Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-26T11:08:18.948Z Has data issue: false hasContentIssue false
  • English
  • Français

The Stokes flow problem for a class of axially symmetric bodies

Published online by Cambridge University Press:  28 March 2006

L. E. Payne  and
W. H. Pell
L. E. Payne
Affiliation:
Institute for Fluid Dynamics and Applied Mathematics, University of Maryland
W. H. Pell
Affiliation:
National Bureau of Standards, Washington 25, D.C.
Get access Rights & Permissions [Opens in a new window]

Abstract

The Stokes flow problem is concerned with fluid motion about an obstacle when the motion is such that inertial effects can be neglected. This problem is considered here for the case in which the obstacle (or configuration of obstacles) has an axis of symmetry, and the flow at distant points is uniform and parallel to this axis. The differential equation for the stream function ψ then assumes the form L2−1ψ = 0, where L−1 is the operator which occurs in axiallysymmetric flows of the incompressible ideal fluid. This is a particular case of the fundamental operator of A. Weinstein's generalized axially symmetric potential theory. Using the results of this theory and theorems regarding representations of the solutions of repeated operator equations, the authors (1) give a general expression for the drag of an axially symmetric configuration in Stokes flow, and (2) indicate a procedure for the determination of the stream function. The stream function is found for the particular case of the lens-shaped body.

Explicit calculation of the drag is difficult for the general lens, without recourse to numerical procedures, but is relatively easy in the case of the hemispherical cup. As examples illustrative of their procedures, the authors briefly consider three Stokes flow problems whose solutions have been given previously.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 7 , Issue 4 , April 1960 , pp. 529 - 549
Copyright
© 1960 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Berry, A. & Swain, L. M. 1923 On the steady motion of a cylinder through infinite viscous fluid. Proc. Roy. Soc. A, 102, 766–78.Google Scholar
Bierens De Haan, D. 1939 Nouvelles tables d'intégrales définies, edn. of 1867 corrected. New York: G. E. Stechert and Co.
Brousse, P. 1956 Resolution de divers problèmes du type Stokes-Beltrami posés par la technique aéronautique. Publ. sci. Minist. Air, no. 323.Google Scholar
Dean, W. R. 1936 Note on the slow motion of a fluid. Phil. Mag. (ser. 7), 21, 724–44.Google Scholar
Dean, W. R. 1944 On the shearing motion of a fluid past a projection. Proc. Camb. Phil. Soc., 40, 19–36.Google Scholar
Dryden, H. L., Murnaghan, F. D. & Bateman, H. 1932 Hydrodynamics, National Research Council Bulletin, no. 84. Reprinted by Dover Publications, 1956.
Finn, R. & Noll, W. 1957 On the uniqueness and non-existence of Stokes flows. Arch. Rat. Mech. Analysis, 1, 97–106.Google Scholar
Green, G. 1944 Solution of some problems in viscous flow. Phil. Mag. (ser. 7), 35, 250–62.Google Scholar
Hobson, E. W. 1931 The Theory of Spherical and Ellipsoidal Harmonics. Cambridge University Press.
Huber, A. 1953 On the uniqueness of generalized axially symmetric potentials. Ann. Math., 60, 351–8.Google Scholar
Hyman, M. A. 1954 Concerning analytic solutions of the generalized potential equation. Proc. Konigl. Nederl. Akad. Wetensch., Amsterdam (A), 57, 408–13.Google Scholar
Ladenberg, R. 1907 über den Einfluss von Wanden Auf die Bewegung einer Kugel in einer reibenden Flussigkeit. Ann. Phys., Lpz., 23, 447–58.Google Scholar
Lamb, H. 1932 Hydrodynamics, 6th edn. Cambridge University Press. Reprinted by Dover Publications, 1945
Lorentz, H. A. 1897 A general theorem concerning the motion of a viscous fluid and a few consequences derived from it. Versl. Konigl. Akad. Wetensch. Amst. 5, 168–74; collected papers, vol. IV, 7–14, Martinus Nijhoff, The Hague, Holland, 1937.Google Scholar
Milne-Thomson, L. M. 1950 Theoretical Hydrodynamics, 2nd edn. New York: Macmillan and Co.
Neumann, C. 1881 über die Mehlersschen Kegelfunktionen und deren Anwendung auf elektrostatische Probleme. Math. Ann., Lpz., 18, 195–236.Google Scholar
Oberbeck, A. 1876 über stationäre Flüssigkeitsbewegungen mit Berücksichtigung der inner Reibung. J. reine angew. Math., 81, 62–80.Google Scholar
Odqvist, F. K. G. 1930 über die Randwertaufgaben der Hydrodynamik Zäher Flüssigkeiten. Math. Z., 32, 329–75.Google Scholar
Payne, L. E. 1958 Representation Formulas for Solutions of a Class of Partial Differential Equations, Techn. Note BN-122. Institute for Fluid Dynamics and Applied Mathematics, University of Maryland.
Ray, M. 1936 Application of Bessel functions to the solution of the problem of motion of a circular disk in a viscous liquid. Phil. Mag. (ser. 7), 21, 546–64.Google Scholar
Stimson, M. & Jeffery, G. B. 1926 The motion of two-spheres in a viscous fluid. Proc. Roy. Soc. A, 111, 110–16.Google Scholar
Stokes, G. G. 1850 On the effect of the internal friction of fluids on pendulums. Trans. Camb. Phil. Soc. 9, 8.Google Scholar
Weinstein, A. 1948 Discontinuous integrals and generalized potential theory. Trans. Amer. Math. Soc., 63, 342–54.Google Scholar
Weinstein, A. 1955 On a class of partial differential equations of even order. Ann. Mat. pura appl. (ser. 4), 39, 245–54.Google Scholar