Elementary convective cell in incompressible viscous fluid and its parameters

Authors

  • Л. С. Бозбей A. N. Podgorny Institute for Mechanical Engineering Problems NAS of Ukraine National Science Center "Kharkiv Institute of Physics and Technology," National Academy of Sciences of Ukraine, Ukraine
  • А. О. Костиков A. N. Podgorny Institute for Mechanical Engineering Problems NAS of Ukraine V.N. Karazin Kharkiv National University, Ukraine
  • В. И. Ткаченко National Science Center "Kharkiv Institute of Physics and Technology," National Academy of Sciences of Ukraine V.N. Karazin Kharkiv National University, Ukraine

Keywords:

elementary convective cell-free border, convective processes, heat transfer, temperature gradient

Abstract

The energy principle of convective structures formation in a layer of viscous incompressible fluid uniformly heated from below is proposed. The energy principle of usage of elementary convective cell of cylindrical shape is proposed and justified. The mathematical model of thermal processes in elementary convective cell with free boundaries is suggested and analytical solutions for the perturbation of velocity and temperature are obtained. The radial wave numbers for the velocity perturbations and eigenvalues of the problem are determined. It is shown that the spectrum of eigenvaluesis discrete both on the mode of the perturbation and the radial wave number. The expression for the radius of an elementary convective cell is obtained. It shows that the radius value can take discrete quantities that correspond to stable convective states. It is shown that maximum heat transfer occurs at the smallest possible radius value. Experimental investigation of forming the convective cells was carried out. It confirmed the correctness of the theoretical results.

Author Biographies

А. О. Костиков, A. N. Podgorny Institute for Mechanical Engineering Problems NAS of Ukraine V.N. Karazin Kharkiv National University

Doctor of Technical Sciences

В. И. Ткаченко, National Science Center "Kharkiv Institute of Physics and Technology," National Academy of Sciences of Ukraine V.N. Karazin Kharkiv National University

Doctor of Physical and Mathematical Sciences

References

Benard H. Les tourbillons cellulaires dans une nappe liquide. Description générale des phénomènes. Revue générale des Sciences pures et appliquées. 11 (24), 1261–1271 (1900).

Benard H. Les tourbillons cellulaires dans une nappe liquide. Procédés mécaniques et optiques d’examen lois numériques des phénomènes. Revue générale des Sciences pures et appliquées. 11 (24). 1309–1328 (1900).

Strutt J. W. (Lord Rауlеigh). On convection currents in a horizontal layer of fluid when the higher temperature is on the under side. Phil. Mag. 32, 529–546 (1916).

Thomson J. On a changing tesselated structure in certain liquid. Proc. Glasgow Philos. Soc. 13, 464–468 (1882).

http://www.ivanov-portal.ru/astron/30.htm

Shishkin N. S. Education cellular structures in a liquid or gas layers. Uspekhi phizicheskih nauk. 31 (4), 462–490 (1991) (in Russian).

http://lifeglobe.net/blogs/details?id=860

http://p-i-f.dreamwidth.org/351593.html

Ray R. J., Krantz W. B., Caine T. N. and Gunn R. D. A model for sorted patterned-ground regularity. J. of Glaciology. 29 (102). 317–337 (1983).

Rychkova E. V., Tychkov S. A. Numerical model of thermal convection in the upper mantle of the earth under continental lithosphere. Vyichislitelnyie tehnologii. 2 (5), 66–81 (1997) (in Russian).

http://en.wikipedia.org/wiki/Supergranulation

Pikel'ner S. B. Dynamics of the solar atmosphere. Successes of physical sciences. 88 (3), 505–523 (1966) (in Russian).

http://opensky.library.ucar.edu/ collections/SOARS-000-000-000-268.

Rieuton M., Rincon F. The Sun’s Supergranulation. Living Rev. Solar Phys. 7 (2), 84 (2010).

Gershuni G. Z, Zhukhovitskii E. M. Convective stability of incompressible fluid. M .: Nauka, 393 (1972) (in Russian).

Chandrasekhar S. Hydrodynamic and hydromagnetic stability. Oxford: University Press, 657 (1970).

Getling A. V. Spatial patterns formed by Rayleigh-Benard convection. Uspekhi phizicheskih nauk. 161 (9), 1–80 (1991) (in Russian).

Bozbiei L. S. An elementary convective cell in layer of incompressible viscous fluid. Conference of young scientists and specialists. IPMach NAN of Ukraine, Kharkov, 29 (2013) (in Russian).

Bozbey L. S., Kostikov A. O., Tkachenko V. I. Elementary convective cell and its thermal properties. Proc. conf. "Physical and technical problems of energy and ways of their solution in 2014". – Kharkiv, Ukraine, on 25-26 June 2014. 6 (2014) (in Russian).

Nekludov I. M., Bortz B. V., Tkachenko V. I. Description Langmuir circulations ordered set of cubic convective cells. Prikladnaya gidromehanika. 14 (86) (2), 29–40 (2012) (in Russian).

Van Dyke M. Atlas flow of liquids and gases. М.: Мir, 184 (1986) (in Russian).

Koschmieder E. L. Bénard Cells and Taylor Vortices: monograph on mechanics. Cambridge: University Press, 350 (1993).

Adelman E. D. Effect of liquid film thickness to cell size ratio convection. Zhurnal tehnicheskoy fiziki. 68 (11), 7–11 (1996) (in Russian).

Eckert K., Bestehorn M., Thess A. E. Square cells in surface-tension-driven Beґnard convection: experiment and theory. 356, 155–197 (1998).

Borts B. V., Kazarinov U. G., Kostikov A. O., Tkachenko V. I. Experimental study of liquid movement in free elementary convective cells. Energetika. 61 (2), 45–56 (2015).

Royal Society Mathematical Tables. Vol. 7. Bessel functions. Cambridge: University Press, 140 (1960).

Zierep J. Über rotationssymmetrische Zellularkonvektionsströmungen. Z. Agev. Mah. Mech. 39 (7/8), 329–333 (1958).

Zierep J. Eine rotationssymmetrische Zellularkonvektionsstromun. Beitr. Phys. Atmos. 30, 215–222 (1958).

G. Korn, Korn T. Mathematical Handbook for Scientists and Engineers. М.: Nauka, 720 (1968) (in Russian).

Conway J. H., Lagarias J. C. Tiling with polyominoes and combinatorial group. Journal of Combinatorial Theory. Series A 53, 183–208 (1990).

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Published

2016-09-30

Issue

Vol. 19 No. 3 (2016)

Section

Heat transfer in engineering constructions

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Copyright (c) 2016 Л. С. Бозбей, А. О. Костиков, В. И. Ткаченко

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