Stability of time-periodic temperature fields
Authors:
Chia-Shun Yih and Jinsong Shi
Journal:
Quart. Appl. Math. 45 (1987), 39-50
MSC:
Primary 76E15
DOI:
https://doi.org/10.1090/qam/885166
MathSciNet review:
885166
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Abstract: The energy method developed by Joseph [4], Davis [2], and Homsy [3] is applied to the time-periodic temperature fields considered by Yih and Li [11] to obtain Rayleigh numbers below which the fluid is stable. This is done to see how far the Rayleigh numbers so determined fall below the critical Rayleigh numbers, above which the flow is unstable, as determined by the linear theory [11]. It is found that, unlike the case of classical Bénard cells, the gray area, or area of ignorance, is quite large, indicating the need for some improvement of the energy method to give sharper lower bounds on the Rayleigh number.
S. Carmi, Energy stability of modulated flows, Phys. Fluids 17, 1951–1955 (1974)
S. H. Davis, Buoyancy-surface tension instability by the method of energy, J. Fluid Mech. 39, 347–359 (1969)
G. M. Homsy, Global stability of time-dependent flows: impulsively heated or cooled fluid layers, J. Fluid Mech. 60, 129–139 (1973)
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D. D. Joseph, Stability of fluid motions, vols. I and II, Springer Verlag, 1976
W. McF. Orr, The stability or instability of the steady motions of a perfect liquid and of a viscous liquid, Part I. A perfect liquid, and Part II, A viscous liquid, Proc. Roy. Irish Acad. Sect. A 27, 9–68 and 69–138 (1907)
- Anne Pellew and R. V. Southwell, On maintained convective motion in a fluid heated from below, Proc. Roy. Soc. London Ser. A 176 (1940), 312–343. MR 3752, DOI https://doi.org/10.1098/rspa.1940.0092
O. Reynolds, On the dynamic theory of incompressible viscous fluid and the determination of the criterion, Philos. Trans .Roy. Soc. London Ser. A 186, 123–164 (1895)
- James Serrin, On the stability of viscous fluid motions, Arch. Rational Mech. Anal. 3 (1959), 1–13. MR 105250, DOI https://doi.org/10.1007/BF00284160
J. T. Stuart, On the nonlinear mechanics of hydrodynamic stability, J. Fluid Mech. 4, 1–21 (1958)
C.-S. Yih and C.-H. Li, Instability of unsteady flows or configurations, Part 2. Convective instability, J. Fluid Mech. 54, 143–152 (1972)
S. Carmi, Energy stability of modulated flows, Phys. Fluids 17, 1951–1955 (1974)
S. H. Davis, Buoyancy-surface tension instability by the method of energy, J. Fluid Mech. 39, 347–359 (1969)
G. M. Homsy, Global stability of time-dependent flows: impulsively heated or cooled fluid layers, J. Fluid Mech. 60, 129–139 (1973)
D. D. Joseph, Nonlinear stability of the Boussinesq equations by the method of energy, Arch. Rat. Mech. Anal. 22, 163–184 (1966)
D. D. Joseph, Stability of fluid motions, vols. I and II, Springer Verlag, 1976
W. McF. Orr, The stability or instability of the steady motions of a perfect liquid and of a viscous liquid, Part I. A perfect liquid, and Part II, A viscous liquid, Proc. Roy. Irish Acad. Sect. A 27, 9–68 and 69–138 (1907)
A. Pellew and R. V. Southwell, On maintained convective motion in a fluid heated from below, Proc. Roy. Soc. London Ser. A. 176, 312–343 (1940)
O. Reynolds, On the dynamic theory of incompressible viscous fluid and the determination of the criterion, Philos. Trans .Roy. Soc. London Ser. A 186, 123–164 (1895)
J. Serrin, On the stability of viscous fluid motions, Arch. Rat. Mech. Anal. 3, 1–13 (1959)
J. T. Stuart, On the nonlinear mechanics of hydrodynamic stability, J. Fluid Mech. 4, 1–21 (1958)
C.-S. Yih and C.-H. Li, Instability of unsteady flows or configurations, Part 2. Convective instability, J. Fluid Mech. 54, 143–152 (1972)
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© Copyright 1987
American Mathematical Society
