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Linear stability of the dissipative, two-fluid, cylindrical Couette problem. Part 1. The stably-stratified hydrodynamic problem

Published online by Cambridge University Press:  29 March 2006

G. P. Schneyer
Affiliation:
Aeronautical Sciences Division, University of California, Berkeley
S. A. Berger
Affiliation:
Aeronautical Sciences Division, University of California, Berkeley

Abstract

The stability of a two-fluid vortex is studied as a step towards understanding the separation and containment problems in a gaseous-core nuclear rocket. In particular, the linear hydrodynamic stability of two incompressible, immiscible, viscous fluids occupying separate annular regions of a cylindrical Couette apparatus is considered. Neglecting surface tension and gravity, a conservative assumption, the governing equations for arbitrary jumps in fluid properties are derived and numerical solutions to the resultant eigenvalue problems obtained. Results are presented for the effect on neutral stability of density and viscosity jumps, varying gap widths, and differing fluid-fluid interfacial positions. The solutions are limited, however, to the case of stably stratified fluids and a stationary outer cylinder.

Two separate modes (multiple eigenvalues) have been discovered for all cases in which two fluids, differing in any property, are present. A rationale is presented for this phenomenon as well as for most of the other observed results.

While most results are believed to be manifestations of the Taylor cylindrical Couette instability phenomenon, evidence is presented for the existence of additional hidden eigenvalues attributable to the classical KelvinHelmholtz and/or the recently reported Yih viscosity-stratification instability phenomena.

Type
Research Article
Copyright
Copyright © Cambridge University Press 1971

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Footnotes

Present address: Mathematics Department, Imperial College, London.

References

Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Oxford: Clarendon.Google Scholar
Coles, D. 1965 Transition in circular Couette flow J. Fluid Mech. 21, 385426.Google Scholar
Davey, A., Di Prima, R. C. & Stuart, J. T. 1968 On the instability of Taylor vortices J. Fluid Mech. 31, 1752.Google Scholar
Di Prima, R. C. 1960 The stability of a viscous fluid between rotating cylinders with an axial flow J. Fluid Mech. 9, 62131.Google Scholar
Donnelly, R. J. & Schwarz, K. W. 1965 Experiments on the stability of viscous flow between rotating cylinders. VI. Proc. Roy. Soc A 283, 53149.Google Scholar
Gross, A. G. 1965 Numerical investigation of the stability of Couette flow. Ph.D. Thesis, Rensselaer Polytechnic Institute, Troy, N.Y.Google Scholar
Harris, D. L. & Reid, W. H. 1964 On the stability of viscous flow between rotating cylinders. Part 2. Numerical analysis J. Fluid Mech. 20, 95102.Google Scholar
Hughes, T. H. & Reid, W. H. 1968 The stability of spiral flow between rotating cylinders. Phil. Trans. Roy. Soc A 263, 5791.Google Scholar
Johnson, K. 1966 A plasma-core nuclear rocket utilizing a magnetohydrodynamically-driven vortex A.I.A.A. J. 4, 63543.Google Scholar
Krueger, E. R., Gross, A. G. & Di Prima, R. C. 1966 On the relative importance of Taylor-vortex and non-axisymmetric modes in flow between rotating cylinders J. Fluid Mech. 24, 52138.Google Scholar
Pao, H. S. 1966 Further results on stability of a swirling fluid with variable density in the presence of a circular magnetic field Phys. Fluids, 9, 12545.Google Scholar
Rayleigh, Lord 1920 On the dynamics of revolving fluids Scientific Papers, 6, 44753.Google Scholar
Reshotko, E. & Monnin, C. F. 1965 Stability of two-fluid wheel flows. NASA TN D-2696.Google Scholar
Roberts, P. H. 1965 The solution of the characteristic value problems. Appendix to Experiments on the stability of viscous flow between rotating cylinders. VI. Proc. Roy. Soc. A 283, 5506.Google Scholar
Schneyer, G. P. 1968 Linear hydrodynamic and hydromagnetic stability of the dissipative, two-fluid cylindrical Couette problem. Ph.D. Thesis, published as University of California, Berkeley Aero. Sci. Rep. AS-68-12.Google Scholar
Snyder, H. A. & Lambert, R. B. 1966 Harmonic generation in Taylor vortices between rotating cylinders J. Fluid Mech. 26, 54562.Google Scholar
Sparrow, E. M., Munro, W. D. & Jonsson, V. K. 1964 Instability of the flow between rotating cylinders: The wide gap problem. J. Fluid Mech. 20, 3546.Google Scholar
Stuart, J. T. 1967 Hydrodynamic stability of fluid flows. Inaugural Lectures, London: The Imperial College of Science and Technology, 111116.Google Scholar
Taylor, G. I. 1923 Stability of a viscous liquid contained between two rotating cylinders. Phil. Trans. Roy. Soc A 223, 289343.Google Scholar
Walowit, J., Tsao, S. & Di Prima, R. C. 1964 Stability of flow between arbitrarily spaced concentric cylindrical surfaces including the effect of a radial temperature gradient J. Appl. Mech. 31, 58593.Google Scholar
Yih, C. S. 1961 Dual role of viscosity in the instability of revolving fluids of variable density Phys. Fluids, 4, 80611.Google Scholar
Yih, C. S. 1967 Instability due to viscosity stratification J. Fluid Mech. 27, 337352.Google Scholar
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