Some Paradoxes of Mathematical Theory of Continues Mechanics

Evelina Prozorova

doi:doi:10.11648/j.ajam.20180601.13

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Some Paradoxes of Mathematical Theory of Continues Mechanics

Evelina Prozorova

Published in American Journal of Applied Mathematics (Volume 6, Issue 1)

Received: 11 January 2018 Accepted: 29 January 2018 Published: 7 March 2018

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Abstract

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.

Published in	American Journal of Applied Mathematics (Volume 6, Issue 1)
DOI	10.11648/j.ajam.20180601.13
Page(s)	15-19
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Angular Momentum, Conservation Laws, Non-Symmetrical Stress Tensor, Boltzmann Equations, Chapman-Enskog Method, Conjugate Problem the Navie-Stokes

References

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[2]	D. N. Zubarev, Nonequilibrium statistical thermodynamics.//M: Nauka, 1971. 414 p.
[3]	E. P Wigner. Symmetries and reflections.//Bloomington-London. 1970.
[4]	Fuxiang Han (2017) Kinetic Theory of Gases. Problems and Solutions in University Physics: pp. 201-223.
[5]	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.
[6]	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.
[7]	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.
[8]	E. V. Prozorova Influence of the Delay and Dispersion in Mechanics.// Journal of Modern Physics 2014. N5, 1796-805.
[9]	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.
[10]	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.
[11]	S. V. Vallander. The equations for movement viscosity gas.// DAN SSSR. 1951. V. LXX ІІІ, N 1.
[12]	Physics of Simple Liquids. Edited by H. N. V Temperley, J. S. Rowlinson, G. S. Rushbrooke. Nouth-Golland publishing company. 1968. Amsterdam.
[13]	C. Cercignani, Mathematical methods in kinetic theory. // Macmillan. 1969.
[14]	J. H. Ferziger, H. G. Kaper, Mathematical theory of transport processes in gases.// Amsterdam-London. 1972.
[15]	J. O. Hirschfelder, C. F. Curtiss, R. B. Bird, The molecular theory of gases and liquids.// New-York, 1954.
[16]	M. N. Kogan. The Dynamics of the Rarefied Gases. // M.: Nauka, 1967. 440 ( Russian).
[17]	Patrick Kee, David Guichard with modifications by Russ Gordon. An Introduction to Higher Mathematics 2010.

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Evelina Prozorova. (2018). Some Paradoxes of Mathematical Theory of Continues Mechanics. American Journal of Applied Mathematics, 6(1), 15-19. https://doi.org/10.11648/j.ajam.20180601.13

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Evelina Prozorova. Some Paradoxes of Mathematical Theory of Continues Mechanics. Am. J. Appl. Math. 2018, 6(1), 15-19. doi: 10.11648/j.ajam.20180601.13

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AMA Style

Evelina Prozorova. Some Paradoxes of Mathematical Theory of Continues Mechanics. Am J Appl Math. 2018;6(1):15-19. doi: 10.11648/j.ajam.20180601.13

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@article{10.11648/j.ajam.20180601.13,
  author = {Evelina Prozorova},
  title = {Some Paradoxes of Mathematical Theory of Continues Mechanics},
  journal = {American Journal of Applied Mathematics},
  volume = {6},
  number = {1},
  pages = {15-19},
  doi = {10.11648/j.ajam.20180601.13},
  url = {https://doi.org/10.11648/j.ajam.20180601.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20180601.13},
  abstract = {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.},
 year = {2018}
}

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TY  - JOUR
T1  - Some Paradoxes of Mathematical Theory of Continues Mechanics
AU  - Evelina Prozorova
Y1  - 2018/03/07
PY  - 2018
N1  - https://doi.org/10.11648/j.ajam.20180601.13
DO  - 10.11648/j.ajam.20180601.13
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 15
EP  - 19
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20180601.13
AB  - 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.
VL  - 6
IS  - 1
ER  -

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Author Information

Evelina Prozorova

Mathematic and Mechanic Faculty, St. Peterburg State University, Peterhof, Russia

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Plain Text BibTeX RIS

APA Style

Evelina Prozorova. (2018). Some Paradoxes of Mathematical Theory of Continues Mechanics. American Journal of Applied Mathematics, 6(1), 15-19. https://doi.org/10.11648/j.ajam.20180601.13

Copy | Download

ACS Style

Evelina Prozorova. Some Paradoxes of Mathematical Theory of Continues Mechanics. Am. J. Appl. Math. 2018, 6(1), 15-19. doi: 10.11648/j.ajam.20180601.13

Copy | Download

AMA Style

Evelina Prozorova. Some Paradoxes of Mathematical Theory of Continues Mechanics. Am J Appl Math. 2018;6(1):15-19. doi: 10.11648/j.ajam.20180601.13

Copy | Download

@article{10.11648/j.ajam.20180601.13,
  author = {Evelina Prozorova},
  title = {Some Paradoxes of Mathematical Theory of Continues Mechanics},
  journal = {American Journal of Applied Mathematics},
  volume = {6},
  number = {1},
  pages = {15-19},
  doi = {10.11648/j.ajam.20180601.13},
  url = {https://doi.org/10.11648/j.ajam.20180601.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20180601.13},
  abstract = {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.},
 year = {2018}
}

Copy | Download

TY  - JOUR
T1  - Some Paradoxes of Mathematical Theory of Continues Mechanics
AU  - Evelina Prozorova
Y1  - 2018/03/07
PY  - 2018
N1  - https://doi.org/10.11648/j.ajam.20180601.13
DO  - 10.11648/j.ajam.20180601.13
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 15
EP  - 19
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20180601.13
AB  - 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.
VL  - 6
IS  - 1
ER  -

Copy | Download