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Exact tensor closures for the three-dimensional Jeffery's equation

Published online by Cambridge University Press:  24 May 2011

STEPHEN MONTGOMERY-SMITH
Affiliation:
Department of Mathematics, University of Missouri, Columbia, MO 65211, USA
WEI HE
Affiliation:
Department of Mathematics, University of Missouri, Columbia, MO 65211, USA
DAVID A. JACK*
Affiliation:
Department of Mechanical Engineering, Baylor University, Waco, TX 76798, USA
DOUGLAS E. SMITH
Affiliation:
Department of Mechanical and Aerospace Engineering, University of Missouri, Columbia, MO 65211, USA
*
Email address for correspondence: david_jack@baylor.edu

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 Cintra & Tucker (J. Rheol., vol. 39, 1995, p. 1095), or equivalently, a natural closure in the sense of Verleye & Dupret (Developments in Non-Newtonian Flow, 1993, p. 139). 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 fibre orientation was isotropic at some point in time. Finally, this paper presents the fast exact closure, a fast and in principle exact method for solving Jeffery's equation, which does not require approximate closures nor the elliptic integral computation.

Type
Papers
Copyright
Copyright © Cambridge University Press 2011

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