References
Chandrasekhar, S., Hydrodynamics and Hydromagnetic Stability. Oxford 1961.
Joseph, D.D., On the stability of the Boussinesq equations. Arch. Rational Mech. Anal. 20, No. 1, 59–71 (1965).
Joseph, D.D., Nonlinear stability of the Boussinesq equations by the method of energy. Arch. Rational Mech. Anal. 22, No. 3, 163–184 (1966).
Joseph, D.D., Parameter and domain dependence of eigenvalues of elliptic partial differential equations. Arch. Rational Mech. Anal. 24, No. 5, 325–351 (1967).
Joseph, D.D., & C.C. Shir, Subcritical convective instability. Part I: Fluid layers heated from below and internally. J. Fluid Mech. 26, pt. 4 (1966).
Mihaljan, J., A rigorous exposition of the Boussinesq approximations applicable to a thin layer of fluid. Astrophys. J. 136, 1126–1133 (1962).
Payne, L., & H. Weinberger, An exact stability bound for Navier-Stokes flow in a sphere. Conference on Non-linear Differential Equations (R. Langer, Editor). University of Wisconsin, 1961.
Pólya, G., & M. Schiffer, Convexity of functionals by transplantation. Journal d'Analyse Mathématique 3, 246–344 (1953).
Sani, R., Ph. D. Thesis, Department of Chemical Engineering, University of Minnesota, Minneapolis, 1963.
Sani, R., On finite amplitude roll cell distrubances in a fluid layer subjected to heat and mass transfer. A.I.Ch.E. Journal 11, No. 6 (1965).
Shir, C.C., Ph. D. Thesis, University of Minnesota, Department of Aeronautics and Engineering Mechanics, 1967.
Serrin, J., On the stability of viscous fluid motions. Arch. Rational Mech. Anal. (1) 3, 1–13 (1959).
Sorger, P., Über ein Variantionsproblem aus der nichtlinearen Stabilitätstheorie zäher, inkompressibler Strömungen. ZAMP 17, H. 2, 201–216 (1966).
Tabor, H., Large-area solar collections for power production. Solar Energy 7, No. 4 (1963).
Velte, W., Über ein Stabilitätskriterium der Hydrodynamik. Arch. Rational Mech. Anal. (1) 9, 9–20 (1962).
Veronis, G., On finite amplitude instability in the thermohaline convection. J. Marine Res. 23, No. 1 (1964).
Weinberger, H., The physics of the solar pond. Solar Energy VIII, No. 2, 45–56 (1964).
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Communicated by C. Truesdell
A portion of this work formed part of the Ph. D. thesis of one of us (C.C.S.) at the University of Minnesota. The work was supported by the U.S. National Aeronautics and Space Administration under NASA Grant (NGR-24-005-063) and also by the National Science Foundation under NSF Grant (GK 1838).
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Shir, C.C., Joseph, D.D. Convective instability in a temperature and concentration field. Arch. Rational Mech. Anal. 30, 38–80 (1968). https://doi.org/10.1007/BF00253245
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DOI: https://doi.org/10.1007/BF00253245