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

| Peer-Reviewed

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

Views: Downloads:

Download PDF

Share This Article

Twitter
Linked In
Facebook

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), 2024. Published by Science Publishing Group

Keywords

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

References

[1]	T. Hayat, A. M. Siddiqu and S. Asghar. Some simple flows of an Oldroyd-B fluid. Int. J. Eng. Sci., 39: 135-147, 2001.
[2]	W. C. Tan and M. Y. Xu. Unsteady flows of a generalized second grade fluid with the fractional derivative model between two parallel plates. Acta. Mech. Sin., 20 (5): 471-476, 2004.
[3]	T. Hayat, Naveed Ahmed, M. Sajid and S. Asghar. On the MHD flow of a second grade fluid in a porous channel. Comput. and Math. with Appl., 54: 407-414, 2007.
[4]	H. Schiessel, R. Metzler, A. Blumen and T. F. Nonnemacher. Generalized viscoelastic models: their fractional equations with solutions. J. Phys. A: Math. Gen., 28: 6567-6584, 1995.
[5]	L. I. Palade, P. Attane, R. R. Huilgol and B. Mena. Anomalous stability behavior of a properly invariant constitutive equation which generalises fractional derivative models. Int. J. Eng. Sci., 37: 315-329, 1999.
[6]	Y. A. Rossihin and M. V. Shitikova. A new method for solving dynamic problems of fractional derivative viscoelasticity. Int. J. Eng. Sci., 39: 149-176, 2001.
[7]	J. P. Bouchaud and A. Georges. Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications. Phys. Rep., 195: 127-293, 1990.
[8]	H. Spohn. Surface dynamics below the roughening transition. J. Phys., 3 (1): 69-81, 1993.
[9]	T. H. Solomon, E. R. Weeks and H. L. Swinney. Observation of anomalous diffusion and Lévy flights in a two-dimensional rotating flow. Phys. Rev. Lett., 71: 3975-3978, 1993.
[10]	J. Buckmaster. Viscous sheets advancing over dry beds. J. Fluid Mech., 81: 735-756, 1977.
[11]	J. R. Drake and J. G. Berryman. Theory of nonlinear diffusion of plasma across the magnetic field of a toroidal multipole. Phys. Fluids, 20 (5): 851-857, 1977.
[12]	F. Bardou, J. P. Bouchaud, O. Emile. A. Aspect and C. Cohen-Tannoudji. Subrecoil laser cooling and Lévy flights. Phys. Rev. Lett., 72: 203-206, 1994.
[13]	R. Hilfer. Applications of Fractional Calculus in Physics. World Scientific, Singapore, 2000.
[14]	R. Metzler and J. Klafter. The random walk’s guide to anomalous diffusion: A fractional dynamics approach. Phys. Rep., 339: 1-77, 2000.
[15]	S. S. Plotkin and P. G. Wolynes. Non-Markovian configurational diffusion and reaction coordinates for protein folding. Phys. Rev. Lett., 80: 5015-5018, 1998.
[16]	W. C. Tan and M. Y. Xu. The impulsive motion of flat plate in a generalized second grade fluid. Mech. Res. Commun., 29: 3-9, 2002.
[17]	Moustafa El-Shahed. On the impulsive motion of flat plate in a generalized second grade fluid. Z. Naturforsch., 59a: 829-837, 2004.
[18]	A. Khan, G. Zaman and O. Algahtani. Unsteady flow of viscoelastic fluid due to impulsive motion of plate. Asian journal of mathematics and applications, Article Id. ama0191, 9 pages, 2014.
[19]	R. S. Rivlin and J. L. Ericksen. Stress deformation relations for isotropic materials. J. Rational Mech. Anal., 4: 323-425, 1955.
[20]	A. M. Siddiqui, T. Hayat and S. Asghar. Periodic flows of a non-Newtonian fluid between two parallel plates. Int. J. Non-Linear Mech., 34: 895-899, 1999.
[21]	N. C. Ghosh, B. C. Ghosh and L. Debnath. The Hydromagnetic flow of a dusty viscoelastic fluid between two infinite parallel plates. Comput. Math. Appl., 39: 103-116, 2000.
[22]	P. D. Ariel. On exact solutions of flow problems of a second grade fluid through two parallel porous walls. Int. J. Eng. Sci., 40: 913-941, 2002.
[23]	R. L. Fosdick and K. R. Rajagopal. Anomalous features in the model of second order fluids. Arch. Ration. Mech. Anal., 70: 145-152, 1979.
[24]	A. Mahmood, C. Fetecau, N. A. Khan and M. Jamil. Some exact solutions of the oscillatory motion of a generalized second grade fluid in an annular region of two cylinders. Acta. Mech. Sin., 26: 541-550, 2010.
[25]	A. M. Mathai and R. K. Saxena. The H-function with applications in statistics and other disciplines. New Delhi-Bangalore-Bombay: Wiley Eastern Limited, 1978.
[26]	R. Gorenflo, Y. Luchkoa and F. Mainardi. Wright functions as scale-invariant solutions of the diffusion-wave equation. J. Comput. Appl. Math., 118 (1): 175-191, 2000.
[27]	G. K. Batchelor. An Introduction to Fluid Dynamics. Cambridge University press, Cambridge, 1967.
[28]	W. Wyss. The fractional diffusion equation. J. Math. Phys., 27 (11): 2782-2785, 1986.
[29]	R. L. Bagley and P. J. Torvik. On the appearence of fractional derivative in the behavior of real materials. J. Appl. Mech. 51, 294, 1984.

Cite This Article

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

Copy | Download

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

Copy | Download

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

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

Download PDF

Sections

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

Copy | Download

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

Copy | Download

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