Abstract
A model of the convective flow of viscous fluid is proposed with regard to possible finite “fluctuations” of thermophysical characteristics in a neighborhood of branch points. A method for the analysis of the influence of averaging on the values of phenomenological variables for the secondary flow and exact bounds on these values are calculated.
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References
V. V. Larchenko, “Application of the Asymptotic Method for Numerical Integration of Singularly Perturbed Equations,” Zh. Vychisl. Mat. Mat. Fiz. 16(2), 460–469 (1976).
V. V. Larchenko, “Bifurcational Indeterminacy and Fluctuation of Branch Points under Conditions of Singular Perturbation,” Zh. Vychisl. Mat. Mat. Fiz. 29, 1538–1551 (1989) [USSR Comput. Math. Math. Phys. 29 (5), 201–211 (1989)].
K. O. Friedrichs and J. J. Stoker, “Buckling of the Circular Plate beyond the Critical Thrust,” J. Appl. Mech. 9(1), 7–14 (1942).
M. Sunakawa and K. Ichida, “A High Precision Experiment on the Buckling of Spherical Caps Subjected to External Pressure,” Report of the University of Tokyo, No. 508, 1974 pp. 87–121.
V. V. Larchenko, “Asymptotic Analysis of Nonaxisymmetric Modes of Equilibrium of a Thin Shallow Spherical Shell,” Prikl. Mat. Mekh. 44, 1076–1086 (1980).
K. A. Cliffe, A. Spence, and S. J. Tavener, “The Numerical Analysis of Bifurcation Problems with Application to Fluid Mechanics,” Acta Numer. 9, 39–131 (2000).
V. V. Larchenko, “Problems which are Ill-Posed in the Bifurcation Sense and Deterministic Phenomena under Singular-Perturbation Conditions,” Dokl. Akad. Nauk SSSR 307, 1349–1354 (1989).
V. V. Vedenyapin, Boltzmann and Vlasov Kinetic Equations (Fizmatlit, Moscow, 2001) [in Russian].
C. Cercignani, Theory and Applications of the Boltzmann Equation (Scottish Academic, Edinburgh, 1975; Mir, Moscow, 1978).
G. Z. Gershuni, E. M. Zhukhovitskii, and A. A. Nepomnyashchii, Stability of Convective Flows (Nauka, Moscow, 1989) [in Russian].
L. D. Landau and E. M. Lifshitz, Fluid Mechanics, 2nd ed. (Nauka, Moscow, 1986; Pergamon Press, Oxford, 1987).
N. E. Kochin, I. A. Kibel’, and N. V. Roze, Theoretical Fluid Dynamics (Moscow, 1963), Vol. 2 [in Russian].
V. A. Dorodnitsyn, Group Properties of Difference Equations (Fizmatlit, Moscow, 2001) [in Russian].
V. V. Larchenko, “Algorithmization of Computations on a Symplectic Basis,” Differ. Uravn. 43, 411–422 (2007) [Differ. Equations 43 426–438 (2007)].
P. I. Golod and A. U. Klimyk, Mathematical Foundations of Symmetry Theory (RKhD, Moscow, 2001) [in Russian].
F. L. Abuev and V. V. Larchenko, “Bifurcational Indeterminacy and Its Proper Boundary,” Zh. Vychisl. Mat. Mat. Fiz. 42, 33–46 (2002) [Comput. Math. Math. Phys. 42, 30–42 (2002)].
N. A. Sidorov, B. V. Loginov, A. Sinitsyn, and M. Falaleev, Lyapunov-Schmidt Methods in Nonlinear Analysis (Kluwer, Dordrecht, 2002).
F. Antoneli, A. P. S. Dias, and C. Paul, “Matthews Invariant, Equivariants and Characters in Symmetric Bifurcation Theory,” Proc. R. Soc. Edinburgh, 138A, 477–512 (2008).
V. V. Larchenko, “Irreproducibility a Secondary Flow in a Neighborhood of the Multiple Bifurcation Point,” Mat. Modelir. 16(12), 44–60 (2004).
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Original Russian Text © V.V. Larchenko, 2011, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2011, Vol. 51, No. 4, pp. 708–722.
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Larchenko, V.V. Archimedes law under bifurcations of solution and partial averaging of phenomenological variables. Comput. Math. and Math. Phys. 51, 661–675 (2011). https://doi.org/10.1134/S0965542511040117
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DOI: https://doi.org/10.1134/S0965542511040117