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Exact solution on unsteady Couette flow of generalized Maxwell fluid with fractional derivative

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In this paper the unsteady Couette flow of a generalized Maxwell fluid with fractional derivative (GMF) is studied. The exact solution is obtained with the help of integral transforms (Laplace transform and Weber transform) and generalized Mittag-Leffler function. It was shown that the distribution and establishment of the velocity is governed by two non-dimensional parameters η, b and fractional derivative α of the model. The result of classical (Newtonian fluid and standard Maxwell fluid) Couette flow can be obtained as a special case of the result given by this paper, and the decaying of the unsteady part of GMF displays power law behavior, which has scale invariance.

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References

  • C. Friedrich (1991) ArticleTitleRelaxation and retardation functions of the Maxwell model with fractional derivatives Rheol. Acta 30 151–158 Occurrence Handle10.1007/BF01134604

    Article  Google Scholar 

  • R. L. Bagley (1983) ArticleTitleA theoretical basis for the application of fractional calculus to viscoelasticity J. Rheology 27 201–210 Occurrence Handle0515.76012 Occurrence Handle10.1122/1.549724

    Article  MATH  Google Scholar 

  • W. G. Glöckle T. F. Nonnenmacher (1994) ArticleTitleFractional relaxation and the time-temperature superposition principle Rheol. Acta 33 337–343 Occurrence Handle10.1007/BF00366960

    Article  Google Scholar 

  • Y. A. Rossikhin M. V. Shitikova (2000) ArticleTitleA new method for solving dynamic problems of fractional derivative viscoelasticity Int. J. Engng. Sci. 39 149–176 Occurrence Handle10.1016/S0020-7225(00)00025-2

    Article  Google Scholar 

  • Y. A. Rossikhin M. V. Shitikova (2001) ArticleTitleAnalysis of dynamic behaviour of viscoelastic rods whose rheological models contain fractional derivatives of two different orders ZAMM 81 IssueID6 363–376 Occurrence Handle1047.74026 Occurrence Handle1834711 Occurrence Handle10.1002/1521-4001(200106)81:6<363::AID-ZAMM363>3.0.CO;2-9

    Article  MATH  MathSciNet  Google Scholar 

  • F. Mainardi (1996) ArticleTitleFractional relaxation-oscillation and fractional diffusion-wave phenomena Chaos, Solitons & Fractals 7 IssueID9 1461–1477 Occurrence Handle1080.26505 Occurrence Handle1409912 Occurrence Handle10.1016/0960-0779(95)00125-5

    Article  MATH  MathSciNet  Google Scholar 

  • F. Mainardi R. Gorenflo (2000) ArticleTitleOn Mittag-Leffler-type functions in fractional evolution processes J. Comput. Appl. Math. 118 IssueID2 283–299 Occurrence Handle0970.45005 Occurrence Handle1765955 Occurrence Handle10.1016/S0377-0427(00)00294-6

    Article  MATH  MathSciNet  Google Scholar 

  • N. Makris M. C. Constantinou (1991) ArticleTitleFractional-derivative Maxwell model for viscous dampers J. Struct. Engng. ASCE 117 IssueID9 2708–2724

    Google Scholar 

  • H. Schiessel C. Friedrich A. Blumen (2000) Applications to problems in polymer physics and rheology R. Hilfer (Eds) Applications of fractional calculus in physics World Scientific Singapore 331

    Google Scholar 

  • W. C. Tan W. X. Pan M. Y. Xu (2003) ArticleTitleA note on unsteady flows of a viscoelatic fluid with the fractional Maxwell model between two parallel plates Int. J. Non-Linear Mech. 38 645–650 Occurrence Handle10.1016/S0020-7462(01)00121-4 Occurrence Handle05138171

    Article  MATH  Google Scholar 

  • D. D. Joseph (1990) Fluid dynamics of viscoelastic liquids Springer New York Occurrence Handle0698.76002

    MATH  Google Scholar 

  • D. Bernardin (1999) ArticleTitleTheoretical study of some transient Couette flows of viscoelastic fluid in inertial devices J. Non-Newtonian Fluid Mech. 88 1–30 Occurrence Handle0971.76010 Occurrence Handle10.1016/S0377-0257(99)00019-1

    Article  MATH  Google Scholar 

  • Y. Demirel (2000) ArticleTitleThermodynamic analysis of thermomechanical coupling in Couette flow Int. J. Heat Mass Transfer 43 4205–4212 Occurrence Handle1064.76525 Occurrence Handle10.1016/S0017-9310(00)00027-2

    Article  MATH  Google Scholar 

  • M. N. Özisik (1980) Heat conduction Wiley New York

    Google Scholar 

  • I. Podlubny (1999) Fractional differential equations Academic Press San Diego Occurrence Handle0924.34008

    MATH  Google Scholar 

  • X. Mingyu T. Wenchang (2001) ArticleTitleTheoretical analysis of the velocity field, stress field and vortex sheet of generalized second order fluid with fractional anomalous diffusion Science In China (Series A) 44 1387–1399 Occurrence Handle10.1007/BF02877067 Occurrence Handle1138.76320

    Article  MATH  Google Scholar 

  • A. M. Mathai R. K. Saxena (1978) The H-function with applications in statistics and other disciplines Wiley Eastern Limited New Delhi Occurrence Handle0382.33001

    MATH  Google Scholar 

  • G. H. Weiss (1994) Aspects and applications of the random walk Amsterdam North-Holland Occurrence Handle0925.60079

    MATH  Google Scholar 

  • C. S. Yih (1977) Fluid mechanics: a concise introduction to the theory, corrected ed West River Press Ann Arbor

    Google Scholar 

  • L. Preziosi D. D. Joseph (1987) ArticleTitleStokes' first problem for viscoelastic fluids J. Non-Newtonian Fluid Mech. 25 239–259 Occurrence Handle0683.76006 Occurrence Handle10.1016/0377-0257(87)85028-0

    Article  MATH  Google Scholar 

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  1. Institute of Applied Mathematics and System Science, Shandong University, Jinan, 250 100, P. R. China

    W. Shaowei & X. Mingyu

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  1. W. Shaowei
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  2. X. Mingyu
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Correspondence to X. Mingyu.

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Shaowei, W., Mingyu, X. Exact solution on unsteady Couette flow of generalized Maxwell fluid with fractional derivative. Acta Mechanica 187, 103–112 (2006). https://doi.org/10.1007/s00707-006-0332-9

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