Kinetic Foundation for the Multimoment Hydrodynamics Equations

Author(s)
Igor V. Lebed

ABSTRACT

The equations for the pair distribution functions are derived directly from the second equation of the Bogolyubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy. The derivation is fulfilled within the frameworks of the multiscale method. The equations for the pair distribution functions are the kinetic foundation for the multimoment hydrodynamics equations. Solutions to the equations for the pair distribution functions predetermine the possibility of constructing the hydrodynamics equations with an arbitrary number of principle hydrodynamic values specified beforehand. The tendency to increase the number of principal hydrodynamic values is caused by the necessity of interpreting the behavior of the system after the loss of stability. Solutions to the classic hydrodynamics equations constructed for only three principle hydrodynamic values are unable to predict the direction of instability evolution. Solutions to the multimoment hydrodynamics equations are capable of reproducing correctly the phenomenon of emergence and development of instability.

The equations for the pair distribution functions are derived directly from the second equation of the Bogolyubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy. The derivation is fulfilled within the frameworks of the multiscale method. The equations for the pair distribution functions are the kinetic foundation for the multimoment hydrodynamics equations. Solutions to the equations for the pair distribution functions predetermine the possibility of constructing the hydrodynamics equations with an arbitrary number of principle hydrodynamic values specified beforehand. The tendency to increase the number of principal hydrodynamic values is caused by the necessity of interpreting the behavior of the system after the loss of stability. Solutions to the classic hydrodynamics equations constructed for only three principle hydrodynamic values are unable to predict the direction of instability evolution. Solutions to the multimoment hydrodynamics equations are capable of reproducing correctly the phenomenon of emergence and development of instability.

Cite this paper

Lebed, I. (2015) Kinetic Foundation for the Multimoment Hydrodynamics Equations.*Open Journal of Fluid Dynamics*, **5**, 76-91. doi: 10.4236/ojfd.2015.51010.

Lebed, I. (2015) Kinetic Foundation for the Multimoment Hydrodynamics Equations.

References

[1] Lebed, I.V. and Umanskii, S.Y. (2007) The Appearance and Development of Turbulence in a Flow Past a Sphere: Problems and the Existing Approaches to Their Solution. Russian Journal of Physical Chemistry B, 1, 52-73.

http://dx.doi.org/10.1134/S1990793107010071

[2] Lebed, I.V. and Umanskii, S.Y. (2012) On the Possibility of Improving Classic Hydrodynamics Equations by an Increase in the Number of Hydrodynamic Values. Russian Journal of Physical Chemistry B, 6, 149-162.

http://dx.doi.org/10.1134/S1990793112010204

[3] Lebed, I.V. (2013) About the Prospects for Passage to Instability. Open Journal of Fluid Dynamics, 3, 214-229.

http://dx.doi.org/10.4236/ojfd.2013.33027

[4] Liboff, R.L. (1969) Introduction to the Theory of Kinetic Equations. Willey, New York.

[5] Lebed, I.V. (1996) Method of Two-Particle Distribution Functions. Hydrodynamic Equations. Chemical Physics Reports, 15, 861-883.

[6] Lebed, I.V. (1997) The Method of Pair Functions as Applied to the Problem of a Flow around a Quiescent Solid Sphere. Chemical Physics Reports, 16, 1263-1301.

[7] Lebed, I.V. (1998) About the Behavior of the Entropy of a Gas Flow Losing Its Stability. Chemical Physics Reports, 17, 411-439.

[8] Lebed, I.V. (2014) Multimoment Hydrodynamics in Problem on Flow around a Sphere: Entropy Interpretation of the Appearance and Development of Instability. Open Journal of Fluid Dynamics, 4, 163-206.

http://dx.doi.org/10.4236/ojfd.2014.42015

[9] Lebed, I.V. (2014) Development of Instability in the Problem of Flow around a Sphere. Russian Journal of Physical Chemistry B, 8, 240-253.

http://dx.doi.org/10.1134/S1990793114020171

[10] Lebed, I.V. (1990) Equations of Pair Distribution Functions. Chemical Physics Letters, 165, 226-228.

http://dx.doi.org/10.1016/0009-2614(90)85433-D

[11] Lebed, I.V. (1995) Derivation of the Equations for Pair Distribution Functions. Chemical Physics Reports, 14, 599-615.

[12] Van Dyke, M.D. (1964) Perturbation Methods in Fluid Mechanics. Academic Press, New York and London.

[13] Bogolyubov, N.N. (1946) The Problems of Dynamic Theory in Statistical Physics. Gostechizdat, Moscow-Leningrad.

[14] Ferziger, J.H. and Kaper, H.G. (1972) Mathematical Theory of Transport Processes in Gases. North-Holland Publishing Company, Amsterdam.

