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The Lorentz reciprocal theorem for micropolar fluids

  • Howard Brenner
  • Ali Nadim

Abstract

A generalization of the Lorentz reciprocal theorem is developed for the creeping flow of micropolar fluids in which the continuum equations involve both the velocity and the internal spin vector fields. In this case, the stress tensor is generally not symmetric and conservation laws for both linear and angular momentum are needed in order to describe the dynamics of the fluid continuum. This necessitates the introduction of constitutive equations for the antisymmetric part of the stress tensor and the so-called couple-stress in the medium as well. The reciprocal theorem, derived herein in the limit of negligible inertia and without external body forces and couples, provides a general integral relationship between the velocity, spin, stress and couple-stress fields of two otherwise unrelated micropolar flow fields occurring in the same fluid domain.

Keywords

Angular Momentum Fluid Domain Micropolar Fluid Reciprocal Theorem Hydrodynamic Resistance 
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References

  1. 1.
    H.A. Lorentz, A general theorem concerning the motion of a viscous fluid and a few consequences derived from it (in Dutch). Zittingsyerslag Koninkl. Akad. van Wetensch. Amsterdam 5 (1986) 168–175. [See also Collected Works, Vol. IV. The Hague: Martinus Nijhoff (1937) pp. 7–14.]Google Scholar
  2. 2.
    J. Happel and H. Brenner, Low Reynolds Number Hydrodynamics. The Hague: Martinus Nijhoff (1983) xii + 553 pp.Google Scholar
  3. 3.
    S.C. Cowin, The theory of polar fluids. Adv. Appl. Mech. 14 (1974) 279–347.CrossRefGoogle Scholar
  4. 4.
    J.S. Dahler and L.E. Scriven, Theory of structured continua I. General consideration of angular momentum and polarization. Proc. Roy. Soc. 215 (1963) 504–527.ADSCrossRefGoogle Scholar
  5. 5.
    V.K. Stokes, Theories of Fluids with Microstructure. Berlin: Springer-Verlag (1984) xi + 209pp.CrossRefGoogle Scholar
  6. 6.
    R.E. Rosensweig and R.J. Johnston, Aspects of magnetic fluid flow with nonequilibrium magnetization. In: G.A.C. Graham and S.K. Malik (eds.), Continuum Mechanics and its Applications. New York: Hemisphere (1989) pp. 707–729.Google Scholar
  7. 7.
    D.W. Condiff and J.S. Dahler, Fluid mechanical aspects of antisymmetric stress. Phys. Fluids 7 (1964) 842–854.ADSCrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    D.R. de Groot and R Mazur, Non-equilibrium Thermodynamics. New York: Dover (1984) pp. 307–308.Google Scholar
  9. 9.
    R.J. Atkin, S.C. Cowin and N. Fox, On boundary conditions for polar materials. J. Appl. Math. Phys. (TAMP) 28 (1977) 1017–1026.CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    R Brunn, The general solution to the equations of creeping motion of a micropolar fluid and its application. Int. J. Eng. Sci. 20 (1982) 575–585.CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    H. Ramkissoon and S.R. Majumdar, Drag on an axially symmetric body in the Stokes flow of micropolar fluids. Phys. Fluids 19 (1976) 16–21.ADSCrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    H. Ramkissoon, Slow steady rotation of an axially symmetric body in a micropolar fluid Appl. Sci. Res. 33 (1977) 243–257.CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    H.S. Sellers and H. Brenner, Translational and rotational motions of a sphere in a dipolar suspension PhysicoChem. Hydrodyn. 11 (1989) 455–466.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1996

Authors and Affiliations

  • Howard Brenner
    • 1
  • Ali Nadim
    • 2
  1. 1.Department of Chemical EngineeringMass. Inst. of Tech.CambridgeUSA
  2. 2.Department of Aerospace and Mechanical EngineeringBoston UniversityBostonUSA

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