# The Lorentz reciprocal theorem for micropolar fluids

## 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## Preview

Unable to display preview. Download preview PDF.

## References

- 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.J. Happel and H. Brenner,
*Low Reynolds Number Hydrodynamics*. The Hague: Martinus Nijhoff (1983) xii + 553 pp.Google Scholar - 3.S.C. Cowin, The theory of polar fluids.
*Adv. Appl. Mech.*14 (1974) 279–347.CrossRefGoogle Scholar - 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.V.K. Stokes, Theories of Fluids with Microstructure. Berlin: Springer-Verlag (1984) xi + 209pp.CrossRefGoogle Scholar
- 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.D.W. Condiff and J.S. Dahler, Fluid mechanical aspects of antisymmetric stress.
*Phys. Fluids*7 (1964) 842–854.ADSCrossRefMATHMathSciNetGoogle Scholar - 8.D.R. de Groot and R Mazur,
*Non-equilibrium Thermodynamics*. New York: Dover (1984) pp. 307–308.Google Scholar - 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.CrossRefMATHMathSciNetGoogle Scholar - 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.CrossRefMATHMathSciNetGoogle Scholar - 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.ADSCrossRefMATHMathSciNetGoogle Scholar - 12.H. Ramkissoon, Slow steady rotation of an axially symmetric body in a micropolar fluid
*Appl. Sci. Res.*33 (1977) 243–257.CrossRefMATHMathSciNetGoogle Scholar - 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