Hostname: page-component-76fb5796d-wq484 Total loading time: 0 Render date: 2024-04-25T09:40:15.563Z Has data issue: false hasContentIssue false
  • English
  • Français

On the finite Dean problem: linear theory

Published online by Cambridge University Press:  17 February 2009

B. J. Kachoyan  and
P. J. Blennerhassett
B. J. Kachoyan
Affiliation:
Department of Applied Mathematics, University of Sydney, Sydney, N.S.W. 2006, Australia.
P. J. Blennerhassett
Affiliation:
School of Mathematics, University of New South Wales, P. O. Box 1, Kensington, N.S.W., 2033, Australia.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The Dean problem of pressure-driven flow between finite-length concentric cylinders is considered. The outer cylinder is at rest and the small-gap approximation is used. In a similar procedure to that of Blennerhassett and Hall [8] in the context of Taylor vortices, special end conditions are imposed in which the ends of the cylinder move with the mean flow, allowing the use of a perturbation analysis from a known basic flow. Difficulties specific to Dean flow (and more generally to non-Taylor-vortex flow) require the use of a parameter α which measures the relative strengths of the velocities due to rotation and the pressure gradient, to trace the solution from Taylor to Dean flow. Asymptotic expansions are derived for axial wavenumbers at a given Taylor number. The calculation of critical Taylor number for a given cylinder height is then carried out. Corresponding stream-function contours clearly show features not evident in infinite flow.

Type
Research Article
Information
The ANZIAM Journal , Volume 30 , Issue 2 , October 1988 , pp. 157 - 178
Copyright
Copyright © Australian Mathematical Society 1988

