The onset of convection in horizontal cylinders
Author:
John P. McHugh
Journal:
Quart. Appl. Math. 58 (2000), 425-436
MSC:
Primary 76E06; Secondary 76E15, 76R10
DOI:
https://doi.org/10.1090/qam/1770647
MathSciNet review:
MR1770647
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Abstract: The convective instability of a fluid that fills a horizontal cylindrical cavity is considered. The boundaries of the cavity are conducting, and the driving force is a linear temperature gradient far from the cylinder. Critical Rayleigh numbers governing the onset of convection are determined for neutral stability. The results show that the critical Rayleigh number depends on the ratio of thermal conductivities of the solid to the fluid $\left ( \lambda \right )$, and a wavenumber. Both two- and three-dimensional disturbances are included. The disturbances are separated into even modes and odd modes. The most unstable odd modes have been found to be two-dimensional, while the most unstable even modes are three-dimensional. The two-dimensional odd modes are most unstable in the vicinity of $\lambda = 1$. The three-dimensional even modes are more unstable for other values of $\lambda$. The results are compared with the previous results of Gershuni and Zhukhovitskii.
T. R. Anthony and H. E. Cline, Thermal migration of liquid droplets through solids, Journal of Applied Physics 42, 3380–3387 (1971)
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G. Z. Gershuni and E. M. Zhukhovitskii, Convective instability of incompressible fluids, Jeter Press, 1976
D. T. J. Hurle, E. Jakeman, and E. R. Pike, On the solution of the Bénard problem with boundaries of finite conductivity, Proc. Royal Soc. London Ser. A 296, 469–475 (1967)
Y. Hwang, T. H. Pigford, P. L. Chambre, and W. Lee, Analysis of mass transport in a nuclear waste repository in salt, Water Resources Research 28, 1857–1868 (1992)
J. I. Kim, K. Gompper, K. D. Closs, G. Kessler, and D. Faude, German approaches to closing the nuclear fuel cycle and final disposal of HLW, Journal of Nuclear Materials 238, 1–10 (1996)
J. P. McHugh, Notes on the onset of convection in horizontal cylinders, FPR report 101, 1998
D. R. Olander, A. J. Machiels, M. Balooch, and S. K. Yagnik, Thermal gradient migration of brine inclusions in synthetic alkali halide single crystals, Journal of Applied Physics 53, 669–681 (1982)
W. Xu and J. Genin, Closure of a nuclear waste repository deeply imbedded in a stratified salt bed, Journal of Engineering Materials and Technology 116, 567–573 (1994)
C. S. Yih, Movement of liquid inclusions in soluble solids: An inverse Stokes’ law, Physics of Fluids 29, 2785–2787 (1986)
C. S. Yih, Convective instability of a spherical liquid inclusion, Physics of Fluids 30, 36–44 (1987)
T. R. Anthony and H. E. Cline, Thermal migration of liquid droplets through solids, Journal of Applied Physics 42, 3380–3387 (1971)
G. Z. Gershuni and E. M. Zhukhovitskii, The stability of equilibrium of fluid within a horizontal cylinder heated from below, J. Appl. Math. Mech. 25, 1551–1558 (1961)
G. Z. Gershuni and E. M. Zhukhovitskii, Convective instability of incompressible fluids, Jeter Press, 1976
D. T. J. Hurle, E. Jakeman, and E. R. Pike, On the solution of the Bénard problem with boundaries of finite conductivity, Proc. Royal Soc. London Ser. A 296, 469–475 (1967)
Y. Hwang, T. H. Pigford, P. L. Chambre, and W. Lee, Analysis of mass transport in a nuclear waste repository in salt, Water Resources Research 28, 1857–1868 (1992)
J. I. Kim, K. Gompper, K. D. Closs, G. Kessler, and D. Faude, German approaches to closing the nuclear fuel cycle and final disposal of HLW, Journal of Nuclear Materials 238, 1–10 (1996)
J. P. McHugh, Notes on the onset of convection in horizontal cylinders, FPR report 101, 1998
D. R. Olander, A. J. Machiels, M. Balooch, and S. K. Yagnik, Thermal gradient migration of brine inclusions in synthetic alkali halide single crystals, Journal of Applied Physics 53, 669–681 (1982)
W. Xu and J. Genin, Closure of a nuclear waste repository deeply imbedded in a stratified salt bed, Journal of Engineering Materials and Technology 116, 567–573 (1994)
C. S. Yih, Movement of liquid inclusions in soluble solids: An inverse Stokes’ law, Physics of Fluids 29, 2785–2787 (1986)
C. S. Yih, Convective instability of a spherical liquid inclusion, Physics of Fluids 30, 36–44 (1987)
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© Copyright 2000
American Mathematical Society