Influence of the Delay and Dispersion in Mechanics

Affiliation(s)

Mathematics & Mechanics Faculty, St. Peterburg State University, St. Peterburg, Russia.

Mathematics & Mechanics Faculty, St. Peterburg State University, St. Peterburg, Russia.

ABSTRACT

The aim of this work is to clarify the new mathematical model describing the mechanics of continuous media and rarefied gas. The present study is associated with the formulation of conservation laws as conditions of equilibrium of angular momentums, while usually formulated in terms of balance of force. The equations for gas are found from the modified Boltzmann equation and the phenomenological theory. For a rigid body, the equations used the phenomenological theory, but changed their interpretation. We elucidate the contribution of cross-effects in the conservation laws of continuum mechanics, including the self-diffusion, thermal diffusion, etc., which indicated S. Wallander. The paradox of Hilbert in the solution of the Boltzmann equation by the Chapman-Enskog method was resolved. Refined model of the boundary conditions for rarefied gas flows and transient flow were near the moving surfaces. We establish conditions for the existence of the A. N. Kolmogorov inertial range on the basis of the proposed theory. Based on the theory, derivation of the Prandtl formula for boundary layer was received. Delay in mechanics plays an important role on commensurability of relaxation times and lateness. New accounting delay option is proposed to consider the difference between the time derivative as a limit and end values of the mean free path in a rarefied gas. The role of individual time delay for each particle velocity and the average time is debated. The Boltzmann equation is written with an additional term. This situation is typical for discrete medium. The transition from discrete to continuous environment is a key issue mechanics. Summary records of all effects lead to a cumbersome system of equations and therefore require the selection of main effects in a particular situation. The role of the time has similar problems in quantum mechanics. Some examples are suggested.

The aim of this work is to clarify the new mathematical model describing the mechanics of continuous media and rarefied gas. The present study is associated with the formulation of conservation laws as conditions of equilibrium of angular momentums, while usually formulated in terms of balance of force. The equations for gas are found from the modified Boltzmann equation and the phenomenological theory. For a rigid body, the equations used the phenomenological theory, but changed their interpretation. We elucidate the contribution of cross-effects in the conservation laws of continuum mechanics, including the self-diffusion, thermal diffusion, etc., which indicated S. Wallander. The paradox of Hilbert in the solution of the Boltzmann equation by the Chapman-Enskog method was resolved. Refined model of the boundary conditions for rarefied gas flows and transient flow were near the moving surfaces. We establish conditions for the existence of the A. N. Kolmogorov inertial range on the basis of the proposed theory. Based on the theory, derivation of the Prandtl formula for boundary layer was received. Delay in mechanics plays an important role on commensurability of relaxation times and lateness. New accounting delay option is proposed to consider the difference between the time derivative as a limit and end values of the mean free path in a rarefied gas. The role of individual time delay for each particle velocity and the average time is debated. The Boltzmann equation is written with an additional term. This situation is typical for discrete medium. The transition from discrete to continuous environment is a key issue mechanics. Summary records of all effects lead to a cumbersome system of equations and therefore require the selection of main effects in a particular situation. The role of the time has similar problems in quantum mechanics. Some examples are suggested.

KEYWORDS

Angular Momentum Conservation Laws, Unbalanced Stress Tensor, The Boltzmann Equation, Chapman-Enskog Method

Angular Momentum Conservation Laws, Unbalanced Stress Tensor, The Boltzmann Equation, Chapman-Enskog Method

Cite this paper

Prozorova, E. (2014) Influence of the Delay and Dispersion in Mechanics.*Journal of Modern Physics*, **5**, 1796-1805. doi: 10.4236/jmp.2014.516177.

Prozorova, E. (2014) Influence of the Delay and Dispersion in Mechanics.

References

[1] Prozorova, E.V. (2013) International Scientific IFNA-ANS Journal Problems of Nonlinear Analysis in Engineering Systems, 19, 45-57.

[2] Alexeev (2004) Generalized Boltzmann Physical Kinetics. Elsevier, Amsterdam.

