#### Cite this article as:

Uchaikin V. V. Fractional models in hydromechanics. Izvestiya VUZ. Applied Nonlinear Dynamics, 2019, vol. 27, iss. 1, pp. 5-40. DOI: https://doi.org/10.18500/0869-6632-2019-27-1-5-40

# Fractional models in hydromechanics

Topic and purpose. The last two decades are marked by wide spreading fractional calculus in theoretical description of the natural processes. Replacement of the integer-order operators by their fractional (and even complex) counterparts opens up a continuous field of new differential equations in which the standard set of equations of theoretical physics (wave, diffusion, etc.) is represented by separate spikelets at points with integer coordinates. But what do the fractional-order derivatives mean physically? What are the common reasons for the appearance of fractional derivatives in the equations? Is it possible to predict in advance the appearance of fractional operators in a particular problem? These questions are not yet removed from the agenda and remain the focus of attention of each of the conferences devoted to the theory and application of fractional calculus. This topic is developing in this review. Models investigated. The fractional calculus is demonstrated in application to various problem of the most, if one may say so, classical field of theoretical physics-hydrodynamic including turbulent diffusion. Results. The review shows how fractional operators appear on the classical field of hydrodynamic problems under the pen of Heisenberg, Weizsacker, Kolmogorov, Obukhov, Monin – theoreticians who can not be suspected of being uncritical of the mathematical tools. Discussion. Actually, the whole review is a continuous discussion of the «inevitability of the strange world» of fractional calculus (Uchaikin V.V. The method of fractional. Ulyanovsk: «Artishok», 2008), and the fact that this is done within the framework of classical hydromechanics only strengthens the convincing conclusions.

