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Variational inequalities for a body in a viscous shearing flow

Published online by Cambridge University Press:  29 March 2006

A. Nir ,
H. F. Weinberger  and
A. Acrivos
A. Nir
Affiliation:
Department of Chemical Engineering, Israel Institute of Technology Haifa
H. F. Weinberger
Affiliation:
Department of Mathematics, University of Minnesota, Mineapolis
A. Acrivos
Affiliation:
Department of Chemical Engineeing, Stanford University, Stanford, California 94305
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Abstract

The slow motion of a body in a viscous shearing field is examined. Variational principles are used to derive inequalities which approximate the elements of the shearing matrix M of a body of arbitrary shape, where M is the matrix relating the force, torque and stresslet exerted by the body on the fluid to the relative translational and rotational velocities of the body and the rate of deformation of the undisturbed linear field. An upper bound for the elements of M is obtained by showing that the quadratic form of M increases monotonically with B, the region occupied by the body, while a lower bound for this form is given in terms of the electrostatic properties of a conductor and a dielectric of the same shape as B. Particular attention is paid to bodies of revolution, for which certain more definitive results are obtained: for example, their resistance to a rotation with axial symmetry is always less than twice their resistance to a rotation perpendicular to their axis.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 68 , Issue 4 , 29 April 1975 , pp. 739 - 755
Copyright
© 1975 Cambridge University Press

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References

Batchelor, G. K. 1970 The stress system in a suspension of force free particles. J. Fluid Mech. 41, 545–570.Google Scholar
Brenner, H. & Oweill, M. E. 1972 On the Stokes resistance of multiparticle systems in a linear shear flow. Chem. Engng Sci. 27, 1421–1439.Google Scholar
Bretherton, F. P. 1962 The motion of rigid particles in a shear flow at low Reynolds numbers. J. Fluid. Mech. 14, 284–304.Google Scholar
Happel, J. & Brenner, H. 1965 Low Reynolds Number Hydrodynamics. Prentice-Hall.
Hashin, Z. 1969 Viscosity of rigid particle suspensions. In Contribution to Mechanics, Reiner Anniversary Volume (ed. D. Abir), pp. 347–359. Pergamon.
Helmholtz, H. 1868 Zur Theorie der stationaren Strome in reibenden Flüssigkeiten. Verh. d. naturhist. med. Vereins zu Heidelberg, 5, 1–7. (See also Wiss. Abh. (Collected Works), vol. 1, pp. 223–230.)Google Scholar
Hill, R. & Power, G. 1956 Extremum principles for slow viscous flow and the approximate calculation of drag. Quart. J. Mech. Appl. Math. 9, 313–319.Google Scholar
Hinch, E. J. 1972 A remark on the symmetries of certain material tensors for a particle in Stokes flow. J. Fluid Mech. 54, 423–425.Google Scholar
Jeffery, G. B. 1922 The motion of ellipsoidal particles immersed in a viscous fluid. Proc. Roy. Soc. A 102, 161–179.Google Scholar
Kearsley, E. A. 1960 Bounds on the dissipation of energy in steady flow of a viscous incompressible fluid around a body rotating within a finite region. Arch. Rat. Mech. Anal. 5, 347–354.Google Scholar
Keller, J. B., Rubenfeld, L. A. & Molyneux, J. E. 1967 Extremum principles for slow viscous flow with application to suspensions. J. Fluid Mech. 30, 97–125.Google Scholar
Korteweg, D. J. 1883 On a general theorem of the stability of the motion of a viscous fluid. Phil. Mag. 16 (5), 112–118.Google Scholar
Nir, A. 1973 Ph.D. dissertation, Stanford University.
Nir, A. & Acrivos, A. 1973 On the creeping motion of two arbitrary-sized touching spheres in a linear shear field. J. Fluid Mech. 59, 209–223.Google Scholar
Polya, G. & Szegö, G. 1951 Isoperimetric inequalities in mathematical physics. Ann. Math. Studies, 27, 8.Google Scholar
Prager, S. 1963 Diffusion and viscous flow in concentrated suspension. Physica, 29, 129–139.Google Scholar
Schiffer, M. 1957 Sur la polarisation et la masse virtuelIe. C.R. Acad. Sci. (Paris), 244, 3118–3121.Google Scholar
Schiffer, M. & Szegö, G. 1949 Virtual mass and polarization. Trans. Am. Math. Soc. 67, 130–205.Google Scholar
Weinberger, H. F. 1972 Variational properties of steady fall in Stokes flow. J. Fluid Mech. 52, 321–344.Google Scholar
Weinberger, H. F. 1973 On the steady fall of a body in a Navier-Stokes fluid. Proc. 23rd Symp. Pure Math., Partial Differential Equations, pp. 421–439. Providence: Am. Math. Soc.
Youngren, G. K. & Acrivos, A. 1975 Stokes flow past a particle of arbitrary shape: a numerical method of solution. J. Fluid Mech. (to appear).Google Scholar