Hostname: page-component-8448b6f56d-mp689 Total loading time: 0 Render date: 2024-04-19T09:29:48.669Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  24 May 2011

STEPHEN MONTGOMERY-SMITH ,
WEI HE ,
DAVID A. JACK  and
DOUGLAS E. SMITH
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
Get access Rights & Permissions [Opens in a new window]

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.

JFM classification

Multiphase and Particle-laden Flows: Particle/fluid flow Materials Processing Flows: Polymer processing Complex Fluids: Suspensions
Type
Papers
Information
Journal of Fluid Mechanics , Volume 680 , 10 August 2011 , pp. 321 - 335
Copyright
Copyright © Cambridge University Press 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Advani, S. G. & Tucker, C. L. 1987 The use of tensors to describe and predict fiber orientation in short fiber composites. J. Rheol. 31 (8), 751–784.CrossRefGoogle Scholar
Altan, M. C., Advani, S. G., Güçeri, S. I. & Pipes, R. B. 1989 On the description of the orientation state for fiber suspensions in homogeneous flows. J. Rheol. 33 (7), 1129–1155.CrossRefGoogle Scholar
Altan, M. C., Subbiah, S., Guceri, S. I. & Pipes, R. B. 1990 Numerical prediction of three-dimensional fiber orientation in Hele-Shaw flows. Polym. Engng Sci. 30 (14), 848–859.CrossRefGoogle Scholar
Altan, M. C. & Tang, L. 1993 Orientation tensors in simple flows of dilute suspensions of non-Brownian rigid ellipsoids, comparison of analytical and approximate solutions. Rheol. Acta 32, 227–244.CrossRefGoogle Scholar
Bay, R. S. 1991 Fiber orientation in injection molded composites: A comparison of theory and experiment. PhD thesis, University of Illinois at Urbana-Champaign.Google Scholar
Bay, R. S. & Tucker, C. L. 1991 Fiber orientation in simple injection moldings. Part 1. Theory and numerical methods. In Plastics and Plastic Composites: Material Properties, Part Performance, and Process Simulation (ed. Stokes, V. K.), ASME, vol. 29, pp. 445–471. American Society of Mechanical Engineers.Google Scholar
Bird, R. B., Armstrong, R. C. & Hassager, O. 1987 Dynamics of Polymeric Liquids, 2nd edn., vol. 1: Fluid Mechanics. Wiley.Google Scholar
Bretherton, F. P. 1962 The motion of rigid particles in a shear flow at low Reynolds number. J. Fluid Mech. 14, 284–304.CrossRefGoogle Scholar
Carlson, B. C. 1995 Numerical computation of real or complex elliptic integrals. Numer. Algorithms 10, 13–26.CrossRefGoogle Scholar
Chung, D. H. & Kwon, T. H. 2002 Invariant-based optimal fitting closure approximation for the numerical prediction of flow-induced fiber orientation. J. Rheol. 46 (1), 169–194.CrossRefGoogle Scholar
Cintra, J. S. & Tucker, C. L. 1995 Orthotropic closure approximations for flow-induced fiber orientation. J. Rheol. 39 (6), 1095–1122.CrossRefGoogle Scholar
Dinh, S. M. & Armstrong, R. C. 1984 A rheological equation of state for semiconcentrated fiber suspensions. J. Rheol. 28 (3), 207–227.CrossRefGoogle Scholar
Doi, M. 1981 Molecular dynamics and rheological properties of concentrated solutions of rodlike polymers in isotropic and liquid crystalline phases. J. Polym. Sci. 19, 229–243.Google Scholar
Folgar, F. P. & Tucker, C. L. 1984 Orientation behavior of fibers in concentrated suspensions. J. Reinf. Plast. Compos. 3, 98–119.CrossRefGoogle Scholar
GNU Scientific Library 2009 GNU Scientific Library. Available at: http://www.gnu.org/software/gsl.Google Scholar
Han, K.-H. & Im, Y.-T. 2002 Numerical simulation of three-dimensional fiber orientation in short-fiber-reinforced injection-molded parts. J. Mater. Process. Technol. 124, 366–371.CrossRefGoogle Scholar
Hinch, E. J. & Leal, L. G. 1976 Constitutive equations in a suspension mechanics. Part 2. Approximate forms for a suspension of rigid particles affected by Brownian rotations. J. Fluid Mech. 76, 187–208.CrossRefGoogle Scholar
Hobson, E. W. 1931 Spherical and Ellipsoid Harmonic. Cambridge University Press.Google Scholar
Jack, D. A., Schache, B. K. & Smith, D. E. 2010 Neural network based closure for modeling short-fiber suspensions. Polym. Compos. 31 (7), 1125–1141.CrossRefGoogle Scholar
Jack, D. A. & Smith, D. E. 2005 An invariant based fitted closure of the sixth-order orientation tensor for modeling short-fiber suspensions. J. Rheol. 49 (5), 1091–1116.CrossRefGoogle Scholar
Jeffery, G. B. 1923 The motion of ellipsoidal particles immersed in a viscous fluid. Proc. R. Soc. Lond. A 102, 161–179.Google Scholar
Lipscomb, G. G. II, Denn, M. M., Hur, D. U. & Boger, D. V. 1988 Flow of fiber suspensions in complex geometries. J. Non-Newtonian Fluid Mech. 26, 297–325.CrossRefGoogle Scholar
Montgomery-Smith, S. J., Jack, D. A. & Smith, D. E. 2010 A systematic approach to obtaining numerical solutions of Jefferys type equations using spherical harmonics. Composites A 41, 827–835.CrossRefGoogle Scholar
Montgomery-Smith, S. J., Jack, D. A. & Smith, D. E. 2011 The fast exact closure for Jeffery's equation with diffusion. J. Non-Newtonian Fluid Mech. 166, 343–353.CrossRefGoogle Scholar
Oberbeck, H. A. 1876 Über stationäre Flüssigkeitsbewegungen mit Berücksichtigung der inneren Reibeng. Crelles J. 81, 62–87.Google Scholar
Press, W. H., Teukolsky, S. A., Vetterling, W. T. & Flannery, B. P. 2003 Numerical Recipes in Fortran 77: The Art of Scientific Computing, 2nd edn., vol. 1. Cambridge University Press.Google Scholar
Szeri, A. J. & Lin, D. J. 1996 A deformation tensor model of Brownian suspensions of orientable particles: the nonlinear dynamics of closure models. J. Non-Newtonian Fluid Mech. 64, 43–69.CrossRefGoogle Scholar
Verleye, V. & Dupret, F. 1993 Prediction of fiber orientation in complex injection molded parts. In Developments in Non-Newtonian Flows (ed. Signer, D. A.), pp. 139–163.Google Scholar
VerWeyst, B. E. 1998 Numerical predictions of flow induced fiber orientation in three-dimensional geometries. PhD thesis, University of Illinois at Urbana Champaign.Google Scholar
VerWeyst, B. E. & Tucker, C. L. III 2002 Fiber suspensions in complex geometries: Flow–orientation coupling. Can. J. Chem. Engng 80, 1093–1106.CrossRefGoogle Scholar
VerWeyst, B. E., Tucker, C. L., Foss, P. H. & O'Gara, J. F. 1999 Fiber orientation in 3-D injection molded features: Prediction and experiment. Intl Polym. Process. 14, 409–420.CrossRefGoogle Scholar
Wetzel, E. D. 1999 Modeling flow-induced microstructure of inhomogeneous liquid–liquid mixtures. PhD thesis, University of Illinois at Urbana Champaign.Google Scholar