Hostname: page-component-848d4c4894-pftt2 Total loading time: 0 Render date: 2024-05-25T15:25:40.274Z Has data issue: false hasContentIssue false

Perturbations of the coupled Jeffery–Stokes equations

Published online by Cambridge University Press:  29 June 2011

Department of Mathematics, University of Missouri, Columbia, MO 65211, USA
Email address for correspondence:


This paper seeks to provide clues as to why experimental evidence for the alignment of slender fibres in semi-dilute suspensions under shear flows does not match theoretical predictions. This paper posits that the hydrodynamic interactions between the different fibres that might be responsible for the deviation from theory, can at least partially be modelled by the coupling between Jeffery's equation and Stokes' equation. It is proposed that if the initial data are slightly non-uniform, in that the probability distribution of the orientation has small spatial variations, then there is feedback via Stokes' equation that causes these non-uniformities to grow significantly in short amounts of time, so that the standard uncoupled Jeffery's equation becomes a poor predictor when the volume ratio of fibres to fluid is not extremely low. This paper provides numerical evidence, involving spectral analysis of the linearization of the perturbation equation, to support this theory.

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.)



Anczurowski, E. & Mason, S. G. 1967 The kinetics of flowing dispersions III. Equilibrium orientation of rods and discs (experimental). J. Colloid Interface Sci. 23, 533546.CrossRefGoogle Scholar
Batchelor, G. K. 1971 Stress generated in a non-dilute suspension of elongated particles by pure straining motion. J. Fluid Mech. 46, 813829.CrossRefGoogle Scholar
Bird, R. B., Curtiss, C. F., Armstrong, R. C. & Hassager, O. 1987 Dynamics of Polymeric Liquids, 2nd edn., vol. 2: Kinetic Theory. John Wiley & Sons, Inc.Google Scholar
Carlson, B. C. 1995 Numerical computation of real or complex elliptic integrals. J. Numer. Algorithms 10, 1326.CrossRefGoogle Scholar
Chinesta, F., Chaidron, G. & Poitou, A. 2003 On the solution of Fokker–Planck equations in steady recirculating flows involving short fiber suspensions. J. Non-Newtonian Fluid Mech. 113, 97125.CrossRefGoogle Scholar
Chinesta, F. & Poitou, A. 2002 Numerical analysis of the coupling between the flow kinematics and the fiber orientation in Eulerian simulations of dilute short fiber suspensions flows. Can. J. Chem. Engng 80, 11071114.CrossRefGoogle Scholar
Davies, R. 2008 Newmat C++ matrix library, Scholar
Dinh, S. M. & Armstrong, R. C. 1984 A rheological equation of state for semiconcentrated fiber suspensions. J. Rheol. 28 (3), 207227.CrossRefGoogle Scholar
Folgar, F. P. & Tucker, C. L. 1984 Orientation behavior of fibers in concentrated suspensions. J. Reinforc. Plast. Compos. 3, 98119.CrossRefGoogle Scholar
GNU Scientific Library 2010 GNU Scientific Library, Scholar
Hairer, E. & Wanner, G. 1996 b Solving Ordinary Differential Equations. Stiff and Differential-Algebraic Problems. Springer-Verlag.Google Scholar
Jeffery, G. B. 1923 The motion of ellipsoidal particles immersed in a viscous fluid. Proc. R. Soc. Lond. A 102, 161179.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, 297325.CrossRefGoogle Scholar
Montgomery-Smith, S. J., He, W., Jack, D. A. & Smith, D. E. 2011 Exact tensor closures for the three dimensional Jeffery's Equation. J. Fluid Mech. doi:10.1017/jfm.2011.165.CrossRefGoogle Scholar
Montgomery-Smith, S. J., Jack, D. A. & Smith, D. E. 2010 a Spherical software package, Scholar
Montgomery-Smith, S. J., Jack, D. A. & Smith, D. E. 2010 b A systematic approach to obtaining numerical solutions of Jeffery's type equations using spherical harmonics. Composites Part A 41, 827835.CrossRefGoogle Scholar
Nguyen, N., Bapanapalli, S. K., Holbery, J. D., Smith, M. T., Kunc, V., Frame, B., Phelps, J. H. & Tucker, C. L. 2008 Fiber length and orientation in long-fiber injection-molded thermoplastics – Part I: Modeling of microstructure and elastic properties. J. Compos. Mater. 42 (10), 10031029.CrossRefGoogle Scholar
Sepehr, M., Carreau, P. J., Grmela, M., Ausias, G. & Lafleur, P. G. 2004 Comparison of rheological properties of fiber suspensions with model predictions. J. Polym. Engng 24 (6), 579610.CrossRefGoogle Scholar
Shaqfeh, E. S. G. & Fredrickson, G. H 1990 The hydrodynamic stress in a suspension. Phys. Fluids A 2, 724.CrossRefGoogle Scholar
Stover, C. A., Koch, D. L. & Cohen, C. 1992 Observations of fibre orientation in simple shear flow of semi-dilute suspensions. J. Fluid Mech. 238, 277296.CrossRefGoogle 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, 4369.CrossRefGoogle Scholar
Verleye, V. & Dupret, F. 1993 Prediction of fiber orientation in complex injection molded parts. In Developments in Non-Newtonian Flows, pp. 139163. ASME.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, 10931106.CrossRefGoogle Scholar
Villermaux, E. 2009 Hesitant nature. J. Fluid Mech. 636, 14.CrossRefGoogle Scholar
Wagner, N. J. & Brady, J. F. 2009 Shear thickening in colloidal dispersions. Phys. Today 62 (10), 2732.CrossRefGoogle Scholar
Wang, J., O'Gara, J. F. & Tucker, C. L. 2008 An objective model for slow orientation kinetics in concentrated fiber suspensions: theory and rheological evidence. J. Rheol. 52, 11791200.CrossRefGoogle Scholar