Multimoment Hydrodynamics in Problem on Flow around a Sphere: Entropy Interpretation of the Appearance and Development of Instability

ABSTRACT

Multimoment hydrodynamics equations are applied to investigate the phenomena of appearance and development of instability in problem on a flow around a solid sphere at rest. The simplest solution to the multimoment hydrodynamics equations coincides with the Stokes solution to the classic hydrodynamics equations in the limit of small Reynolds number values, . Solution to the multimoment hydrodynamics equations reproduces recirculating zone in the wake behind the sphere having the form of an axisymmetric toroidal vortex ring. The solution remains stable while the entropy production in the system exceeds the entropy outflow through the surface confining the system. The passage of the first critical value is accompanied by the solution stability loss. The solution, when loses its stability, reproduces periodic pulsations of the periphery of the recirculating zone in the wake behind the sphere. The and solutions to the multimoment hydrodynamics equations interpret a vortex shedding. After the second critical value is reached, the solution at the periphery of the recirculating zone and in the far wake is replaced by the solution. In accordance with the solution, the periphery of the recirculating zone periodically detached from the core and moves downstream in the form of a vortex ring. After the attainment of the third critical value , the solution at the periphery of the recirculating zone and in the far wake is replaced by the solution. In accordance with the solution, vortex rings penetrate into each other and form the continuous vortex sheet in the wake behind the sphere. The replacement of one unstable flow regime by another unstable regime is governed the tendency of the system to discover the fastest path to depart from the state of statistical equilibrium. Having lost the stability, the system does not reach a new stable position. Such a scenario differs from the ideas of classic hydrodynamics, which interprets the development of instability in terms of bifurcations from one stable state to another stable state. Solutions to the multimoment hydrodynamics equations indicate the direction of instability development, which qualitatively reproduces the experimental data in a wide range of Re values. The problems encountered by classic hydrodynamics when interpreting the observed instability development process are solved on the way toward an increase in the number of principle hydrodynamic values.

Cite this paper

Lebed, I. (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. doi: 10.4236/ojfd.2014.42015.

Lebed, I. (2014) Multimoment Hydrodynamics in Problem on Flow around a Sphere: Entropy Interpretation of the Appearance and Development of Instability.

References

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[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.

[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] Lebed, I.V. (2002) On the Inapplicability of Navier-Stokes Equations to Interpreting the Turbulence. Physica A, 315, 228-235. http://dx.doi.org/10.1016/S0378-4371(02)01254-2

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[11] Chomaz, J.M., Bonneton, P. and Hopfinger, E.J. (1993) The Structure of the Near Wake of a Sphere Moving Horizontally in a Stratified Fluid. Journal of Fluid Mechanics, 254, 1-21.

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http://dx.doi.org/10.1017/S0022112000008880

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[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.

[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.

[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] Lebed, I.V. (2002) On the Inapplicability of Navier-Stokes Equations to Interpreting the Turbulence. Physica A, 315, 228-235. http://dx.doi.org/10.1016/S0378-4371(02)01254-2

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

[6] Lebed, I.V. (1996) Hydrodynamic Equations Stemming from Two Particle Distributions in the Limit of Weak Non-Equilibrium. Analysis of Invertibility of Equations. Chemical Physics Reports, 15, 1725-1750.

[7] Loitsyanskii, L.G. (1966) Mechanics of Liquids and Gases. Pergamon, Oxford.

[8] 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.

[9] Tikhonov, A.N. and Samarskii, A.A. (1953) Equations of Mathematical Physics. Gostechizdat, Moscow.

[10] Taneda, S. (1956) Experimental Investigation of the Wake behind a Sphere at Low Reynolds Numbers. Journal of the Physical Society of Japan, 11, 1104-1108. http://dx.doi.org/10.1143/JPSJ.11.1104

[11] Chomaz, J.M., Bonneton, P. and Hopfinger, E.J. (1993) The Structure of the Near Wake of a Sphere Moving Horizontally in a Stratified Fluid. Journal of Fluid Mechanics, 254, 1-21.

http://dx.doi.org/10.1017/S0022112093002009

[12] Glansdorff, P. and Prigogine, I. (1971) Thermodynamic Theory of Structure, Stability and Fluctuations. Wiley-Interscience, A Division of John Wiley & Sons, Ltd., London.

[13] Prigogine, I. (1980) From Being to Becoming: Time and Complexity in the Physical Sciences. W.H. Freeman and Company, San Francisco.

[14] Schuster, H.G. (1984) Deterministic Chaos. Physik_Verlag, Weinheim.

[15] Zubarev, D.N. (1971) Nonequilibrium Statistical Thremodynamics. Nauka, Moscow.

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

[17] Lifshitz, E.M. and Pitaevskii, L.P. (1980) Course of Theoretical Physics. Vol. 9, Statistical Physics, Part 2, Pergamon, New York.

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

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

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

[21] Johnson, T.A. and Patel, V.S. (1999) Flow Past a Sphere Up to a Reynolds Number of 300. Journal of Fluid Mechanics, 378, 19-70. http://dx.doi.org/10.1017/S0022112098003206

[22] Boltzmann, L. (1896) Entgegnung auf die Warmetheoretischen Betrachtungen des Hrn. Zermelo. Ann. Phys. Chem., 57, 773-784.

[23] Zermelo, E. (1896) Uber einen Satz der Dynamik and die Mechanische Warmetheorie. Ann. Phys. Chem., 57, 485-494.

[24] Orban, J. and Bellemance, A. (1967) Velocity-Inversion and Irreversibility in a Dilute Gas of Hard Discs. Physics Letters A, 24, 620-621. http://dx.doi.org/10.1016/0375-9601(67)90651-2

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

[26] Loschmidt, J. (1876) Uber den Zustand des Warmegleichgewichtes eines Systems von Korpern mit Rücksicht auf die Schwerkraft. Wien. Ber., 73, 128-142.

[27] Tomboulides, A.G. and Orszag, S.A. (2000) Numerical Investigation of Transitional and Weak Turbulent Flow Past a Sphere. Journal of Fluid Mechanics, 416, 45-73.

http://dx.doi.org/10.1017/S0022112000008880

[28] Hannemann, K. and Oertel Jr., H. (1989) Numerical Simulation of the Absolutely and Convectively Unstable Wake. Journal of Fluid Mechanics, 199, 55-88. http://dx.doi.org/10.1017/S0022112089000297

[29] Isihara, A. (1971) Statistical Physics. Academic Press, New York.

[30] Grad, H. (1949) About Kinetic Theory of Rarefied Gases. Communications on Pure and Applied Mathematics, 2, 331- 407. http://dx.doi.org/10.1002/cpa.3160020403