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Archimedes law under bifurcations of solution and partial averaging of phenomenological variables

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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

  1. 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).

    MATH  Google Scholar 

  2. 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)].

    MathSciNet  Google Scholar 

  3. K. O. Friedrichs and J. J. Stoker, “Buckling of the Circular Plate beyond the Critical Thrust,” J. Appl. Mech. 9(1), 7–14 (1942).

    MathSciNet  Google Scholar 

  4. 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.

  5. V. V. Larchenko, “Asymptotic Analysis of Nonaxisymmetric Modes of Equilibrium of a Thin Shallow Spherical Shell,” Prikl. Mat. Mekh. 44, 1076–1086 (1980).

    Google Scholar 

  6. 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).

    Article  MathSciNet  Google Scholar 

  7. 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).

    MathSciNet  Google Scholar 

  8. V. V. Vedenyapin, Boltzmann and Vlasov Kinetic Equations (Fizmatlit, Moscow, 2001) [in Russian].

    Google Scholar 

  9. C. Cercignani, Theory and Applications of the Boltzmann Equation (Scottish Academic, Edinburgh, 1975; Mir, Moscow, 1978).

    Google Scholar 

  10. G. Z. Gershuni, E. M. Zhukhovitskii, and A. A. Nepomnyashchii, Stability of Convective Flows (Nauka, Moscow, 1989) [in Russian].

    MATH  Google Scholar 

  11. L. D. Landau and E. M. Lifshitz, Fluid Mechanics, 2nd ed. (Nauka, Moscow, 1986; Pergamon Press, Oxford, 1987).

    Google Scholar 

  12. N. E. Kochin, I. A. Kibel’, and N. V. Roze, Theoretical Fluid Dynamics (Moscow, 1963), Vol. 2 [in Russian].

  13. V. A. Dorodnitsyn, Group Properties of Difference Equations (Fizmatlit, Moscow, 2001) [in Russian].

    MATH  Google Scholar 

  14. V. V. Larchenko, “Algorithmization of Computations on a Symplectic Basis,” Differ. Uravn. 43, 411–422 (2007) [Differ. Equations 43 426–438 (2007)].

    MathSciNet  Google Scholar 

  15. P. I. Golod and A. U. Klimyk, Mathematical Foundations of Symmetry Theory (RKhD, Moscow, 2001) [in Russian].

    Google Scholar 

  16. 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)].

    MathSciNet  MATH  Google Scholar 

  17. N. A. Sidorov, B. V. Loginov, A. Sinitsyn, and M. Falaleev, Lyapunov-Schmidt Methods in Nonlinear Analysis (Kluwer, Dordrecht, 2002).

    MATH  Google Scholar 

  18. 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).

    Google Scholar 

  19. V. V. Larchenko, “Irreproducibility a Secondary Flow in a Neighborhood of the Multiple Bifurcation Point,” Mat. Modelir. 16(12), 44–60 (2004).

    MathSciNet  MATH  Google Scholar 

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Correspondence to V. V. Larchenko.

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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

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