Skip to main content Accessibility help

Microstructure suspended in three-dimensional flows

  • Andrew J. Szeri (a1) and L. Gary Leal (a2)


The dynamical behaviour of stretchable, orientable microstructure suspended in a general three-dimensional fluid flow is investigated. Model equations given by Olbricht, Rallison & Leal (1982) are examined in the case of microstructure travelling through arbitrarily complicated flows of the carrier fluid. As in the two-dimensional analysis of Szeri, Wiggins & Leal (1991), one must first treat the orientation dynamics problem; only then can the equation for stretch of the microstructure be analyzed rationally. In three-dimensional flows that are steady in the Lagrangian frame, attractors for the orientation dynamics are shown to be equilibria or limit cycles; this asymptotic behaviour was first deduced by Bretherton (1962). In three-dimensional flows that are time periodic in the Lagrangian frame (e.g. recirculating flows), the orientation dynamics may be characterized by periodic or quasi-periodic attractors. Thus, robust (generic) behaviour in these cases is always characterized by a single global attractor; there is no asymptotic dependence of orientation dynamics on the initial orientation. The type of asymptotic orientation dynamics – steady, periodic, or quasi-periodic - is signified by a simple criterion. Details of the relevant bifurcations, as well as history-dependent strong flow criteria are developed. Examples which illustrate the various types of behaviour are given.



Hide All
Arnol'd, V. I. 1973 Ordinary Differential Equations. MIT Press.
Bretherton, F. P. 1962 The motion of rigid particles in a shear flow at low Reynolds number. J. Fluid Mech. 14, 284304.
Cullen, C. G. 1979 Linear Algebra and Differential Equations. Prindle, Weber and Schmidt.
Jeffery, G. B. 1922 The motion of ellipsoidal particles immersed in a fluid. Proc. R. Soc. Lond. A 102, 161179.
Lipscomb, G. G., Denn, M. M., Hur, D. U. & Boger, D. V. 1988 The flow of fiber suspensions in complex geometries. J. Non-Newtonian Fluid Mech. 26, 297325.
Olbricht, W. L., Rallison, J. M. & Leal, L. G. 1982 Strong flow criterion based on microstructure deformation. J. Non-Newtonian Fluid Mech. 10, 291318.
Szeri, A. J. 1993 Pattern formation in recirculating flows of suspensions of orientable particles. Phil. Trans. R. Soc. Lond., to appear.
Szeri, A. J., Milliken, W. J. & Leal, L. G. 1992 Rigid particles suspended in time-dependent flows: irregular versus regular motion, disorder versus order. J. Fluid Mech. 237, 3356.
Szeri, A. J., Wiggins, S. W. & Leal, L. G. 1991 On the dynamics of microstructure in unsteady, spatially inhomogeneous, two-dimensional fluid flows. J. Fluid Mec. 228, 207241 (referred to herein as SWL.)
Thompson, J. M. T. & Stewart, H. B. 1986 Nonlinear Dynamics and Chaos. J. Wiley & Sons.
MathJax is a JavaScript display engine for mathematics. For more information see

Related content

Powered by UNSILO

Microstructure suspended in three-dimensional flows

  • Andrew J. Szeri (a1) and L. Gary Leal (a2)


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.