American Journal of Applied Mathematics
Volume 6, Issue 1, February 2018, Pages: 15-19
Received: Jan. 11, 2018;
Accepted: Jan. 29, 2018;
Published: Mar. 7, 2018
Views 738 Downloads 50
Evelina Prozorova, Mathematic and Mechanic Faculty, St. Peterburg State University, Peterhof, Russia
The classic theory of continuum mechanics does not preserve the continuity of the environment due to the use of the conditions of equilibrium of forces and the symmetry of the stress tensor. We used many unreasonable mathematical approximations when by the Boltzmann equation is solved to describe the equations of continuum mechanics. The paper presents an analysis of mathematical approximations underlying description in different environments, and new models, to avoid the resulting misunderstandings. For rarefied gas the self-diffusion and thermo-diffusion which were foretold by S. V. Vallander are obtained from kinetic theory.
Some Paradoxes of Mathematical Theory of Continues Mechanics, American Journal of Applied Mathematics.
Vol. 6, No. 1,
2018, pp. 15-19.
Copyright © 2018 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/
) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
A. Ye. Ishlinskii, D. D. Ivlev. The mathematical theory of plasticity.// M.: Fizmatlit, 2003, 702s.
D. N. Zubarev, Nonequilibrium statistical thermodynamics.//M: Nauka, 1971. 414 p.
E. P Wigner. Symmetries and reflections.//Bloomington-London. 1970.
Fuxiang Han (2017) Kinetic Theory of Gases. Problems and Solutions in University Physics: pp. 201-223.
L. Boudin, B. Grec, K. Okuda. Molecular kinetic analysis of a local equilibrium Carnot cycle. PHYSICAL REVIEW E. V. 96 N1, N. art. 012123, JUL. 2017.
Evelina Prozorova. Effect of Mathematical Models on Experimental Data for the Gas and Liquids.//Journal of Mechanics Engineering and Automation. N 6, 2016, 313-318.
E. Prozorova, The Role of Dispersion Effects and Delay for Continuum Mechanics.// Proceedings of 16th International Workshop on New Approaches to High-Tech: Nano-Design, Technology, Computer Simulations. NDTCS-2015, 136-8.
E. V. Prozorova Influence of the Delay and Dispersion in Mechanics.// Journal of Modern Physics 2014. N5, 1796-805.
E. V. Prozorova The Influence of the Dispersion of Non-equilibrium Continuum Mechanics.// Problems Environment. Moscow State University Electronic Journal: Physical and Chemical Kinetics in Gas Dynamics 13: 30. URL: http://www.chemphys.edu.ru/pdf/2012-10-30-001.pdf.
Oleg Galaev, Evelina Prozorova. Dispersion Effects in the Falkner-Skan Problem and in the Kinetic Theory. Journal of Applied Mathematics and Physics, 2017, 5, 522-537.
S. V. Vallander. The equations for movement viscosity gas.// DAN SSSR. 1951. V. LXX ІІІ, N 1.
Physics of Simple Liquids. Edited by H. N. V Temperley, J. S. Rowlinson, G. S. Rushbrooke. Nouth-Golland publishing company. 1968. Amsterdam.
C. Cercignani, Mathematical methods in kinetic theory. // Macmillan. 1969.
J. H. Ferziger, H. G. Kaper, Mathematical theory of transport processes in gases.// Amsterdam-London. 1972.
J. O. Hirschfelder, C. F. Curtiss, R. B. Bird, The molecular theory of gases and liquids.// New-York, 1954.
M. N. Kogan. The Dynamics of the Rarefied Gases. // M.: Nauka, 1967. 440 ( Russian).
Patrick Kee, David Guichard with modifications by Russ Gordon. An Introduction to Higher Mathematics 2010.