Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-18T19:09:31.086Z Has data issue: false hasContentIssue false
  • English
  • Français

The steady flow due to a rotating sphere at low and moderate Reynolds numbers

Published online by Cambridge University Press:  19 April 2006

S. C. R. Dennis ,
S. N. Singh  and
D. B. Ingham
S. C. R. Dennis
Affiliation:
Department of Applied Mathematics, University of Western Ontario, London, Canada
S. N. Singh
Affiliation:
Department of Mechanical Engineering, University of Kentucky, Lexington, U.S.A.
D. B. Ingham
Affiliation:
Department of Applied Mathematical Studies, University of Leeds, England
Get access Rights & Permissions [Opens in a new window]

Abstract

The problem of determining the steady axially symmetrical motion induced by a sphere rotating with constant angular velocity about a diameter in an incompressible viscous fluid which is at rest at large distances from it is considered. The basic independent variables are the polar co-ordinates (r, θ) in a plane through the axis of rotation and with origin at the centre of the sphere. The equations of motion are reduced to three sets of nonlinear second-order ordinary differential equations in the radial variable by expanding the flow variables as series of orthogonal Gegenbauer functions with argument μ = cosθ. Numerical solutions of the finite set of equations obtained by truncating the series after a given number of terms are obtained. The calculations are carried out for Reynolds numbers in the range R = 1 to R = 100, and the results are compared with various other theoretical results and with experimental observations.

The torque exerted by the fluid on the sphere is found to be in good agreement with theory at low Reynolds numbers and appears to tend towards the results of steady boundary-layer theory for increasing Reynolds number. There is excellent agreement with experimental results over the range considered. A region of inflow to the sphere near the poles is balanced by a region of outflow near the equator and as the Reynolds number increases the inflow region increases and the region of outflow becomes narrower. The radial velocity increases with Reynolds number at the equator, indicating the formation of a radial jet over the narrowing region of outflow. There is no evidence of any separation of the flow from the surface of the sphere near the equator over the range of Reynolds numbers considered.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 101 , Issue 2 , 27 November 1980 , pp. 257 - 279
Copyright
© 1980 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

Banks, W. H. H. 1965 Quart. J. Mech. Appl. Math. 18, 443.
Banks, W. H. H. 1976 Acta Mechanica 24, 273.
Bickley, W. G. 1938 Phil. Mag. 25, 746.
Bowden, F. P. & Lord, R. G. 1963 Proc. Roy. Soc. A 271, 143.
Brabston, D. C. & Keller, H. B. 1975 J. Fluid Mech. 69, 179.
Brison, J.-F. & Mathieu, J. 1973 C. R. Acad. Sci. Paris A 276, 871.
Collins, W. D. 1955 Mathematika 2, 42.
Dennis, S. C. R. & Chang, G.-Z. 1969a Math. Res. Center, Univ. Wisconsin, Tech. Summary Rep. no. 859.
Dennis, S. C. R. & Chang, G.-Z. 1969b Phys. Fluids Suppl. 12, II 88.
Dennis, S. C. R. & Chang, G.-Z. 1970 J. Fluid Mech. 42, 471.
Dennis, S. C. R. & Ingham, D. B. 1979 Phys. Fluids 22, 1.
Dennis, S. C. R. & Singh, S. N. 1978 J. Comp. Phys. 28, 297.
Dennis, S. C. R. & Walker, J. D. A. 1971 J. Fluid Mech. 48, 771.
Dennis, S. C. R., Walker, J. D. A. & Hudson, J. D. 1973 J. Fluid Mech. 60, 273.
Fox, J. 1964 Boundary layer on rotating spheres and other axisymmetric shapes. N.A.S.A. TN D-2491.Google Scholar
Happel, J. & Brenner, H. 1965 Low Reynolds Number Hydrodynamics, pp. 134–138. Englewood Cliffs, New Jersey: Prentice-Hall.
Howarth, L. 1951 Phil. Mag. 42, 1308.
Kobashi, Y. 1957 J. Sci. Hiroshima Univ., Japan, Ser. A 20, 149.
Kreith, F., Roberts, L. G., Sullivan, J. A. & Sinha, S. N. 1963 Int. J. Heat Mass Transfer 6, 881.
Lamb, H. 1932 Hydrodynamics, pp. 588–589. Cambridge University Press.
Manohar, R. 1967 Z. angew. Math. Phys. 18, 320.
Munson, B. R. & Joseph, D. D. 1971 J. Fluid Mech. 49, 289.
Nigam, S. D. 1954 Z. angew. Math. Phys. 5, 151.
Ovseenko, Iu. C. 1963 Izv. Vysshikh Uchevnykh Zavedenii Mathematika 35, 129.
Rotenberg, M., Bivins, M., Metropolis, N. & Wooten, J. K. 1959 The 3-j and 6-j symbols. Massachusetts Institute of Technology Press.
Sampson, R. A. 1891 Phil. Trans. Roy. Soc. A 182, 449.
Sawatzki, O. 1970 Acta Mechanica 9, 159.
Singh, S. N. 1970 Phys. Fluids 13, 2452.
Smith, F. T. & Duck, P. W. 1977 Quart. J. Mech. Appl. Math. 30, 143.
Stewartson, K. 1958 In Boundary Layer Research, p. 59. Berlin: Springer.
Stokes, G. G. 1845 Trans. Camb. Phil. Soc. 8, 287.
Takagi, H. 1977 J. Phys. Soc. Japan 42, 319.
Talman, J. D. 1968 Special Functions. New York: Benjamin.
Thomas, R. H. & Walters, K. 1964 Quart. J. Mech. Appl. Math. 17, 39.
Van Dyke, M. D. 1964 Proc. 11th Int. Cong. Appl. Mech., Munich, pp. 1165.
Van Dyke, M. D. 1965 Stanford Univ. SUDAER no. 247.