Systems of Navier-Stokes equations on cantor sets

Mathematical Problems in Engineering

Volumn 2013, Issue , 2013, Pages

  Yang, Xiao Jun ^a   Baleanu, Dumitru ^b,c,d   Tenreiro Machado, J A ^e  

^a CHINA UNIVERSITY OF MINING AND TECHNOLOGY   (China)

^b CANKAYA UNIVERSITY   (Turkey)

^c INSTITUTE OF SPACE SCIENCES   (Romania)

^d KING ABDULAZIZ UNIVERSITY   (Saudi Arabia)

^e INSTITUTO SUPERIOR DE ENGENHARIA DO PORTO   (Portugal)

Author keywords

[No Author keywords available]

Indexed keywords

BOUNDED DOMAIN; CANTOR SETS; FRACTAL MEDIA; VECTOR CALCULUS;

FRACTALS; TOPOLOGY; VISCOUS FLOW;

NAVIER STOKES EQUATIONS;

EID: 84880084367     PISSN: 1024123X     EISSN: 15635147     Source Type: Journal    
DOI: 10.1155/2013/769724     Document Type: Article

References (39)

1. Foias C., Manley O., Rosa R., Temam R., Navier-Stokes Equations and Turbulence 2001 83 Cambridge, UK Cambridge University Press 10.1017/ CBO9780511546754 MR1855030 ZBL1239.47012
2. Galdi G. P., An Introduction to the Mathematical Theory of the Navier-Stokes Equations 1994 1-2 New York, NY, USA Springer 10.1007/978-1-4612- 5364-8 MR1284205 ZBL0949.35005
3. Wang C. Y., Exact solutions of the steady-state Navier-Stokes equations. Annual Review of Fluid Mechanics 1991 23 Palo Alto, CA, USA Annual Reviews 159 177 MR1090331 ZBL0717.76033
4. Dubrulle B., Laval J.-P., Nazarenko S., Zaboronski O., A model for rapid stochastic distortions of small-scale turbulence. Journal of Fluid Mechanics 2004 520 1 21 10.1017/S0022112004001417 MR2257850 ZBL1065.76112 (Pubitemid 40107028)
5. Alexakis A., Mininni P. D., Pouquet A., Imprint of large-scale flows on turbulence. Physical Review Letters 2005 95 26 2-s2.0-30344436703 10.1103/PhysRevLett.95.264503 264503 (Pubitemid 43061555)
6. Mininni P. D., Alexakis A., Pouquet A., Large-scale flow effects, energy transfer, and self-similarity on turbulence. Physical Review E 2006 74 1 2-s2.0-33746253248 10.1103/PhysRevE.74.016303 016303 (Pubitemid 44091268)
7. Benzi R., Biferale L., Ciliberto S., Struglia M. V., Tripiccione R., Scaling property of turbulent flows. Physical Review E 1996 53 4 R3025 R3027 2-s2.0-0009477961 (Pubitemid 126583979)
8. She Z.-S., Leveque E., Universal scaling laws in fully developed turbulence. Physical Review Letters 1994 72 3 336 339 2-s2.0-12044250916 10.1103/PhysRevLett.72.336 (Pubitemid 24975607)
9. Shraiman B. I., Siggia E. D., Scalar turbulence. Nature 2000 405 6787 639 646 2-s2.0-0034621898 10.1038/35015000 (Pubitemid 30428643)
10. Benzi R., Ciliberto S., Tripiccione R., Baudet C., Massaioli F., Succi S., Extended self-similarity in turbulent flows. Physical Review E 1993 48 1 R29 R32 2-s2.0-0002894412 10.1103/PhysRevE.48.R29
11. Meneveau C., Katz J., Scale-invariance and turbulence models for large-eddy simulation. Annual Review of Fluid Mechanics 2000 32 Palo Alto, CA, USA Annual Reviews 1 32 10.1146/annurev.fluid.32.1.1 MR1744302 ZBL0988.76044 (Pubitemid 36144991)
12. Constantin P., Foias C., Manley O. P., Temam R., Determining modes and fractal dimension of turbulent flows. Journal of Fluid Mechanics 1985 150 427 440 10.1017/S0022112085000209 MR794051 ZBL0607.76054 (Pubitemid 15479967)
13. Chepyzhov V. V., Ilyin A. A., On the fractal dimension of invariant sets: applications to Navier-Stokes equations. Discrete and Continuous Dynamical Systems A 2004 10 1-2 117 135 MR2026186 (Pubitemid 38544747)
14. Scheffer V., Turbulence and Hausdorff dimension, in Turbulence and the Navier-Stokes equations. Lecture Notes in Mathematics 1976 565 Berlin, Germany Springer 94 112 MR0452123 ZBL0394.76029
15. Scheffer V., Hausdorff measure and the Navier-Stokes equations. Communications in Mathematical Physics 1977 55 2 97 112 MR0510154 10.1007/BF01626512 ZBL0357.35071
16. Boffetta G., Mazzino A., Vulpiani A., Twenty-five years of multifractals in fully developed turbulence: a tribute to Giovanni Paladin. Journal of Physics A 2008 41 363001 10.1088/1751-8113/41/36/363001 MR2447856
