Hostname: page-component-77c89778f8-fv566 Total loading time: 0 Render date: 2024-07-19T10:14:12.581Z Has data issue: false hasContentIssue false
  • English
  • Français

On the principle of exchange of stabilities for the magnetohydrodynamic thermal stability problem in completely confined fluids

Published online by Cambridge University Press:  28 March 2006

Michael Sherman  and
Simon Ostrach
Michael Sherman
Affiliation:
Engineering Division, Case Institute of Technology, Cleveland, Ohio
Simon Ostrach
Affiliation:
Engineering Division, Case Institute of Technology, Cleveland, Ohio
Get access Rights & Permissions [Opens in a new window]

Abstract

The principle of exchange of stabilities (exchange principle) for the thermal stability problem has been proved by Pellew & Southwell (1940) for fluids bounded by two infinite, horizontal parallel planes. Chandrasekhar (1952) discussed the establishment of the exchange principle for the same geometry when the fluid is an electrical conductor and when an arbitrary oriented, uniform, external magnetic field is applied in the vertical direction.

In this paper, the exchange principle is examined for fluids completely confined in an arbitrary region with rigid bounding surfaces that are good electrical conductors with respect to the fluid. The uniform magnetic field is applied in an arbitrary direction. A generalized thermal boundary condition is imposed which includes the fixed temperature and prescribed heat-flux conditions as special cases.

If no magnetic field is applied to the fluid, the present work reduces to a generalization (for completely confined fluids) of the Pellew & Southwell proof of the exchange principle. In the magnetohydrodynamic (MHD) thermal stability problem, the exchange principle is found to be valid if the total kinetic energy associated with an arbitrary disturbance is greater than or equal to its total magnetic energy. In a special case it is demonstrated that a sufficient condition which will establish the exchange principle is k ≤ η, where k is the fluid thermal diffusivity and η is the fluid electrical resistivity.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 24 , Issue 4 , April 1966 , pp. 661 - 671
Copyright
© 1966 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

Chandrasekhar, S. 1952 Phil. Mag. Series 7, 43, 501.
Chandrasekhar, S. 1955 Proc. Camb. Phil. Soc. 51, 162.
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Oxford University Press.
Ostrach, S. & Pnueli, D. 1963 J. Heat Transfer, Trans. ASME, Series C, vol. 85, no. 4, p. 346.
Pellew, A. & Southwell, R. V. 1940 Proc. Roy. Soc., A 176, 312.
Rayleigh, Lord 1916 Phil. Mag., Series 6, 32, 529.
Selig, F. 1964 Phys. Fluids, 7, 1114.
Sherman, M. 1965 Ph.D. Thesis, Case Institute of Technology, Cleveland, Ohio.
Sparrow, E. M., Goldstein, R. J. & Jonsson, V. K. 1964 J. Fluid Mech. 18, 51.
Thompson, W. B. 1951 Phil. Mag., Series 7, 42, 1417.
Velte, W. 1964 Arch. Rat. Mech. & Anal. 16, 9.
Weinbaum, S. 1964 J. Fluid Mech. 18, 40.
Yih, C. S. 1959 Quart. Appl. Math. 17, 2.