[3] Prozorova, E.V. (2012) Electronic Journal Physico-Chemical Kinetics in Gas Dynamics, 13.

http://www.chemphys.edu.ru/pdf/2012-10-30-001.pdf (in Russian)

[4] Prozorova E.V. and Shadrin, A. (2013) Influence Dispersion in Gas and Solid for Moving Body. 5th European Conference for Aeronautics and Space Sciences, Munich, 1-5 July 2013/Holiday Inn Munich City Centre, Munich.

[5] Prozorova, E.V. (2013) Influence of the Dispersion in Model of Continuous Mechanics. VVM Publishing Ltd., St.- Peterburg. (in Russian)

[6] Bogolubov, N.N. (1946) Problems of Dynamic Theory in Statistical Physics. Gostexizdat, M-L. (in Russian)

[7] Cercignani, C. (1969) Mathematical Methods in Kinetic Theory. Macmillan.

[8] Ferziger, J.H. and Kaper, H.G. (1972) Mathematical Theory of Transport Processes in Gases. Amsterdam-London.

[9] Hirschfelder, J.O., Curtiss, C.F. and Bird, R.B. (1954) The Molecular Theory of Gases and Liquids. New York.

[10] Kogan, M.N. (1967) The Dynamics of the Rarefied Gases. Nauka, M. (in Russian)

[11] Lazic, I. (2009) Atomic Scale Simulation of Oxide and Metal Film Growth. Chapter 3. General Performance Testing, Chapter 4. An Improved Molecular Dynamics Potential for Studying Aluminum Oxidation. Part1-Parameter Optimization for the Electrostatic Part of the Potential. College Voor Promoties.

[12] Zubarev, D.N. (1971) Nonequilibrium Statistical Thermodynamics. Nauka, Moscow, 414 p.

[13] Fermi, E. (1968) Quantum Mechanics. Wiley, New York.

[14] Eugene, P. (1970) Wigner Symmetries and Reflections. Bloomigston-London.

[1] Prozorova, E.V. (2013) International Scientific IFNA-ANS Journal Problems of Nonlinear Analysis in Engineering Systems, 19, 45-57.

[2] Alexeev (2004) Generalized Boltzmann Physical Kinetics. Elsevier, Amsterdam.

[3] Prozorova, E.V. (2012) Electronic Journal Physico-Chemical Kinetics in Gas Dynamics, 13.

http://www.chemphys.edu.ru/pdf/2012-10-30-001.pdf (in Russian)

[4] Prozorova E.V. and Shadrin, A. (2013) Influence Dispersion in Gas and Solid for Moving Body. 5th European Conference for Aeronautics and Space Sciences, Munich, 1-5 July 2013/Holiday Inn Munich City Centre, Munich.

[5] Prozorova, E.V. (2013) Influence of the Dispersion in Model of Continuous Mechanics. VVM Publishing Ltd., St.- Peterburg. (in Russian)

[6] Bogolubov, N.N. (1946) Problems of Dynamic Theory in Statistical Physics. Gostexizdat, M-L. (in Russian)

[7] Cercignani, C. (1969) Mathematical Methods in Kinetic Theory. Macmillan.

[8] Ferziger, J.H. and Kaper, H.G. (1972) Mathematical Theory of Transport Processes in Gases. Amsterdam-London.

[9] Hirschfelder, J.O., Curtiss, C.F. and Bird, R.B. (1954) The Molecular Theory of Gases and Liquids. New York.

[10] Kogan, M.N. (1967) The Dynamics of the Rarefied Gases. Nauka, M. (in Russian)

[11] Lazic, I. (2009) Atomic Scale Simulation of Oxide and Metal Film Growth. Chapter 3. General Performance Testing, Chapter 4. An Improved Molecular Dynamics Potential for Studying Aluminum Oxidation. Part1-Parameter Optimization for the Electrostatic Part of the Potential. College Voor Promoties.

[12] Zubarev, D.N. (1971) Nonequilibrium Statistical Thermodynamics. Nauka, Moscow, 414 p.

[13] Fermi, E. (1968) Quantum Mechanics. Wiley, New York.

[14] Eugene, P. (1970) Wigner Symmetries and Reflections. Bloomigston-London.