- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review E

Export citation

Export citation

Choose format for download:

Download Citation
  • Rapid Communication
  • Access by Xinjiang University

Hydrodynamics of fractal continuum flow

Alexander S. Balankin and Benjamin Espinoza Elizarraraz

  • Grupo “Mecánica Fractal,” Instituto Politécnico Nacional, México Distrito Federal, México 07738

PDF

Phys. Rev. E 85, 025302(R) – Published 13 February, 2012

DOI: https://doi.org/10.1103/PhysRevE.85.025302

Abstract

A model of fractal continuum flow employing local fractional differential operators is suggested. The generalizations of the Green-Gauss divergence and Reynolds transport theorems for a fractal continuum are suggested. The fundamental conservation laws and hydrodynamic equations for an anisotropic fractal continuum flow are derived. Some physical implications of the long-range correlations in the fractal continuum flow are briefly discussed. It is noteworthy to point out that the fractal (quasi)metric defined in this paper implies that the flow of an isotropic fractal continuum obeying the Mandelbrot rule of thumb for intersection is governed by conventional hydrodynamic equations.

View figure in article

Comments & Replies

Comment on “Hydrodynamics of fractal continuum flow” and “Map of fluid flow in fractal porous medium into fractal continuum flow”

Jun Li and Martin Ostoja-Starzewski
Phys. Rev. E 88, 057001 (2013)

Reply to “Comment on ‘Hydrodynamics of fractal continuum flow’ and ‘Map of fluid flow in fractal porous medium into fractal continuum flow’ ”

Alexander S. Balankin and Benjamin Espinoza Elizarraraz
Phys. Rev. E 88, 057002 (2013)

Article Text

Original Article

Map of fluid flow in fractal porous medium into fractal continuum flow

Alexander S. Balankin and Benjamin Espinoza Elizarraraz
Phys. Rev. E 85, 056314 (2012)

References (33)

