On the Mathematical Theory of Turbulence and Its Relation to Chaos and Fractals

Volume 1, Issue 3, pp 187--215 http://dx.doi.org/10.22436/jmcs.001.03.07

Download PDF

Download XML

• Export citations
  • CITATIONS & REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • ARTICLE CITATION
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)

• 2241 Downloads
• 4023 Views

Authors

Bertrand Wong - Eurotech, Spore Branch

Abstract

The Navier-Stokes differential equations describe the motion of fluids which are incompressible. The three-dimensional Navier-Stokes equations misbehave very badly although they are relatively simple-looking. The solutions could wind up being extremely unstable even with nice, smooth, reasonably harmless initial conditions. A mathematical understanding of the outrageous behaviour of these equations would dramatically alter the field of fluid mechanics. The Orr-Sommerfeld equation is also described. In this paper the author adopts a reasoned, practical approach towards resolving the issue and proposes a practical, statistical kind of mathematical solution.

Share and Cite

Keywords

• unpredictable
• probability
• estimate

MSC

•  37D45
•  76F02
•  76F20
•  76F99
•  97K50

References

• [1] A. Beiser, Concepts Of Modern Physics (Fifth Edition), McGraw-Hill, Japan (1995)
  • View Article
  • Google Scholar


• [2] M. Courbage, Intrinsic Irreversibility Of Kolmogorov Dynamical Systems, Phys. A, 122 (1983), 459--482
  • View Article
  • MathSciNet
  • Google Scholar


• [3] M. Courbage, I. Progogine, Intrinsic Randomness And Intrinsic Irreversibility In Classical Dynamical Systems, Proc. Nat. Acad. Sci. , 80 (1983), 2412--2416
  • View Article
  • Google Scholar


• [4] R. L. Devaney, An Introduction To Chaotic Dynamical Systems, The Benjamin/Cummings Publishing Co., Menlo Park (1986)
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [5] P. A. M. Dirac, The Principles Of Quantum Mechanics, Oxford University Press, Oxford (1958)
  • View Article
  • Google Scholar


• [6] P. G. Drazin, W. H. Reid, Hydrodynamic Stability, Cambridge University Press, Cambridge (1981)
  • View Article
  • Google Scholar


• [7] M. J. Feigenbaum, Universal Behavior In Nonlinear Systems, Los Alamos Sci., 1 (1980), 4--27
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [8] U. Frisch, Turbulence, Cambridge University Press, Cambridge (1955)
  • View Article
  • Google Scholar


• [9] J. Gleick, Chaos: Making A New Science, Viking Press, U.S.A. (1987)
  • View Article
  • Google Scholar


• [10] H. A. Lauwerier, Fractals-Images Of Chaos, Penguin books, Canada (1991)
  • View Article
  • Google Scholar


• [11] B. B. Mandelbrot, The Fractal Geometry Of Nature, W. H. Freeman, U.S.A. (1977)
  • View Article


• [12] R. M. May, Simple Mathematical Models With Very Complicated Dynamics, Nature, 261 (1976), 459--467
  • View Article
  • Google Scholar


• [13] B. Misra, I. Prigogine, M. Courbage, From Deterministic Dynamics To Probabilistic Descriptions, Phys. A, 98 (1979), 1--26
  • View Article
  • Google Scholar


• [14] A. Modinos, Quantum Theory of Matter: A Novel Introduction, John Wiley & Sons, U. K. (1996)
  • View Article
  • Google Scholar


• [15] J. Moser, Stable And Random Motions In Dynamical Systems, Princeton University Press, Princeton (1974)
  • View Article
  • MathSciNet


• [16] A. M. Mukhamedov, Towards a gauge theory of turbulence, Chaos Solitons Fractals, 29 (2006), 253--261
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [17] A. M. Mukhamedov, Dynamic Paradigm Of Turbulence, Chaos Solitons Fractals , 30 (2006), 278--289
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [18] M. S. El Naschie, Statistical Mechanics Of Multidimensional Cantor Sets, Godel’s Theorem And Quantum Space-time, Journal of the Franklin Institute, 330 (1993), 199--211
  • View Article
  • Google Scholar


• [19] S. A. Orszag, Accurate Solution Of The Orr-Sommerfeld Stability Equation, J. Fluid Mech., 50 (1971), 689--703
  • View Article
  • Google Scholar


• [20] T. Petrosky, I. Progogine, Poincare Resonances And The Limits Of Trajectory Dynamics, Proc. Nat. Acad. Sci. U.S.A., 90 (1993), 9393--9397
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [21] T. Petrosky, I. Progogine, Poincare Resonances And The Extension Of Classical Dynamics, Chaos Solitons Fractals , 7 (1996), 441--497
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [22] H. L. Swinney, J. P. Gollub, The Transition To Turbulence, Phys. Today, 41 (1978), 41--49
  • View Article
  • Google Scholar


• [23] J. M. T. Thompson, H. B. Stewart, Nonlinear Dynamics And Chaos, John Wiley, New York (1986)
  • Google Scholar