Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-23T05:35:29.830Z Has data issue: false hasContentIssue false
  • English
  • Français

On the slow viscous rotation of a body straddling the interface between two immiscible semi-infinite fluids

Part of: Rotating fluids Incompressible viscous fluids

Published online by Cambridge University Press:  26 February 2010

Joanne C. Schneider ,
Michael E. O'Neill  and
Howard Brenner
Joanne C. Schneider
Affiliation:
Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh, Pennsylvania, 15213, U.S.A.
Michael E. O'Neill
Affiliation:
Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh, Pennsylvania, 15213, U.S.A.
Howard Brenner
Affiliation:
Department of Mathematics, University College London, London, WC1E 6BT.
Get access

Abstract

Using toroidal coordinates, an exact solution is derived for the velocity field induced in two immiscible semi-infinite fluids possessing a plane interface, by the slow rotation of an axially symmetric body partly immersed in each fluid. The surface of the body is assumed to be formed from two intersecting spheres, or a sphere and a circular disc, with the circle of intersection of the composite surfaces lying in th interface.

It is shown that when the rotating body possesses reflection symmetry about the plane of the interface of the fluids, the velocity field in either fluid is independent of the viscosities of the fluids. The torque exerted on the body is then proportional to the sum of the viscosities. Analytic closed-form expressions are derived for the torque when the body is either a sphere, a circular disc, or a tangent-sphere dumbbell, and for a hemisphere rotating in an infinite homogeneous fluid. Closed-form results are also given for an immersed sphere, tangent to a free surface. For other geometrical configurations, numerical values of the torque are provided for a variety of body shapes and two-fluid systems of various viscosity ratios.

MSC classification

Secondary: 76D99: None of the above, but in this section 76U05: Rotating fluids
Type
Research Article
Information
Mathematika , Volume 20 , Issue 2 , December 1973 , pp. 175 - 196
Copyright
Copyright © University College London 1973

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

Brenner, H., Advances in Chem. Engrg., 6 (Academic Press, New York, 1966).Google Scholar
Haan, D. Bierens de, Nouvelles Tables d'Integrales Defines (G. E. Steckert & Co., New York, 1939).Google Scholar
Cox, R. G. and Brenner, H., Chem. Engrg. Sci., 22 (1967), 1753.Google Scholar
Erdelyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F. G., Higher transcendental functions 1 (McGraw-Hill, New York, 1953).Google Scholar
Happel, J. and Brenner, H., Low Reynolds number hydrodynamics (Prentice-Hall, Englewood Cliffs, New Jersey, 1965).Google Scholar
Jeffery, G. B., Proc. London Math. Soc, 14 (1915), 327.CrossRefGoogle Scholar
Kanwal, R. P., J. Fluid Meek., 10 (1961), 17.CrossRefGoogle Scholar
Kunesh, J. G., O'Neill, M. E. and Brenner, H. (to appear).Google Scholar
Lamb, H., Hydrodynamics (Dover, New York, 1945).Google Scholar
Majumdar, S. R., Mathematika, 14 (1967), 43.CrossRefGoogle Scholar
Robin, L., Fonctions spheriques de Legendre et fonctions spheroidales, III (Gauther-Villars, Paris, 1959).Google Scholar
Snow, C., U.S. Nat. Bureau of Standards J. of Research, 43 (1949), 377.Google Scholar