Stephen Montgomery-Smith, Wei He, David Jack and Douglas E. Smith, Exact tensor closures for the three dimensional Jeffery's Equation.
Journal of Fluid Mechanics, volume 680, (2011), pp. 321-335.
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 the quadratic, hybrid or current orthotropic closures. This closure is orthotropic in the sense of Cintra and Tucker, or equivalently, a natural closure in the sense of Verleye and Dupret. The existence of these explicit formulae has been asserted by previous authors, but as far as the authors know, the explicit forms have yet to be published. The formulae involve elliptic integrals, and are valid whenever the fiber orientation tensor 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 computation of the elliptic integrals.
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