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Microstructure suspended in three-dimensional flows

Published online by Cambridge University Press:  26 April 2006

Andrew J. Szeri  and
L. Gary Leal
Andrew J. Szeri
Affiliation:
Department of Mechanical and Aerospace Engineering, University of California, Irvine, CA 92717-3975, USA
L. Gary Leal
Affiliation:
Department of Chemical and Nuclear Engineering, University of California, Santa Barbara, CA 93106, USA
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Abstract

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.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 250 , May 1993 , pp. 143 - 167
Copyright
© 1993 Cambridge University Press

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References

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, 284–304.Google Scholar
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, 161–179.Google Scholar
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, 297–325.Google Scholar
Olbricht, W. L., Rallison, J. M. & Leal, L. G. 1982 Strong flow criterion based on microstructure deformation. J. Non-Newtonian Fluid Mech. 10, 291–318.Google Scholar
Szeri, A. J. 1993 Pattern formation in recirculating flows of suspensions of orientable particles. Phil. Trans. R. Soc. Lond., to appear.Google Scholar
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, 33–56.Google Scholar
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, 207–241 (referred to herein as SWL.)Google Scholar
Thompson, J. M. T. & Stewart, H. B. 1986 Nonlinear Dynamics and Chaos. J. Wiley & Sons.