Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-20T16:05:52.360Z Has data issue: false hasContentIssue false
  • English
  • Français

Torque measurements on a stationary axially positioned sphere partially and fully submerged beneath the free surface of a slowly rotating viscous fluid

Published online by Cambridge University Press:  20 April 2006

J. G. Kunesh ,
H. Brenner ,
M. E. O'Neill  and
A. Falade
J. G. Kunesh
Affiliation:
Fractionation Research Inc., 1517 Fair Oaks Ave., South Pasadena, CA 91030
H. Brenner
Affiliation:
Department of Chemical Engineering, Massachusetts Institute of Technology Cambridge, MA 02139
M. E. O'Neill
Affiliation:
Department of Mathematics, University College, London WC1E 6BT
A. Falade
Affiliation:
Department of Mechanical Engineering, University of Lagos, Lago
Get access Rights & Permissions [Opens in a new window]

Abstract

Experimental measurements are presented for the hydrodynamic torque exerted on a stationary sphere situated at the axis of a slowly rotating viscous liquid at small rotary sphere Reynolds numbers (Re < 0.1) as a function of depth of submersion of the sphere below the free surface. Effects of free-surface proximity on the torque furnished the impetus for the study. Experiments were performed for different depths of sphere immersion beneath the free surface, varying from full to partial submersion. Rotation rates were maintained sufficiently low to approximate a planar interface. Torque measurements agreed well with existing theoretical predictions for both the interface-straddling and fully submerged sphere cases. In particular, the predicted continuity of the torque and its derivative at the interface-penetration point (where the sphere first starts to protrude through the free surface) was observed. Free-surface curvature as well as meniscus-curvature effects upon the torque were found to be negligible in the experiments, including even the extreme case where the sphere was in the almost fully withdrawn configuration.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 154 , May 1985 , pp. 29 - 42
Copyright
© 1985 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

