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.
Similar content being viewed by others
References
H.A. Lorentz, A general theorem concerning the motion of a viscous fluid and a few consequences derived from it (in Dutch). Zittingsverslag Koninkl. Akad. van Wetensch. Amsterdam 5 (1986) 168–175. [See also Collected Works, Vol. IV. The Hague: Martinus Nijhoff (1937) pp. 7–14.]
J. Happel and H. Brenner, Low Reynolds Number Hydrodynamics. The Hague: Martinus Nijhoff (1983) xii + 553 pp.
S.C. Cowin, The theory of polar fluids. Adv. Appl. Mech. 14 (1974) 279–347.
J.S. Dahler and L.E. Scriven, Theory of structured continua I. General consideration of angular momentum and polarization. Proc. Roy. Soc. 275 (1963) 504–527.
V.K. Stokes, Theories of Fluids with Microstructure. Berlin: Springer-Verlag (1984) xi + 209pp.
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.
D.W. Condiff and J.S.Dahler, Fluid mechanical aspects of antisymmetric stress. Phys. Fluids 7 (1964) 842–854.
D.R.de Groot and P. Mazur, Non-equilibrium Thermodynamics. New York: Dover (1984) pp. 307–308.
R.J. Atkin, S.C. Cowin and N. Fox, On boundary conditions for polar materials. J. Appl. Math. Phys. (ZAMP) 28 (1977) 1017–1026.
P. 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.
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.
H. Ramkissoon, Slow steady rotation of an axially symmetric body in a micropolar fluid Appl. Sci. Res. 33 (1977) 243–257.
H.S. Sellers and H. Brenner, Translational and rotational motions of a sphere in a dipolar suspension PhysicoChem. Hydrodyn. 11 (1989) 455–466.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Brenner, H., Nadim, A. The Lorentz reciprocal theorem for micropolar fluids. J Eng Math 30, 169–176 (1996). https://doi.org/10.1007/BF00118829
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00118829