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

Mohammad Tanzil Hasan; Md. Shafiqul Islam; Mir Shariful Islam

doi:doi:10.11648/j.ajam.20200806.15

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

Mohammad Tanzil Hasan, Md. Shafiqul Islam, Mir Shariful Islam

Published in American Journal of Applied Mathematics (Volume 8, Issue 6)

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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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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TY  - JOUR
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
PY  - 2020
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
SP  - 327
EP  - 333
PB  - Science Publishing Group
SN  - 2330-006X
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
IS  - 6
ER  -

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Author Information

Mohammad Tanzil Hasan

Department of Mathematics, Bangabandhu Sheikh Mujibur Rahman Maritime University, Dhaka, Bangladesh
Md. Shafiqul Islam

Department of Applied Mathematics, Dhaka University, Dhaka, Bangladesh
Mir Shariful Islam

Department of Oceanography, Dhaka University, Dhaka, Bangladesh

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Plain Text BibTeX RIS

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

Copy | Download

@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}
}

Copy | Download

TY  - JOUR
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
PY  - 2020
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
SP  - 327
EP  - 333
PB  - Science Publishing Group
SN  - 2330-006X
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
IS  - 6
ER  -

Copy | Download