Abstract
The convection of an isothermally incompressible fluid in a horizontal layer with free undeformable boundaries kept at a constant temperature is considered. Under the fairly common assumptions of the temperature dependence of the specific volume, it is shown that the monotonicity principle holds and that the spectrum of critical Rayleigh numbers is countable and prime. Models with linear and quadratic temperature dependences of the specific volume are given as examples. The results on the spectrum of the critical Rayleigh numbers are also valid for some other boundary conditions.
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References
L. D. Landau and E. M. Lifshits, Course of Theoretical Physics, Vol. 6: Fluid Mechanics, Pergamon Press, Oxford-Elmsford, New York (1987).
V. I. Yudovich, Equations of Free Convection for an Isothermally Incompressible Fluid [in Russian], Rostov State Univ., Rostov-on-Don (1983).
G. Veronis, “Penetrative convection,” Astrophys. J., 137, No. 2, 641–663 (1963).
K. A. Nadolin, “Numerical study of mathematical models of free convection for an isothermally incompressible fluid,” Doct. Dissertation in Phys.-Math. Sci., Rostov-on-Don (1989).
K. A. Nadolin, “Convection in a horizontal fluid layer with specific volume inversion,” Izv. Akad. Nauk SSSR, Mekh. Zhidk. Gaza, No. 1, 43–49 (1989).
V. K. Andreev and V. B. Bekezhanova “Stability of the equilibrium of a flat layer in a microconvection model,” J. Appl. Mech. Tech. Phys., 43, No. 2. 208–216 (2002).
G. Polya and G. Szego, Problem and Theorems in Analysis, Springer, Berlin-Heidelberg-New York (1976).
P. D. Kalafati, “On Green’s functions of ordinary differential equations,” Dokl. Akad. Nauk SSSR, 26, No. 6, 535–539 (1940).
F. R. Gantmakher and M. G. Krein, Oscillation Matrices and Kernels and Small Fluctuations of Mechanical Systems [in Russian], Gostekhteoretizdat (1950).
Yu. S. Barkovskii and V. I. Yudovich, “Spectral properties of one class of boundary-value problems,” Mat. Sb., 114, No. 3, 438–450 (1981).
J. M. Mihaljan, “A rigorous exposition of the Boussinesq approximation applicable to a thin layer of fluid,” Astrophys. J., 136, No. 3, 1126–1133 (1962).
V. I. Yudovich, “Convection of an isothermally incompressible fluid,” Moscow (1999). Deposited at VINITI 05.28.99, No. 1699-B99.
V. V. Pukhnachev, “Model of convective motion under reduced gravity,” Model. Mekh., 6, No. 4, 47–56 (1992).
V. V. Pukhnachev, “Hierarchy of models in convection theory,” in: Zap. Nauch. Seminar. St. Petersburg. Otd. Mat. Inst. Steklova, 288, 152–177 (2002).
V. I. Yudovich, “On the occurrence of convection,” Prikl. Mat. Mekh., 30, 1000–1005 (1966).
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Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 5, pp. 35–42, September–October, 2007.
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Tyaglov, M.Y. Monotonicity principle in the rayleigh problem for an isothermally incompressible fluid. J Appl Mech Tech Phys 48, 649–655 (2007). https://doi.org/10.1007/s10808-007-0083-y
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DOI: https://doi.org/10.1007/s10808-007-0083-y