A journal of IEEE and CAA , publishes high-quality papers in English on original theoretical/experimental research and development in all areas of automation
Advanced Search
Volume 3 Issue 3
Jul.  2016

IEEE/CAA Journal of Automatica Sinica

  • JCR Impact Factor: 15.3, Top 1 (SCI Q1)
    CiteScore: 23.5, Top 2% (Q1)
    Google Scholar h5-index: 77， TOP 5
Turn off MathJax
Article Contents
Article Navigation >  IEEE/CAA Journal of Automatica Sinica  > 2016  >  3(3): 257-260
Bruce J. West and Malgorzata Turalska, "The Fractional Landau Model," IEEE/CAA J. of Autom. Sinica, vol. 3, no. 3, pp. 257-260, 2016.
Citation: Bruce J. West and Malgorzata Turalska, "The Fractional Landau Model," IEEE/CAA J. of Autom. Sinica, vol. 3, no. 3, pp. 257-260, 2016.
shu

The Fractional Landau Model

  • 1. Information Science Directorate, Army Research Office, Research Triangle Park, NC 27708, USA;

  • 2. Department of Physics, Duke University, Durham, NC 27708, USA

    Abstract
  • Herein the Landau model of the transition from laminar to turbulent fluid flow is generalized to include the effect of memory. The original Landau model is quadratically nonlinear and memoryless, with turbulent fluctuations decaying exponentially. However, recent experiments show a dependence of the decay of fluctuations on memory, with the exponential being replaced by an inverse power law. This transition is explained herein as being due to critical slowing down. The fractional calculus is used to model this memory and to relate the index of the inverse power law decay to that of the fractional derivative in time.

     

    • FullText(HTML)
  • loading
    • References(25)
  • [1]
    Arnold V I. Mathematical Methods of Classical Mechanics(Second Edition). New York: Springer-Verlag, 1989.
    [2]
    Lichtenberg A J, Lieberman M A. Regular and Stochastic Motion. New York: Springer-Verlag, 1983.
    [3]
    Koopman B O. Hamiltonian systems and transformation in Hilbert space. Proceedings of the National Academy of Sciences of the United States of America, 1931, 17: 315-318
    [4]
    von Neumann J. Zur Operatorenmethode in der klassischen Mechanik. Annals of Mathematics, 1932, 33: 587-642
    [5]
    von Neumann J. Züsatze zur Arbeit zur Operatorenmethode. Annals of Mathematics, 1932, 33: 789-791
    [6]
    Kowalski K. Methods of Hilbert Spaces in the Theory of Nonlinear Dynamical Systems. Singapore: World Scientific, 1994.
    [7]
    Lindenberg K, West B J. The Nonequilibrium Statistical Mechanics of Open and Closed Systems. New York: Wiley-VCH, 1990.
    [8]
    Stiassnie M. On the application of fractional calculus for the formulation of viscoelastic models. Applied Mathematical Modelling, 1979, 3(4): 300-302
    [9]
    Magin R L. Fractional calculus models of complex dynamics in biological tissues. Computers & Mathematics with Applications, 2010, 59(5): 1586-1593
    [10]
    Henry B I, Langlands T A M, Wearne S L. Fractional cable models for spiny neuronal dendrites. Physical Review Letters, 2008, 100(12): 128103
    [11]
    Garra R. Fractional-calculus model for temperature and pressure waves in fluid-saturated porous rocks. Physical Review E, 2011, 84(3): 036605
    [12]
    Prajapati J C, Patel A D, Pathak K N, Shukla A K. Fractional calculus approach in the study of instability phenomenon in fluid dynamics. Palestine Journal of Mathematics, 2012, 1(2): 95-103
    [13]
    West B J. Colloquium: fractional calculus view of complexity: a tutorial. Reviews of Modern Physics, 2014, 86(4): 1169-1186
    [14]
    West B J. Fractional Calculus View of Complexity: Tomorrow's Science. New York: CRC Press, 2015.
    [15]
    Grigolini P, Rocco A, West B J. Fractional calculus as a macroscopic manifestation of randomness. Physical Review E, 1999, 59(3): 2603-2613
    [16]
    Monin A S, Yaglom A M, Lumley J L. Statistical Fluid Mechanics. Vol. 1: Mechanics of Turbulence. Cambridge, MA: The MIT Press, 1971.
    [17]
    Landau L D, Lifshitz E M. Fluid Mechanics (Second Edition). Vol. 6. Course of Theoretical Physics. U.S.A.: Butterworth-Heinemann, 1987.
    [18]
    Diethelm K, Ford N J, Freed A D, Luchko Y. Algorithms for the fractional calculus: a selection of numerical methods. Computer Methods in Applied Mechanics and Engineering, 2005, 194(6-8): 743-773
    [19]
    Cooper N G. Putting design into turbulence. 1663, Los Alamos Science & Technology Magazine. 2009. 10-15
    [20]
    Podlubny I. Fractional Differential Equations. New York: Academic Press, 1999.
    [21]
    Svenkeson A, Glaz B, Stanton S, West B J. Spectral decomposition of nonlinear systems with memory. to be published
    [22]
    West B J, Bologna M, Grigolini P. Physics of Fractal Operators. New York: Springer, 2003.
    [23]
    Turalska M, West B J. A search for a spectral technique to solve nonlinear fractional differential equations. to be published
    [24]
    Hald O H, Stinis P. Optimal prediction and the rate of decay for solutions of the Euler equations in two and three dimensions. Proceedings of the National Academy of Sciences of the United States of America, 2007, 104: 6527-6532
    [25]
    Stanislavsky A A. Hamiltonian formalism of fractional systems. The European Physical Journal B, 2006, 49(1): 93-101
    • Related Articles
    • Supplements(0)

Catalog

    通讯作者: 陈斌, bchen63@163.com
    • 1. 

      沈阳化工大学材料科学与工程学院 沈阳 110142

    1. 本站搜索
    2. 百度学术搜索
    3. 万方数据库搜索
    4. CNKI搜索
    Get Citation
    PDF
    XML

    Article Metrics

    Article views (1283) PDF downloads(165) Cited by()

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return