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Theoretical analysis of the velocity field, stress field and vortex sheet of generalized second order fluid with fractional anomalous diffusion

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The velocity field of generalized second order fluid with fractional anomalous diiusion caused by a plate moving impulsively in its own plane is investigated and the anomalous diffusion problems of the stress field and vortex sheet caused by this process are studied. Many previous and classical results can be considered as particular cases of this paper, such as the solutions of the fractional diffusion equations obtained by Wyss; the classical Rayleigh’s time-space similarity solution; the relationship between stress field and velocity field obtained by Bagley and co-worker and Podlubny’s results on the fractional motion equation of a plate. In addition, a lot of significant results also are obtained. For example, the necessary condition for causing the vortex sheet is that the time fractional diffusion index β must be greater than that of generalized second order fluid α; the establiihment of the vorticity distribution function depends on the time history of the velocity profile at a given point, and the time history can be described by the fractional calculus.

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  1. Nonnemacher, T. F., Metzler, R., On the Riemann-Liouville fractional calculus and some recent applications, Fractals, 1995, 3(3): 557–566.

    Article  MathSciNet  Google Scholar 

  2. Mainardi, F., Fractional calculus: some basic problems in continuum and statistical mechanics, in Fractals and Fractional Calculus in Continuum Mechanics (eds. Cappinteri, A., Mainardi, F.), New York: Springer Wien, 1997, 291–348.

    Google Scholar 

  3. Roasikhin, Y. A., Shitikova, M. V., Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids, Appl. Mech. Rev., 1997, 50(1): 15–67.

    Article  Google Scholar 

  4. Podlubny, I., Fractional Differential Equations, San Diego: Academic Press, 1999, 86–231.

    MATH  Google Scholar 

  5. Henry, B. I., Wearne, S. L., Fractional reaction-diffusion, Physica A, 2000, 276(3): 448–455.

    Article  MathSciNet  Google Scholar 

  6. Wyss, W., The fractional diffusion equation, J. Math. Phys., 1986, 27(11): 2782–2785.

    Article  MATH  MathSciNet  Google Scholar 

  7. Bagley, R. L., Torvik, P. J., On the appearance of the fractional derivative in the behavior of real materials, J. Appl. Mech., 1984. 51(2): 294–298.

    Article  Google Scholar 

  8. Mathai, A. M., Saxena, R. K., The H-function with Applications in Statistics and Other Disciplines, New Delhi-Bangalore- Bombay: Wiley Eastern Limited, 1978, 1–12.

    MATH  Google Scholar 

  9. Gorenflo, R., Luchko. Y., Mainardi, F., Wright function as scale-invariant solutions of the diffusion-wave equation, J. Comput. Appl. Math., 2000, 118(1): 175–191.

    Article  MATH  MathSciNet  Google Scholar 

  10. Yih, C. S., Fluid Mechanics: A Concise Introduction to the Theory, New York: MeGraw-Hill, Inc., 1969, 321–324.

    Google Scholar 

  11. Wu Wangyi, Fluid Mechanics (in Chinese), Beijing: Peking Univ. Press, 1983, 226–230.

    Google Scholar 

  12. Mainad, F., Gorenflo, R., On Mittag-Leffler-Type function in fractional evolution processes. J. Comput. Appl. Math. 2000, 118(2): 283–299.

    Article  MathSciNet  Google Scholar 

  13. Anhand, V. V., Leonenko, N. N., Scaling law for fractional diffusion-wave equations with singular data, Statistics and Probability Letters, 2000, 48(3): 239–252.

    Article  MathSciNet  Google Scholar 

  14. Kivyakova, V., Multiple (multiindex) Mittag-Leffler functions and relations to generalized fractional calculus. J. Comput. Appl. Math., 2000, 118(1): 241–259.

    Article  MathSciNet  Google Scholar 

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Correspondence to Mingyu Xu.

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Xu, M., Tan, W. Theoretical analysis of the velocity field, stress field and vortex sheet of generalized second order fluid with fractional anomalous diffusion. Sci. China Ser. A-Math. 44, 1387–1399 (2001).

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