category of a moment tensor closure, it does not rely upon an approximation, either analytic or curve fit, of the fourth-moment tensor as do previous closures. This closure is orthotropic in the sense of \cite{cintra:95}, or equivalently, a natural closure in the sense of \cite{verleye:93}. The existence of these explicit formulae has been asserted previously, but as far as the authors know, the explicit forms have yet to be published. The formulae involve elliptic integrals, and are valid whenever fiber orientation was isotropic at some point in time. Finally, this paper presents the Fast Exact Closure (FEC), a fast and in principle exact method for solving Jeffery's equation, which does not require approximate closures, nor the elliptic integral computation.