Abramowitz, M. & Stegun, I. A. 1964 Handbook of Mathematical Functions, pp. 803–811. U.S. National Bureau of Standards.
Ashare, E., Bird, R. B. & Lescarboura, J. A. 1965 Falling-cylinder viscometers for non-Newtonian fluids. AIChE J. 11, 910–916.Google Scholar
Berdan, C. & Leal, L. G. 1984 Motion of a sphere in the presence of a deformable interface. Part 3. An experimental study of translation normal to the interface. J. Coll. Interface Sci. (in press).Google Scholar
Bowden, F. P. & Lord, R. G. 1963 The aerodynamic resistance to a sphere rotating at high speed. Proc. R. Soc. Lond. A 271, 143–153.Google Scholar
Brenner, H. 1961 The slow motion of a sphere through a viscous fluid towards a plane surface. Chem. Engng Sci. 16, 242–251.Google Scholar
Brenner, H. 1964a Slow viscous rotation of an axisymmetric body in a circular cylinder of finite length. Appl. Sci. Res. A13, 81–120.Google Scholar
Brenner, H. 1964b The Stokes resistance of an arbitrary particle II. An extension. Chem. Engng Sci. 19, 599–629.Google Scholar
Brenner, H. & Leal, L. G. 1978 A micromechanical derivation of Fick's law for interfacial diffusion of surfactant molecules. J. Colloid Interface Sci. 65, 191–209.Google Scholar
Brenner, H. & Leal, L. G. 1982 Conservation and constitutive equations for adsorbed species undergoing surface diffusion and convection at a fluid-fluid interface. J. Colloid Interface Sci. 88, 136–184.Google Scholar
Caswell, B. 1970 Effect of finite boundaries on the motion of particles in non-Newtonian fluids. Chem. Engng Sci. 25, 1167–1176.Google Scholar
Childress S. 1964 The slow motion of a sphere in a rotating viscous fluid. J. Fluid Mech. 20, 305–314.Google Scholar
Cho, Y. I. & Hartnett, J. P. 1979 Falling ball viscometer — a new instrument for viscoelastic fluids. Lett. Heat Mass Transfer 6, 335–342.Google Scholar
Cox, R. G. & Brenner, H. 1967a Effect of finite boundaries on the Stokes resistance of an arbitrary particle. Part 3. Translation and rotation. J. Fluid Mech. 28, 391–411.Google Scholar
Cox, R. G. & Brenner, H. 1967b The slow motion of a sphere through a viscous fluid towards a plane surface. II. Small gap widths, including inertial effects. Chem. Engng Sci. 22, 1753–1777.Google Scholar
Davis, A. M. J. & O'Neill, M. E.1979 Slow rotation of a sphere submerged in a fluid with a surfactant surface layer. Intl J. Multiphase Flow 5, 413–425.Google Scholar
Davis, P. K. 1965 Motion of a sphere in a rotating fluid at low Reynolds number. Phys. Fluids 8, 560–567.Google Scholar
Dennis, S. C. R. & Ingham, D. B. 1982 The boundary layer on a fixed sphere on the axis of an unbounded viscous fluid. J. Fluid Mech. 123, 210–236.Google Scholar
Dennis, S. C. R., Ingham, D. B. & Singh, S. N. 1982 The slow translation of a sphere in a rotating fluid. J. Fluid Mech. 117, 251–267.Google Scholar
Dennis, S. C. R., Singh, S. N. & Ingham, D. B. 1980 The steady flow due to a rotating sphere at low and moderate Reynolds numbers. J. Fluid Mech. 101, 257–279.Google Scholar
Falade, A. 1982 Arbitrary motion of an elliptic disk at a fluid interface. Intl J. Multiphase Flow 8, 543–551.Google Scholar
Fluide, M. J. C. & Daborn, J. E. 1982 Viscosity measurement by means of falling spheres compared with capillary viscometry. J. Phys. E: Sci. Instrum. 15, 1313–1321Google Scholar
Geils, R. H. 1977 Small-volume inclined falling-ball viscometer. Rev. Sci. Instrum. 48, 783–785.Google Scholar
Hirschfeld, B. R., Brenner, H. & Falade, A. 1984 First- and second-order wall effects upon the slow viscous asymmetric motion of an arbitrarily-shaped,-positioned and -oriented particle within a circular cylinder. Physicochem. Hydrodyn. 5, 99–133.Google Scholar
Hocking, L. M., Moore, D. W. & Walton, I. C. 1979 Drag on a sphere moving axially in a long rotating cylinder. J. Fluid Mech. 90, 781–793.Google Scholar
Jeffery, G. B. 1915 On the steady rotation of a solid of revolution in a viscous fluid. Proc. Lond. Math. Soc. 14, 327–338.Google Scholar
Kunesh, J. G. 1971 Experiments on the hydrodynamic resistance of translating-rotating particles. Ph.D. thesis, Carnegie-Mellon University.
Lamb, H. 1932 Hydrodynamics, 6th edn, p. 585. Cambridge University Press.
Leal, L. G. & Lee, S. H. 1982 Particle motion near a deformable fluid interface. Adv. Coll. Interface Sci. 17, 61–81.Google Scholar
Majumdar, S. R., O'Neill, M. E. & Brenner, H.1974 Note on the slow rotation of a concave spherical lens or bowl in two immiscible semi-infinite viscous fluids. Mathematika 21, 147–154.Google Scholar
Maxworthy, T. 1965 An experimental determination of the slow motion of a sphere in a viscous rotating fluid. J. Fluid Mech. 23, 373–384.Google Scholar
Maxworthy, T. 1970 The flow created by a sphere moving along the axis of a rotating slightly-viscous fluid. J. Fluid Mech. 40, 453–479.Google Scholar
Mena, B., Levinson, E. & Caswell, B. 1972 Torque on a sphere inside a rotating cylinder. Z. angew. Math. Phys. 23, 173–181.Google Scholar
Munro, R. G., Piermarini, G. J. & Block, S. 1979 Wall effects in diamond-anvil pressure-cell falling-sphere viscometers. J. Appl. Phys. 50, 3180–3184.Google Scholar
O'Neill, M. E. & Ranger, K. B.1979 On the rotation of a rotlet or sphere in the presence of an interface. Intl J. Multiphase Flow 5, 143–148.Google Scholar
Ranger, K. B. 1978 Circular disk straddling the interface of two-phase flow. Intl J. Multiphase Flow 4, 263–277.Google Scholar
Richardson, P. D. 1976 Flow beyond an isolated rotating disk. Intl J. Heat Mass Transfer 19, 1189–1195.Google Scholar
Sawatzki, O. 1970 Flow field around a rotating sphere. Acta Mech. 9, 159–214.Google Scholar
Schneider, J. C., O'Neill, M. E. & Brenner, H.1973 On the slow viscous rotation of a body straddling the interface between two immiscible semi-infinite fluids. Mathematika 20, 175–196.Google Scholar
Sneddon, I. N. 1951 Fourier Transforms, p. 516. McGraw-Hill.
Sonshine, R. M., Cox, R. G. & Brenner, H. 1966 The Stokes translation of a particle of arbitrary shape along the axis of a circular cylinder filled to a finite depth with viscous liquid. Appl. Sci. Res. 16, 273–300; 325–360.Google Scholar
Taylor, G. I. 1923 The motion of a sphere in a rotating liquid. Proc. R. Soc. Lond. A 102, 180–189.Google Scholar
Thomas, R. H. & Walters, K. 1964 Motion of elasto-viscous fluid due to sphere rotating about its diameter. Q. J. Mech. Appl. Maths 17, 39–53.Google Scholar
Walters, K. & Waters, N. D. 1963 On the use of a rotating sphere in the measurement of elasto-viscous parameters. Brit. J. Appl. Phys. 14, 667–671.Google Scholar
Waters, N. D. & Gooden, D. K. 1980 Couple on a rotating oblate spheroid in an elasto-viscous fluid. Q. J. Mech. Appl. Maths 33, 189–206.Google Scholar
Wein, O. 1979 Rotational quasi-viscometric flows around a rotating sphere. J. Non-Newtonian Fluid Mech. 5, 297–313.Google Scholar