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Variational properties of steady fall in Stokes flow

Published online by Cambridge University Press:  29 March 2006

H. F. Weinberger
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Abstract

It is shown that for a given body and a given orientation g there is always a position of the centre of mass which produces a stable falling motion in a very viscous fluid with g vertical and, in general, with a spin about the vertical axis. The corresponding terminal settling speed is bounded by means of several variational principles.

Relations between the terminal speeds for falls with different downward directions and between the terminal speed and the geometry of the body are deduced. In particular, it is proved that for a large class of slender bodies the first approximation to the drag obtained from the slender-body theory of Burgers (1938) is correct. It follows that the ratio of the terminal speeds for falls with the long axis vertical and horizontal is near two.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 52 , Issue 2 , 28 March 1972 , pp. 321 - 344
Copyright
© 1972 Cambridge University Press

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References

Batchelor, G. K. 1970 Slender-body theory for particles of arbitrary cross-section in Stokes flow. J. Fluid Mech. 44, 419–440.Google Scholar
Brenner, H. 1964 The Stokes resistance of an arbitrary particle II. CAem. Engng. Sci. 19, 599–624.Google Scholar
Burgers, J. M. 1938 On the motion of small particles of elongated form suspended in a viscous liquid. In Second Report on Viscosity and Plasticity, Kon. Ned. Akad. Wet., Amsterdam, 16, 113.Google Scholar
Chinn, W. G. & Steenrod, N. E. 1966 First Concepts of Topology. New York: Random House.
Cox, R. G. 1970 The motion of long slender bodies in a viscous fluid 1. General theory. J. Fluid Mech. 43, 641–660.Google Scholar
Diaz, J. B. 1951 Upper and lower bounds for quadratic functionals. Proc. Symp. Spectral Theory & Differential Problems, Stillwater, Oklahoma, pp. 279–289.
Goldman, A. J. 1966 Investigations in low Reynolds number fluid particle dynamics. Ph.D. Dissertation, Graduate Division, New York University School of Engineering.
Gröbner, W. & Hofreiter, N. 1966 Integraltafel. Zweiter Ted: Bestimmte Integrale. Springer.
Happel, V. & Brenner, H. 1965 Low Reynolds Number Hydrodynamics. Prentice-Hall.
Helmholtz, H. 1968 Zur Theorie der stationären Ströme in reibenden Flüssigkeiten. Verh. d. naturhist. med. Vereins zu Heidelberg, 5, 1–7. (See also Wiss. Abh. (collected works), 1, 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
Jahnke, E. & Emde, F. 1945 Tables of Functions. Dover.
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. & Rubenbeld, L. A. & Molyneux, J. E. 1967 Extremum principles for slow viscous flows with applications 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
Lamb, H. 1932 Hydrodynamics. Cambridge University Press.
Mikhlin, S. K. 1957 Variational Methods in Mathematical Physics. Moscow: Gostekhizdat (Trans. Pergamon, 1964).
Moran, J. P. 1963 Line source distributions and slender-body theory. J. Fluid Mech. 17, 285–304.Google Scholar
Oberbeck, H. A. 1876 Über stationiire Flussigkeitsbewegungen mit Beruchsiclitigung der inneren Reibung. Crelle, 81, 62.Google Scholar
Polya, G. & Szego, G. 1951 Isoperimetric Inequalities in Mathematical Physics, p. 205. Ann. Math. Studies, vol. 27.
Skalak, R. 1970 Extensions of extremum principles for slow viscous flows. J. Fluid Mech. 42, 527–548.Google Scholar
Synge, J. L. 1957 The Hypercircle in. Mathematical Physics. Cambridge University Press.
Taylor, G. I. 1967 Film: Low Reynolds number fiows. Chicago: Educational Services Inc.
Taylor, G. I. 1969 Motion of axisymmetric bodies in viscous fluids. In Problems of Hydrodynamics and Continuum Mechanics in Honor of Academician L. I. Sedov, English edn., pp. 718–724. Philadelphia: Siam.
Tillett, J. P. K. 1970 Axial and transverse Stokes flow past slender axisymmetric bodies. J. Fluid Mech. 44, 401–417.Google Scholar
Tuck, E. O. 1964 Some methods for flows past slender bodies. J. Fluicl Mech. 18, 619.Google Scholar
Tuck, E. O. 1970 Toward the calculation and minimization of Stokes drag on bodies of arbitrary shape. Proc. 3rd Aust. Conf. on Hydraulics and Fluid Mech. 1968, pp. 29–32. Sidney: Institution of Engineers of Australia.
Weinberger, H. F. 1961 Variational methods in boundary-value problems. Lecture notes, University of Minnesota Engineering Bookstore.
Weinberger, H. F. 1972 On the steady fall of a body in a Navier-Stokes fluid. Proc. Symp. Pure Math. Partial Diffeerential Eqn., Providence: Amer. Math. Soc. (in press).
Weinstein, A. 1953 Generalized axially symmetric potential theory. Bull. Am. Math. Soc. 59, 20–38.Google Scholar