[15] Landau, L.D. and Lifshitz, E.M. (1976) Course of Theoretical Physics, Vol. 5: Statistical Physics, Part 1. Nauka, Moscow.

[16] Grad, H. (1958) Principles of the Kinetic Theory of Gases. Handbuch der Physik, 3/12, 205-294.

[17] Lebed, I.V. (2014) About Appearance of the Irreversibility. Open Journal of Fluid Dynamics, 4, 298-320.

http://dx.doi.org/10.4236/ojfd.2014.43023

[18] Sandri, G. (1963) The Foundations of Nonequilibrium Statistical Mechanics. Annals of Physics, 24, 332-379, 380-418.

[19] Freeman, E.A. (1963) On a New Method in the Theory of Irreversible Processes. Journal of Mathematical Physics, 4, 410-427.

http://dx.doi.org/10.1063/1.1703968

[20] Kogan, M.N. (1967) Rarefied Gas Dynamics. Nauka, Moscow.

[1] Lebed, I.V. and Umanskii, S.Y. (2007) The Appearance and Development of Turbulence in a Flow Past a Sphere: Problems and the Existing Approaches to Their Solution. Russian Journal of Physical Chemistry B, 1, 52-73.

http://dx.doi.org/10.1134/S1990793107010071

[2] Lebed, I.V. and Umanskii, S.Y. (2012) On the Possibility of Improving Classic Hydrodynamics Equations by an Increase in the Number of Hydrodynamic Values. Russian Journal of Physical Chemistry B, 6, 149-162.

http://dx.doi.org/10.1134/S1990793112010204

[3] Lebed, I.V. (2013) About the Prospects for Passage to Instability. Open Journal of Fluid Dynamics, 3, 214-229.

http://dx.doi.org/10.4236/ojfd.2013.33027

[4] Liboff, R.L. (1969) Introduction to the Theory of Kinetic Equations. Willey, New York.

[5] Lebed, I.V. (1996) Method of Two-Particle Distribution Functions. Hydrodynamic Equations. Chemical Physics Reports, 15, 861-883.

[6] Lebed, I.V. (1997) The Method of Pair Functions as Applied to the Problem of a Flow around a Quiescent Solid Sphere. Chemical Physics Reports, 16, 1263-1301.

[7] Lebed, I.V. (1998) About the Behavior of the Entropy of a Gas Flow Losing Its Stability. Chemical Physics Reports, 17, 411-439.

[8] Lebed, I.V. (2014) Multimoment Hydrodynamics in Problem on Flow around a Sphere: Entropy Interpretation of the Appearance and Development of Instability. Open Journal of Fluid Dynamics, 4, 163-206.

http://dx.doi.org/10.4236/ojfd.2014.42015

[9] Lebed, I.V. (2014) Development of Instability in the Problem of Flow around a Sphere. Russian Journal of Physical Chemistry B, 8, 240-253.

http://dx.doi.org/10.1134/S1990793114020171

[10] Lebed, I.V. (1990) Equations of Pair Distribution Functions. Chemical Physics Letters, 165, 226-228.

http://dx.doi.org/10.1016/0009-2614(90)85433-D

[11] Lebed, I.V. (1995) Derivation of the Equations for Pair Distribution Functions. Chemical Physics Reports, 14, 599-615.

[12] Van Dyke, M.D. (1964) Perturbation Methods in Fluid Mechanics. Academic Press, New York and London.

[13] Bogolyubov, N.N. (1946) The Problems of Dynamic Theory in Statistical Physics. Gostechizdat, Moscow-Leningrad.

[14] Ferziger, J.H. and Kaper, H.G. (1972) Mathematical Theory of Transport Processes in Gases. North-Holland Publishing Company, Amsterdam.

[15] Landau, L.D. and Lifshitz, E.M. (1976) Course of Theoretical Physics, Vol. 5: Statistical Physics, Part 1. Nauka, Moscow.

[16] Grad, H. (1958) Principles of the Kinetic Theory of Gases. Handbuch der Physik, 3/12, 205-294.

[17] Lebed, I.V. (2014) About Appearance of the Irreversibility. Open Journal of Fluid Dynamics, 4, 298-320.

http://dx.doi.org/10.4236/ojfd.2014.43023

[18] Sandri, G. (1963) The Foundations of Nonequilibrium Statistical Mechanics. Annals of Physics, 24, 332-379, 380-418.

[19] Freeman, E.A. (1963) On a New Method in the Theory of Irreversible Processes. Journal of Mathematical Physics, 4, 410-427.

http://dx.doi.org/10.1063/1.1703968

[20] Kogan, M.N. (1967) Rarefied Gas Dynamics. Nauka, Moscow.