Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-19T15:06:15.146Z Has data issue: false hasContentIssue false
  • English
  • Français

The slow translation of a sphere in a rotating viscous fluid

Published online by Cambridge University Press:  20 April 2006

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

Abstract

The motion of a sphere along the axis of rotation of an incompressible viscous fluid that is rotating as a solid mass is investigated by means of numerical methods for small values of the Reynolds and Taylor numbers. The Navier–Stokes equations governing the steady axisymmetric flow can be written as three coupled, nonlinear, elliptic partial differential equations for the stream function, vorticity and rotational velocity component. Two numerical methods are employed to solve these equations. The first is the method of series truncation in which the dependent variables are expressed as series of orthogonal Gegenbauer functions and the equations of motion are then reduced to three coupled sets of ordinary differential equations, which are integrated numerically subject to their boundary conditions. In the second method, specialized finite–difference techniques of solution are applied to the two-dimensional partial differential equations. These techniques employ finite-difference equations with coefficients that depend upon the exponential function; a particularly suitable form of approximation for use in calculating numerical solutions is obtained by expanding the exponential coefficients in powers of their exponents.

Calculated results obtained by the two methods are in good agreement with each other. The calculations have been carried out according to theoretical assumptions that simulate the experiments of Maxworthy (1965) in which the sphere experiences no resultant torque exerted by the surrounding fluid and is free to rotate with constant angular velocity. Numerical estimates of this angular velocity and of the drag exerted by the fluid on the sphere are found to agree well with the experimental results for Reynolds and Taylor numbers in the range from zero to unity. The results for small values of the Reynolds number are also consistent with theoretical work of Childress (1963, 1964) which is valid as the Reynolds number tends to zero.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 117 , April 1982 , pp. 251 - 267
Copyright
© 1982 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

Allen, D. N. De G. & Southwell, R. V. 1955 Quart. J. Mech. Appl. Math. 8, 129.
Allen, D. N. De G. 1962 Quart. J. Mech. Appl. Math. 15, 11.
Barnard, B. J. S. & Pritchard, W. G. 1975 J. Fluid Mech. 71, 43.
Childress, W. S. 1963 Jet Propulsion Laboratory Space Programs Summary 37–18, vol. IV, p. 46.
Childress, W. S. 1964 J. Fluid Mech. 20, 305.
Dennis, S. C. R. 1960 Quart. J. Mech. Appl. Math. 13, 487.
Dennis, S. C. R. 1973 In Proc. 3rd Int. Conf. on Numerical Methods in Fluid Mech., Paris 1972 (ed. H. Cabannes & R. Teman). Lect. notes in Phys., vol. 19, p. 120. Springer.
Dennis, S. C. R. & Hudson, J. D. 1978 In Proc. Int. Conf. on Numerical Methods in Laminar and Turbulent Flow, Swansea, p. 69. Pentech.
Dennis, S. C. R. & Ingham, D. B. 1979 Phys. Fluids 22, 1.
Dennis, S. C. R. & Ingham, D. B. 1981 Lect. notes in Phys., vol. 141, p. 151. Springer.
Dennis, S. C. R., Ingham, D. B. & Cook, R. N. 1979 J. Comp. Phys. 33, 325.
Dennis, S. C. R. & Singh, S. N. 1978 J. Comp. Phys. 28, 297.
Dennis, S. C. R., Singh, S. N. & Ingham, D. B. 1980 J. Fluid Mech. 101, 257.
Dennis, S. C. R. & Walker, J. D. A. 1971 J. Fluid Mech. 48, 771.
Gosman, A. D., Pun, W. M., Runchal, A. K., Spalding, D. G. & Wolfshtein, M. 1969 Heat and Mass Transfer in Recirculating Flows. Academic.
Grace, S. F. 1926 Proc. R. Soc. Lond. A 113, 46.
Greenspan, D. 1968a Lectures on the Numerical Solution of Linear, Singular and Non-linear Differential Equations. Prentice-Hall.
Greenspan, H. P. 1968b The Theory of Rotating Fluids. Cambridge University Press.
Hocking, L. M., Moore, D. W. & Walton, I. C. 1979 J. Fluid Mech. 90, 781.
Long, R. R. 1953 J. Met. 10, 197.
Maxworthy, T. 1965 J. Fluid Mech. 23, 373.
Maxworthy, T. 1970 J. Fluid Mech. 40, 453.
Miles, J. W. 1969 J. Fluid Mech. 36, 265.
Miles, J. W. 1971 J. Fluid Mech. 45, 513.
Moore, D. W. & Saffman, P. G. 1968 J. Fluid Mech. 31, 635.
Moore, D. W. & Saffman, P. G. 1969 Phil. Trans. R. Soc. Lond. A 264, 597.
Morrison, J. W. & Morgan, G. W. 1956 Div. Appl. Math., Brown University, Rep. no. 56207/8.
Proudman, J. 1916 Proc. R. Soc. Lond. A 92, 408.
Roscoe, D. F. 1975 J. Inst. Math. Appl. 16, 291.
Roscoe, D. F. 1976 Int. J. Num. Meth. Engng 10, 1299.
Rotenberg, M., Bivins, M., Metropolis, N. & Wooten, J. K. 1959 The 3-j and 6-j Symbols. M.I.T. Press.
Runchal, A. K., Spalding, D. B. & Wolfshtein, M. 1969 Phys. Fluids Suppl. 12, 11–21.
Sampson, R. A. 1891 Phil. Trans. R. Soc. Lond. A 182, 449.
Singh, S. N. 1975a J. Appl. Math. & Phys. 26, 415.
Singh, S. N. 1975b Int. J. Engng Sci. 13, 1085.
Spalding, D. B. 1972 Int. J. Num. Meth. Engng 4, 551.
Stewartson, K. 1952 Proc. Camb. Phil. Soc. 48, 168.
Stewartson, K. 1958 Quart. J. Mech. Appl. Math. 11, 39.
Stewartson, K. 1968 Quart. J. Mech. Appl. Math. 21, 353.
Talman, J. D. 1968 Special Functions, chap. 9. Benjamin.
Taylor, G. I. 1917 Proc. R. Soc. Lond. A 93, 99.
Taylor, G. I. 1921 Proc. R. Soc. Lond. A 100, 114.
Taylor, G. I. 1922 Proc. R. Soc. Lond. A 102, 180.