Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-18T22:38:48.808Z Has data issue: false hasContentIssue false
  • English
  • Français

A theory of transversely isotropic fluids

Published online by Cambridge University Press:  28 March 2006

S. J. Allen  and
C. N. DeSilva
S. J. Allen
Affiliation:
Department of Aeronautics and Engineering Mechanics, University of Minnesota, Minneapolis Present address: Rosemount Engineering Company, Minneapolis, Minnesota.
C. N. DeSilva
Affiliation:
Department of Aeronautics and Engineering Mechanics, University of Minnesota, Minneapolis
Get access Rights & Permissions [Opens in a new window]

Abstract

The present paper proposes a theory for the mechanical behaviour of a fluid with a rigid microstructure. The microstructure is described by a director frame of three vectors and a second-order tensor W and its gradient are proposed as measures of the kinematics of this frame. When the frame is spinning without deforming, W reduces to the director spin velocity. Postulating the existence of a couple stress in addition to the classical Cauchy stress, the linear constitutive equations for such a structured fluid are derived and then specialized to the case of transverse isotropy.

These equations are used to study rectilinear shearing flow. When [dtri ]W = 0, the condition for a non-interacting substructure, the results of the theory are shown to be in agreement with the work of Jeffery and of Ericksen. For mutually interacting substructure particles, [dtri ]W ≠ 0, a geometric analysis of the non-linear differential equations is performed in order to exhibit the effects of particle concentration on the flow kinematics.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 24 , Issue 4 , April 1966 , pp. 801 - 821
Copyright
© 1966 Cambridge University Press

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

Anzelius, A. 1931 Über die Bewegung der anisotropen Flüssigkeiten. The Annual of the University of Uppsala.Google Scholar
Born, M. 1920 Z. Phys. 1, 22.
Burnside, W. S. & Panton, A. W. 1960 The Theory of Equations, vol. 1. New York: Dover.
Condiff, D. W. & Dahler, J. S. 1964 Phys. Fluids, 7, 842–54.
Cosserat, E. & F. 1909 Théorie des corps déformables. Paris: Hermann et Fils.
Dahler, J. S. 1965 High-density phenomena. Research Frontiers of Fluid Dynamics. New York: Interscience.
Dahler, J. S. & Scriven, L. E. 1963 Proc. Roy. Soc., A 275, 504–27.
Ericksen, J. L. 1960a Arch. Ration. Mech. Analysis, 4, 231–7.
Ericksen, J. L. 1960b Kolloidzeitschrift, 173, 117–22.
Ericksen, J. L. & Truesdell, C. 1958 Arch. Ration. Mech. Analysis, 1, 295–323.
Eringen, A. C. 1964 Int. J. Engng Sci. 2, 205–17.
Eringen, A. C. & Suhubi, E. S. 1964 Int. J. Engng Sci. 2, 189–203.
Grad, H. 1952 Comm. Pure Appl. Math. 5, 455–94.
Hand, G. L. 1962 J. Fluid Mech. 13, 33–46.
Haynes, R. H. 1961 Trans. Soc. Rheology, 5, 85–102.
Jeffery, G. B. 1922 Proc. Roy. Soc., A 102, 171.
Mindlin, R. D. 1964 Arch. Ration. Mech. Analysis, 16, 51–78.
Oseen, C. W. 1933 Trans. Faraday Soc. 29, 883–99.
Synge, J. L. & Griffith, B. A. 1959 Principles of Mechanics. New York: McGraw-Hill.
Toupin, R. A. 1962 Arch. Ration. Mech. Analysis, 11, 385–414.
Toupin, R. A. 1964 Arch. Ration. Mech. Analysis, 17, 85–112.