References

[1]Alziary de Roquefort, T. and Grillaud, G., “Computation of Taylor vortex flow in finite cylinders: linear theory”, Comput. & Fluids 6 (1978) 259–269.CrossRefGoogle Scholar
[2]Benjamin, T. B., “Bifurcation phenomena in steady flows of a viscous fluid. I. Theory”, Proc. Roy. Soc. London Ser. A 359 (1978) 1–26.Google Scholar
[3]Benjamin, T. B., “Bifurcation phenomena in steady flows of a viscous fluid. II. Experiments”, Proc. Roy. Soc. London Ser. A 359 (1978) 27–43.Google Scholar
[4]Benjamin, T. B., “New observations in the Taylor experiment.” in Transition and Thrbulence. (ed. Meyer, R. E.). (Academic Press, 1981)Google Scholar
[5]Benjamin, T. B. and Mullin, T., “Anomalous modes in the Taylor experiment”, Proc. Roy. Soc. London Ser. A 377 (1981) 221–249.Google Scholar
[6]Benjamin, T. B. and Mullin, T., “Notes on the multiplicity of flows in the Taylor experiment”, J. Fluid Mech. 121 (1982) 219–230.CrossRefGoogle Scholar
[7]Bergé, P., Rayleigh-Bénard instability: experimental findings obtained by light scattering and other optical methods. in Fluctuations, Instabilities and Phase Transitions. (ed. Riste, T.). (Plenum Press, 1975)Google Scholar
[8]Blennerhassett, P. J. and Hall, P., “Centrifugal instabilities of circumferential flows in finite cylinders: linear theory”, Proc. Roy. Soc. London Ser. A 365 (1979) 191–207.Google Scholar
[9]Brewster, D. B., Grosberg, P. and Nissan, A. H., “The stability of viscous flow between horizontal concentric cylinders”, Proc. Roy. Soc. London Ser. A 251 (1959) 76–91.Google Scholar
[10]Burkhalter, J. E. and Koschmieder, E. L., “Steady supercritical Taylor vortices after sudden starts”, Phys. Fluids 17 (1974) 1929–1935.CrossRefGoogle Scholar
[11]Chandrasekhar, S., “The stability of viscous flow between rotating cylinders”, Mathematika 1 (1954) 5–13.CrossRefGoogle Scholar
[12]Chandrasekhar, S., Hydrodynamic and hydromagnetic stability (Oxford University Press, 1961)Google Scholar
[13]Cheng, K. C., Line, R.-C., and Ou, J.-W., “Fully developed laminar flow in curved rectangular channels”, Trans. ASME J. Fluids Eng. 98 (1976) 41–48.CrossRefGoogle Scholar
[14]Cole, J. A., “Taylor vortices with short rotating cylinders”, Trans. ASME J. Fluids Eng. 96 (1974) 69–70.CrossRefGoogle Scholar
[15]Daniels, P. G., “Asymptotic sidewall effects in rotating Bénard convection”, Z. Angew. Math. Phys. 28 (1977) 577–584.CrossRefGoogle Scholar
[16]Daniels, P. G., “The effect of distant sidewalls on the transition to finite amplitude Bénard convection”, Proc. Roy. Soc. London Ser. A 358 (1977) 173–197.Google Scholar
[17]Dean, W. R., “Fluid motion in a curved channel”, Proc. Roy. Soc. London Ser. A 121 (1928) 402–420.Google Scholar
[18]De Vriend, J. H., “Velocity redistribution in curved rectangular channels”, J. Fluid Mech. 107 (1981) 423–439.CrossRefGoogle Scholar
[19]DiPrima, R. C., “The stability of viscous flow between rotating concentric cylinders with a pressure gradient acting round the cylinders”, J. Fluid Mech. 6 (1959) 462–468.CrossRefGoogle Scholar
[20]Drazin, P. G., “On the effects of sidewalls on Bénard convection”, Z. Angew. Math. Phys. 26 (1975) 239–243.CrossRefGoogle Scholar
[21]Ghia, K. N. and Sokhey, J. S., “Laminar incompressible viscous flow in ducts of regular cross-section”, Trans. ASME J. Fluids Eng. 99 (1977) 640–648.CrossRefGoogle Scholar
[22]Hall, P. and Walton, I. C., “The smooth transition to a convective régime in a two- dimensional box”, Proc. Roy. Soc. London Ser. A 358 (1977) 199–221.Google Scholar
[23]Hall, P., “Centrifugal instabilities in finite containers: a periodic model”, J. Fluid Mech. 99 (1980) 575–596.CrossRefGoogle Scholar
[24]Hall, P., “Centrifugal instabilities of circumferential flows in finite cylinders: the wide gap problem”, Proc. Roy. Soc. London Ser. A 384 (1982) 359–379.Google Scholar
[25]Hille, P., Vehrenkamp, R. and Schultz-Dubois, E. O., “The development and structure of primary and secondary flow in a curved square duct”, J. Fluid Mech. 151 (1985) 219–241.CrossRefGoogle Scholar
[26]Hillmand, A. P. and Salzer, H. E., “Roots of sin z = z”, Phil. Mag. 34 (1943) 575.CrossRefGoogle Scholar
[27]Joseph, B., Smith, E. P. and Adler, R. J., “Numerical treatment of laminar flow in a helically coiled tube of finit pitch”, Chem. Eng. Comm. 7 (1975) 57–78.Google Scholar
[28]Kachoyan, B. J., “The finite Dean problem: nonlinear theory”, IMA J. Appl. Math 38 (1987) 71–86.CrossRefGoogle Scholar
[29]Kachoyan, B. J., “Neutral curve behaviour in Taylor-Dean flow”, Z. Angew. Math. Phys. 38 (1987) 905–924.CrossRefGoogle Scholar
[30]Pfister, G. and Rehberg, I., “Space-dependent order parameter in circular Couette flow”, Phys. Lett. A 83 (1981) 19–22.CrossRefGoogle Scholar
[31]Reid, W. H., “On the stability of viscous flow in a curved channel”, Proc. Roy. Soc. London Ser. A 244 (1958) 186–198.Google Scholar
[32]Reiss, E. L., “Imperfect bifurcation”, in Applications of Bifurcation Theory, (ed. Rabinowitz, P. H.), (Academic Press, New York, 1977), 37–71.Google Scholar
[33]Seminara, G. and Hall, P., “Linear stability of slowly varying unsteady flows in a curved channel”, Proc. Roy. Soc. London Ser. A 346 (1975) 279–303.Google Scholar
[34]Schaeffer, E. H., “Qualitative analysis of a model for boundary effects in the Taylor problem”, Math. Poc. Camb. Phil. Soc. 87 (1980) 307–337.CrossRefGoogle Scholar
[35]Singelton, R. C., “On computing the fast Fourier transform”, Comm. ACM 10 (10) (1967) 647–654.CrossRefGoogle Scholar
[36]Snyder, H. A. and Lambert, R. B., “Harmonic generation in Taylor vortices between rotating cylinders”, J. Fluid Mech. 26 (1966) 546–562.CrossRefGoogle Scholar
[37]Stuart, J. T. and DiPrima, R. C., “On the mathematics of Taylor vortex flows in cylinders of finite length”, Proc. Roy. Soc. Lond. Ser. A 372 (1980) 357–365.Google Scholar
[38]Winters, K. H., “A bifurcation study of laminar flow in a curved tube of rectangular cross-section”, Harwell Report AERE TP. 1104. (1984)Google Scholar