1. Rutman R.S. On physical interpretations of fractional integration and differentiation. Theoretical and Mathematical Physics, 1995, vol. 105, no. 3, pp. 1509–1519. 2. Uchaikin V.V. Self-similar anomalous diffusion and Levy-stable laws. Physics–Uspekhi, 2003, vol. 46, no. 8, pp. 821. 3. Sibatov R.T., Uchaikin V.V. Fractional differential approach to dispersive transport in semiconductors. Physics–Uspekhi, 2009, vol. 52, no. 10, pp. 1019–1043. 4. Uchaikin V.V. Fractional phenomenology of cosmic ray anomalous diffusion. Physics–Uspekhi. 2013. vol. 56, no. 11, pp. 1074–1119. 5. Uchaikin V.V. The Method of Fractional Derivatives. Ulyanovsk: «Artishok», 2008 (in Russian). 6. Uchaikin V.V. Fractional Derivatives for Physicists and Engineers. Vol’s I–II, Springer Berlin, HEP Beijing, 2013. 7. Boltzmann L. Zur Theorie der Elastishen Nachwirkungen. Akad. Wiss. Wien, Math. Naturwiss. 1874, vol.70, no. 2, pp. 275–306. 8. Volterra V. Lectures in the Mathematical Theory of the Struggle for Existence. Gauthier-Villars, Paris, 1931. 9. Volterra V. Theory of Functionals of Integral and Integro-Differential Equations, Dover, New York, 1959. 10. Rabotnov Y.N. Elementy nasledstvennoy mekhaniki tverdykh tel. Moscow, Nauka, 1977 (in Russian). 11. Valanis K.C. and Lee C.F. Endochronic theory of cyclic plasticity with application. J. Appl. Mech., 1984, vol. 51, pp. 367–374. 12. Blatt J.M. An alternative approach to the ergodic problem. Progress in Theoretical Physics, 1959, vol. 22, pp. 745–756. 13. Maugin G.A. and Muschik W. Thermodynamics with internal variables. Part I. General concepts. Journal of Non-Equilibrium Thermodynamics, 1994, vol. 19, pp. 217–249. 14. Maugin G.A. and Muschik W. Thermodynamics with internal variables. Part II. Applications. Journal of Non-Equilibrium Thermodynamics, 1994, vol. 19, pp. 250–289. 15. Maugin G. The Thermomechanics of Nonlinear Irreversible Behaviors (An Introduction). World Scientific, Singapure–New Jersey–London–Hong Kong, 1999. 16. Gemant A. A method of analyzing experimental results obtained from elastoviscous bodies. Physics, 1936, vol. 7, pp. 311–317. 17. Gerasimov A.N. A generalization of linear laws of deformation and its application to internal friction problem. Prikl. Mat. Mekh., 1948, vol. 12, no. 3. pp. 251–260 (in Russian). 18. Nigmatullin R.R. Fractional integral and its physical interpretation. Тheoretical and Mathematical Physics, 1992, vol. 90, no. 3, pp. 242–251. 19. Hilfer R. Classification theory for anequilibrium phase transitions. Phys Rev E., 1993, vol. 48, pp. 2466–2475. 20. Hilfer R. Fractional time evolution, In Applications of Fractional Calculus in Physics. R. Hilfer (ed.), World Scientific, Singapore, 2000, pp. 87–131. 21. Lukashchuk S.Yu. Time-fractional extensions of the Liouville and Zwanzig equation. Cent. Eur. J. Phys., 2013, vol. 11, no. 6, p. 740. 22. Kwok Sau Fa. A falling body problem through the air in view of the fractional derivative approach. Physica A, 2005, vol. 350, pp. 199–206. 23. Narahari Achar B.N., Hanneken J.W., Enck T., Clarke T. Dynamics of the fractional oscillator. Physica A, 2001, vol. 297, pp. 361–367. 24. Ryabov Ya.E. and Puzenko A. Damped oscillations in view of the fractional oscillator equation. Phys. Rev. B, 2002, vol. 66, 184201. 25. Baleanu D., Golmankhaneh A.K., Nigmatullin R., Golmankhaneh Ali K. Fractional Newtonian mechanics Cent. Eur. J. Phys., 2010, vol. 8, no. 1, pp. 120–125. 26. Slezkin N.A. Dynamics of Viscous Incompressible Fluid, Moscow, Gostekhizdat, 1955 (in Russian). 27. Boussinesq V.J. Sur la resistance quoppose un fluide indefini au repose...Compt.Rend.de l’Academ. des Sci. 1885, vol. 100, pp. 935–937. 28. Basset A.B. Treatise on Hydrodynamics 2.,Deighton, Bell and Co., Cambridge. UK, 1888. 29. Zener C.M. Anelasticity of metals. Suppl. Nuovo Cimento, 1958, vol. 7, p. 544. 30. Uchaikin V.V., Sibatov R.T. Fractional differential kinetics of dispersive transport as the consequence of its self-similarity. JETP Letters, 2007, vol. 86, no. 8, pp. 512–516. 31. Uchaikin V.V., Sibatov R.T. Fractional Kinetics in Solids, World Scientific, 2013. 32. Landau L.D., Lifschitz E.M. Fluid Mechanics. New York: Pergamon Press, 1987. 33. van Hove L. The approach to equilibrium in quantum statistics: A perturbation treatment to general order. Physica, 1957, vol. 23, pp. 441–480. 34. Prigogine I., Resibois P. On the kinetics of the approach to equilibrium. Physica, 1961, vol. 27, pp. 629–646. 35. Brout R., Prigogine I. Statistical mechanics of irreversible processes, Physica, 1956, vol. 22, pp. 35–47, 263–272, 621–636. 36. Resibua P., De Lener M. Classical Kinetic Theory of Liquids and Gases. Moscow, Mir, 1980 (in Russian). 37. Zwanzig R. Nonequilibrium Statistical Mechanics. New York: Oxford University Press, 2001. 38. Montroll E . W. In Fundamental Problems in Statistical Mechanics, ed. E. Cohen, North-Holland, Amsterdam, 1962. 39. Chester G.V. The theory of irreversible processes, Rep. on Progress in Physics, 1963, vol. 26, p. 411. 40. Alder B.J., Alley W.E. Generalized Hydrodynamics, Physics Today, 1984, vol. 37, pp. 56–83. 41. Richardson L.F. Atmospheric diffusion on a distance-neighbor graph. Proc. Roy Soc. London, Ser A, 1926, vol. 110, pp. 709–737. 42. Kolmogorov A.N. Scattering of energy for locally isotropic turbulence. Dokl. Acad. Sci. USSR, 1941, vol. 32, pp. 16–18 (in Russian). 43. Obukhov A.M. Energy distribution in the spectrum of a turbulent flow. Dokl. Acad. Sci. USSR, 1941, vol. 32, pp. 22–24 (in Russian). 44. Jullien M.C., Paret J., Tabeling P. Richardson pair dispersion in two-dimensional turbulence. Phys. Rev. Lett., 1999, vol. 82, p. 2872. 45. Tchen C.M. Diffusion of Particles in Turbulent Flow. Advances in Geophysics, 1959, vol. 9, pp. 165–174. 46. Heisenberg W. Zur Statistischen Theorie der Turbulenz. Zeitschrift fuer Physik 1948, vol. 124, pp. 628–657. 47. Monin A.S. Yaglom A.M. Statistical Hydromechanics. Part I. M.: Nauka, 1965; Part II. Moscow: Nauka, 1967 (in Russian). 48. Shlesinger M., Klafter J., West B. Levy walks with applications to turbulence and chaos. Physica, 1986, vol. 140A, pp. 212–218. 49. Schonfeld J.C. Integral diffusivity. ? Journal of Geophysical Research, 1962, vol. 67, no. 8, pp. 3187–3199. 50. Tchen C.M. Transport processes as foundations of the Heisenberg and Obukhoff theories of turbulence. Phys Rev., 1954, vol. 93, no. 1, pp. 4–14. 51. Uchaikin V.V. Mechanics. Fundamentals of Continuum Mechanics. SPb, Lan, 2016 (in Russian). 52. Uchaikin V.V. On time-fractional representation of an open system response. Fractional Calculus and Applied Analysis, 2016, vol. 19, no. 5, pp. 1306–1315. 53. Klimontovich Yu.L. Introduction to the Physics of Open Systems. M., Janus-K, 2002 (in Russian). 54. Lindenberg K., West B.J. The Nonequilibrium Statistical Mechanics of Open and Closed Systems, Wiley, VCH Publishers, New York, 1990. 55. Di-Ventra M. Electrical Transport in Nanoscale Systems. Cambridge University Press, 2008. 56. Uchaikin V.V. On the fractional-differential Liouville equation as an equation of dynamics of an open system. Scientific bulletins of the Belgorod University, series: Mathematics. Physics, 2014, vol. 25(196), no. 37, pp. 58–67 (in Russian). 57. Slonimsky G.L. On the law of deformation of highly elastic polymer bodies. Dokl. AN SSSR, 1961, vol. 140, no. 2, pp. 343–346 (in Russian).

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