17. Cheskidov A., Holm D. D., Olson E., Titi E. S., On a Leray- α model of turbulence. Proceedings of The Royal Society of London A 2005 461 2055 629 649 10.1098/rspa.2004.1373 MR2121928 (Pubitemid 40522441)
18. Ilyin A. A., Lunasin E. M., Titi E. S., A modified-Leray- α subgrid scale model of turbulence. Nonlinearity 2006 19 4 879 897 10.1088/0951-7715/19/ 4/006 MR2214948 (Pubitemid 43429696)
19. Holm D. D., Kármán-Howarth theorem for the Lagrangian-averaged Navier-Stokes-alpha model of turbulence. Journal of Fluid Mechanics 2002 467 205 214 10.1017/S002211200200160X MR1928936 ZBL1064.76058
20. Hilfer R., Applications of fractional calculus in physics 2000 River Edge, NJ, USA World Scientific Publishing 10.1142/9789812817747 MR1890104
21. Sabatier J., Agrawal O. P., Machado J. A. T., Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering 2007 New York, NY, USA Springer
22. Kilbas A. A., Srivastava H. M., Trujillo J. J., Theory and Applications of Fractional Differential Equations 2006 204 Amsterdam, The Netherlands Elsevier Science MR2218073
23. Podlubny I., Fractional Differential Equations 1999 198 New York, NY, USA Academic Press MR1658022
24. El-Shahed M., Salem A., On the generalized Navier-Stokes equations. Applied Mathematics and Computation 2004 156 1 287 293 10.1016/j.amc.2003.07.022 MR2087428 ZBL1134.76323
25. Momani S., Odibat Z., Analytical solution of a time-fractional Navier-Stokes equation by Adomian decomposition method. Applied Mathematics and Computation 2006 177 2 488 494 10.1016/j.amc.2005.11.025 MR2291973 ZBL1096.65131 (Pubitemid 43866876)
26. Ganji Z. Z., Ganji D. D., Ganji A. D., Rostamian M., Analytical solution of time-fractional Navier-Stokes equation in polar coordinate by homotopy perturbation method. Numerical Methods for Partial Differential Equations 2010 26 1 117 124 10.1002/num.20420 MR2588910
27. Yang X. J., Advanced Local Fractional Calculus and Its Applications 2012 New York, NY, USA World Science Publisher
28. Yang X. J., Baleanu D., Machado J. A. T., Mathematical aspects of Heisenberg uncertainty principle within local fractional Fourier analysis. Boundary Value Problems 2013 1 131 150
29. Yang X. J., Baleanu D., Fractal heat conduction problem solved by local fractional variation iteration method. Thermal Science 2013 17 2 625 628
30. Hu M.-S., Agarwal R. P., Yang X.-J., Local fractional Fourier series with application to wave equation in fractal vibrating string. Abstract and Applied Analysis 2012 2012 15 10.1155/2012/567401 567401 MR3004869 ZBL1257.35193
31. Yang X. J., Baleanu D., Zhong W. P., Approximation solution to diffusion equation on Cantor time-space. Proceedings of the Romanian Academy A 2013 14 2 127 133
32. Yang X. J., Local Fractional Functional Analysis and Its Applications 2011 Hong Kong, China Asian Academic publisher Limited
33. Yang X. J., Local fractional integral transforms. Progress in Nonlinear Science 2011 4 1 225
34. Carpinteri A., Sapora A., Diffusion problems in fractal media defined on Cantor sets. ZAMM Zeitschrift fur Angewandte Mathematik und Mechanik 2010 90 3 203 210 2-s2.0-77049111800 10.1002/zamm.200900376
35. Kolwankar K. M., Gangal A. D., Local fractional Fokker-Planck equation. Physical Review Letters 1998 80 2 214 217 10.1103/PhysRevLett.80.214 MR1604435 ZBL0945.82005 (Pubitemid 128621921)
36. Li J., Ostoja-Starzewski M., Micropolar continuum mechanics of fractal media. International Journal of Engineering Science 2011 49 12 1302 1310 10.1016/j.ijengsci.2011.03.010 MR2893953
37. Tarasov V. E., Continuous medium model for fractal media. Physics Letters, Section A 2005 336 2-3 167 174 2-s2.0-13444252540 10.1016/j.physleta. 2005.01.024 (Pubitemid 40215401)
38. Drapaca C. S., Sivaloganathan S., A fractional model of continuum mechanics. Journal of Elasticity 2012 107 2 105 123 10.1007/s10659-011-9346-1 MR2899005
39. Tarasov V. E., Fractional hydrodynamic equations for fractal media. Annals of Physics 2005 318 2 286 307 10.1016/j.aop.2005.01.004 MR2150460 ZBL1071.76002 (Pubitemid 40864807)