  1. M. Sahimi, Flow and Transport in Porous Media and Fractured Rock (VCH, Weinheim, 1995); Flow and Transport in Fractured Porous Media, edited by P. Dietrich, R. Helmig, M. Sauter, H. Hötzl, J. Köngeter, and G. Teutsch (Springer, Berlin, 2005); M. E. Winters, Basic Clinical Pharmacokinetics (Applied Therapeutics, Vancouver, WA, 1994).
  2. J. S. Andrade, M. P. Almeida, J. Mendes, S. Havlin, B. Suki, and H. E. Stanley, Phys. Rev. Lett. 79, 3901 (1997); J. S. Andrade, A. M. Alencar, M. P. Almeida, J. Mendes, S. V. Buldyrev, S. Zapperi, H. E. Stanley, and B. Suki, ibid. 81, 926 (1998); M. P. Almeida, J. S. Andrade, S. V. Buldyrev, F. S. A. Cavalcante, H. E. Stanley, and B. Suki, Phys. Rev. E 60, 5486 (1999).
  3. V. V. Mourzenko, J.-F. Thovert, and P. M. Adler, Phys. Rev. E 59, 4265 (1999); G. Drazer and J. Koplik, ibid. 62, 8076 (2000); I. Bogdanov, V. V. Mourzenko, J.-F. Thovert, and P. M. Adler, ibid. 76, 036309 (2007); J. A. Ochoa, F. J. Valdes, and J. Alvarez, Physica A 374, 1 (2007); Y. Yan and J. Koplik, Phys. Rev. E 77, 036315 (2008); S. W. Coleman and J. C. Vassilicos, ibid. 78, 016308 (2008); A. Cortis and J. G. Berryman, Phys. Fluids 22, 053603 (2010).
  4. M. A. Zaks and A. V. Straube, Phys. Rev. Lett. 89, 244101 (2002); H. A. Knudsen and A. Hansen, Phys. Rev. E 65, 056310 (2002); H. P. Huinink and M. A. J. Michels, ibid. 66, 046301 (2002).
  5. A. S. Balankin, O. Susarrey, and J. Márquez, Phys. Rev. Lett. 90, 096101 (2003); A. S. Balankin, R. García, O. Susarrey, D. Morales, and F. Castrejon, ibid. 96, 056101 (2006); A. S. Balankin, S. M. Gutierres, D. Samayoa, J. Patiño, B. Espinoza, and C. L. Martínez, Phys. Rev. E 83, 036310 (2011).
  6. E. López, S. V. Buldyrev, N. V. Dokholyan, L. Goldmakher, S. Havlin, P. R. King, and H. E. Stanley, Phys. Rev. E 67, 056314 (2003).
  7. P. M. Adler, Phys. Fluids 29, 15 (1986); C. G. Jacquin and P. M. Adler, Transp. Porous Media 2, 553 (1987); 2, 571 (1987); P. M. Adler, ibid. 3, 185 (1988); R. Lemaitre and P. M. Adler, ibid. 5, 325 (1990); R. M. Adler and J.-E Thovert, ibid. 13, 41 (1993); H. E. Stanley and J. S. Andrade, Physica A 295, 17 (2001); R. B. Kazerooni and S. K. Hannani, Sci. Iran. Trans. Mech. Eng. 16, 159 (2009); Y. Chen, C. Zhang, M. Shi, and G. P. Peterson, Phys. Rev. E 80, 026301 (2009); R. S. Oliveira, J. S. Andrade, and R. F. S. Andrade, ibid. 81, 047302 (2010); S. Laizet and J. Ch. Vassilicos, Flow, Turbul. Combust. 87, 673 (2011).
  8. J. A. Barker, Water Resour. Res. 24, 1796 (1988); J. Chang and Y. C. Yortsos, Trans. AIME 9, 31 (1990); P. Xu, B. Yu, S. Qiu, and J. Cai, Physica A 387 6471 (2008); K. X. Yan, L. D. Lun, and L. U. DeTang, Sci. China, Ser. E: Techol. Sci. 52, 2700 (2009).
  9. H. W. Park, J. Choe, and J. M. Kang, Energy Sources 22, 881 (2000); 23, 619 (2001); T. Ji and T. Deng-ke, J. Hydrodynamics B 18, 287 (2006); S. Lian-tao, T. Deng-ke, and X. Li-li, ibid. 21, 705 (2009); Q. Wang and D. Tong, Transp. Porous Med. 81, 295 (2010).
  10. S. W. Wheatcraft and M. M. Meerschaert, Adv. Water Res. 31, 1377 (2008).
  11. V. E. Tarasov, Adv. Ann. Phys. 318, 286 (2005).
  12. G. S. F. Frederico and D. F. M. Torres, Nonlinear Dyn. 53, 215 (2008).
  13. J. Li and M. Ostoja-Starzewski, Proc. R. Soc. London, Ser. A 465, 2521 (2009); Formamente 4, 93 (2011).
  14. P. N. Demmie and M. Ostoja-Starzewski, J. Elasticity 104, 187 (2011).
  15. E.g., the failure of the Reynolds transport theorem for fractal continuum flow (see Ref. [13]), the conclusion that the velocity field of stationary incompressible fractal continuum flow is nonsolenoidal, the statement that the Bernoulli integral does not exists for the fractal continuum flow in the gravitational (potential) field, among others misstatements.
  16. In fact, the author of Ref. [11] has postulated the expressions of fractional differential operators in terms of conventional partial derivatives without any definition of the fractional derivative. Moreover, the local character of the fractional operators was not outlined explicitly. This has led to some confusion in subsequent works devoted to the fractal continuum. For example, in Refs. [13] and [14] a somewhat modified form of the fractional partial derivative introduced in Ref. [11] is explicitly referred to as a nonlocal fractional differential operator related to the right Jumarie fractional derivative. Consequently, the authors of Refs. [13] and [14] have introduced the upper limit of the Jumarie integral in the relation for density of states. However, it is worth noting that the operators related to the nonlocal Jumarie fractional derivative cannot be expressed in terms of the conventional partial derivatives as used in Refs. [13] and [14].
  17. W. Chen, Chaos Soliton Fractals 28, 923 (2006); W. Chen, X. D. Zhang, and D. Korosak, Int. J. Nonlin. Sci. 2, 3 (2010); W. Chen, H. Sun, X. D. Zhang, and D. Korosak, Comput. Math. Appl. 59, 1754 (2010).
  18. F. M. Borodich and Z. Feng, Z. Angew. Math. Phys. 61, 21 (2010).
  19. V. E. Tarasov, Fractional Dynamics (Springer, New York, 2010).
  20. J. Feder, Fractals (Plenum, New York, 1988).
  21. S. Miyazima and H. E. Stanley, Phys. Rev. B 35, 8898 (1987).
  22. K. Falconer, Fractal Geometry—Mathematical Foundations and Applications (Wiley, New York, 1990).
  23. C. Türk, A. Carbone, and B. M. Chiaia, Phys. Rev. E 81, 026706 (2010).
  24. It is worth noting that any quantity accompanied by flow should satisfy the generalized Green-Gauss divergence theorem (6).
  25. Except for the indexes of coordinates xi, lower cutoffs ℓi, exponents di, αi, ζi, and indexes in the parentheses, for which the Einstein summation rule is not applied.
  26. Accordingly, the numerical factor ζ−1ℓi is not used in definition (7).
  27. F. H. Stillinger, J. Math. Phys. 18, 1224 (1977); C. Palmer, and P. N. Stavrinou, J. Phys. A 37, 6987 (2004).
  28. M. Zubair, M. J. Mughal, and Q. A. Naqvi, Prog. Electromagn. Res. Lett. 19, 137 (2010); M. Zubair, M. J. Mughal, Q. A. Naqvi,and A. A. Rizvi, ibid. 114, 255 (2011).
  29. Notice the difference between Eqs. (14) and (15) and the forms of the material time derivative and the Reynolds transport theorem adopted in the fractal continuum model of Ref. [13].
  30. T. E. Faber, Fluid Dynamics for Physicists (Cambridge University Press, Cambridge, UK, 1995).
  31. H. E. Stanley and A. Coniglio, Phys. Rev. B 29, 522 (1984); M. Porto, A. Bunde, S. Havlin, and H. E. Roman, Phys. Rev. E 56, 1667 (1997).
  32. H. H. Hardi and R. E. Beier, Fractals in Reservoir Engineering (World Scientific, Singapore, 1994).
  33. L. M. Bangoy and C. Drogue, J. Hydrol. 158, 47 (1994); J. A. Acuña, I. Ershaghi, and Y. C. Yortsos, Trans. AIME 299, 173 (1995); F. Flamenco-López and R. Camacho-Velázues, SPE Res. Eval. Eng. 6, 39 (2003); R. Camacho-Velazquez, G. Fuentes-Cruz, and M. Vasquez-Cruz, ibid. 11, 606 (2008).

Outline

Information

Phys. Rev. E 85, 025302(R)– Published 13 February, 2012
  • Received 7 December 2011

©2012 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation