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The Impulsive Motion of Flat Plate in Generalized Second Grade Fluid with Anomalous Diffusion

Received: 26 September 2020     Accepted: 3 November 2020     Published: 11 December 2020
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Abstract

The flow adjacent to a wall rapidly set in motion for a generalized second-grade fluid with anomalous diffusion is examined. For the elucidation of such a fluid, the fractional-order derivative approach in the constitutive relationship model is presented because models based on ordinary differential equations have a relatively limited class of solutions, which does not provide compatible description of the complex systems in general. The current model of second-order fluid involving fractional calculus is based on the formal replacement of the first-order derivative in ordinary rheological constitutive equation by fractional derivative of a non-integer order. In addition, the time-fractional equation considered in this article describes the anomalous sub-diffusion. In this article, the velocity and stress field of generalized second-grade fluid with fractional anomalous diffusion are studied by fractional partial differential equations. Analytic solutions are given in closed form, from these differential equations in terms of the generalized G-functions or Fox's H-function with the discrete Laplace transform technique. Thus, many previous and classical results, namely, the solution of fractional diffusion equation obtained by Wyss, the classical Rayleigh’s time-space regularity solution, the relationship between velocity field and stress field obtained by Bagley and Torvik, are represented by particular cases of our proposed derivation.

Published in American Journal of Applied Mathematics (Volume 8, Issue 6)
DOI 10.11648/j.ajam.20200806.15
Page(s) 327-333
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2020. Published by Science Publishing Group

Keywords

Time Fractional Navier-Stokes Equation, Generalized Second Grade Fluid, Anomalous Diffusion, Fox's H-function

References
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Cite This Article
  • APA Style

    Mohammad Tanzil Hasan, Md. Shafiqul Islam, Mir Shariful Islam. (2020). The Impulsive Motion of Flat Plate in Generalized Second Grade Fluid with Anomalous Diffusion. American Journal of Applied Mathematics, 8(6), 327-333. https://doi.org/10.11648/j.ajam.20200806.15

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    ACS Style

    Mohammad Tanzil Hasan; Md. Shafiqul Islam; Mir Shariful Islam. The Impulsive Motion of Flat Plate in Generalized Second Grade Fluid with Anomalous Diffusion. Am. J. Appl. Math. 2020, 8(6), 327-333. doi: 10.11648/j.ajam.20200806.15

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    AMA Style

    Mohammad Tanzil Hasan, Md. Shafiqul Islam, Mir Shariful Islam. The Impulsive Motion of Flat Plate in Generalized Second Grade Fluid with Anomalous Diffusion. Am J Appl Math. 2020;8(6):327-333. doi: 10.11648/j.ajam.20200806.15

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  • @article{10.11648/j.ajam.20200806.15,
      author = {Mohammad Tanzil Hasan and Md. Shafiqul Islam and Mir Shariful Islam},
      title = {The Impulsive Motion of Flat Plate in Generalized Second Grade Fluid with Anomalous Diffusion},
      journal = {American Journal of Applied Mathematics},
      volume = {8},
      number = {6},
      pages = {327-333},
      doi = {10.11648/j.ajam.20200806.15},
      url = {https://doi.org/10.11648/j.ajam.20200806.15},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20200806.15},
      abstract = {The flow adjacent to a wall rapidly set in motion for a generalized second-grade fluid with anomalous diffusion is examined. For the elucidation of such a fluid, the fractional-order derivative approach in the constitutive relationship model is presented because models based on ordinary differential equations have a relatively limited class of solutions, which does not provide compatible description of the complex systems in general. The current model of second-order fluid involving fractional calculus is based on the formal replacement of the first-order derivative in ordinary rheological constitutive equation by fractional derivative of a non-integer order. In addition, the time-fractional equation considered in this article describes the anomalous sub-diffusion. In this article, the velocity and stress field of generalized second-grade fluid with fractional anomalous diffusion are studied by fractional partial differential equations. Analytic solutions are given in closed form, from these differential equations in terms of the generalized G-functions or Fox's H-function with the discrete Laplace transform technique. Thus, many previous and classical results, namely, the solution of fractional diffusion equation obtained by Wyss, the classical Rayleigh’s time-space regularity solution, the relationship between velocity field and stress field obtained by Bagley and Torvik, are represented by particular cases of our proposed derivation.},
     year = {2020}
    }
    

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    T1  - The Impulsive Motion of Flat Plate in Generalized Second Grade Fluid with Anomalous Diffusion
    AU  - Mohammad Tanzil Hasan
    AU  - Md. Shafiqul Islam
    AU  - Mir Shariful Islam
    Y1  - 2020/12/11
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    N1  - https://doi.org/10.11648/j.ajam.20200806.15
    DO  - 10.11648/j.ajam.20200806.15
    T2  - American Journal of Applied Mathematics
    JF  - American Journal of Applied Mathematics
    JO  - American Journal of Applied Mathematics
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    PB  - Science Publishing Group
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    UR  - https://doi.org/10.11648/j.ajam.20200806.15
    AB  - The flow adjacent to a wall rapidly set in motion for a generalized second-grade fluid with anomalous diffusion is examined. For the elucidation of such a fluid, the fractional-order derivative approach in the constitutive relationship model is presented because models based on ordinary differential equations have a relatively limited class of solutions, which does not provide compatible description of the complex systems in general. The current model of second-order fluid involving fractional calculus is based on the formal replacement of the first-order derivative in ordinary rheological constitutive equation by fractional derivative of a non-integer order. In addition, the time-fractional equation considered in this article describes the anomalous sub-diffusion. In this article, the velocity and stress field of generalized second-grade fluid with fractional anomalous diffusion are studied by fractional partial differential equations. Analytic solutions are given in closed form, from these differential equations in terms of the generalized G-functions or Fox's H-function with the discrete Laplace transform technique. Thus, many previous and classical results, namely, the solution of fractional diffusion equation obtained by Wyss, the classical Rayleigh’s time-space regularity solution, the relationship between velocity field and stress field obtained by Bagley and Torvik, are represented by particular cases of our proposed derivation.
    VL  - 8
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    ER  - 

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Author Information
  • Department of Mathematics, Bangabandhu Sheikh Mujibur Rahman Maritime University, Dhaka, Bangladesh

  • Department of Applied Mathematics, Dhaka University, Dhaka, Bangladesh

  • Department of Oceanography, Dhaka University, Dhaka, Bangladesh

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