Numerical investigation of fractional-order unsteady natural convective radiating flow of nanofluid in a vertical channel

M. Hamid; T. Zubair; M. Usman; R. U. Haq; M. Hamid; T. Zubair; M. Usman; R. U. Haq

doi:10.3934/math.2019.5.1416

AIMS Mathematics

2019, Volume 4, Issue 5: 1416-1429. doi: 10.3934/math.2019.5.1416

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Research article

Numerical investigation of fractional-order unsteady natural convective radiating flow of nanofluid in a vertical channel

M. Hamid ^{1
,
,},
T. Zubair ¹,
M. Usman ^{2,3,4
,
,},
R. U. Haq ⁵

1.
School of Mathematical Sciences, Peking University, Beijing 100871, China;
2.
BIC-ESAT, College of Engineering, Peking University, Beijing 100871, China;
3.
State Key Laboratory for Turbulence and Complex Systems, Department of Mechanics and Engineering Science, Peking University, Beijing 100871, China;
4.
Institute of Ocean Research, Peking University, Beijing 100871, China;
5.
Department of Electrical Engineering, Bahria University, Islamabad Campus, Islamabad 44000, Pakistan

Received: 15 April 2019 Accepted: 14 August 2019 Published: 18 September 2019
MSC : 34A08

In the current article, we analyzed the unsteady natural convection with the help of fractional approach. Firstly, the unsteady natural convection radiating flow in an open ended vertical channel beside the magnetic effects. We assumed the channel is stationary with non-uniform temperature. Secondly, we utilized a fractional calculus approach for the constitutive relationship of a fluid model. The modeled problem is transformed into nondimensional form via viable non-dimensional variables. In order to investigate the numerical solutions of non-dimensional system of partial differential equations finite difference approach coupled with Crank Nicolson method is developed and successfully applied. The beauty of Crank Nicolson finite difference scheme is, this scheme is unconditionally stable. A very careful survey of literate witnesses that this scheme has never been reported in the literary for fluid problems. The physical changes are discussed with the help of graphics. The expression for both velocity field and temperature distribution has been made via said scheme. A comprehensive discussion about the influence of various related dimensionless parameters upon the flow properties disclosed our work. It is observed that velocity field decreases as enhancing the magnetic field effects. Heat transfer enhanced as enhancing the nanoparticle volume fraction parameter. Velocity field and heat transfer shows the dominant behavior for the case of Cu-based nanofluid as compare to Al₂O₃ based nanofluid. Comparative study also included to show the accuracy of the proposed finite difference scheme. It is to be highlighted that the proposed scheme is very efficient and well-matched to investigate the solutions of modeled problem and can be extended to diversify problems of physical nature.
- finite difference method,
- fractional calculus,
- nanofluid,
- magnetic effects,
- thermal radiation
Citation: M. Hamid, T. Zubair, M. Usman, R. U. Haq. Numerical investigation of fractional-order unsteady natural convective radiating flow of nanofluid in a vertical channel[J]. AIMS Mathematics, 2019, 4(5): 1416-1429. doi: 10.3934/math.2019.5.1416

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Abstract

In the current article, we analyzed the unsteady natural convection with the help of fractional approach. Firstly, the unsteady natural convection radiating flow in an open ended vertical channel beside the magnetic effects. We assumed the channel is stationary with non-uniform temperature. Secondly, we utilized a fractional calculus approach for the constitutive relationship of a fluid model. The modeled problem is transformed into nondimensional form via viable non-dimensional variables. In order to investigate the numerical solutions of non-dimensional system of partial differential equations finite difference approach coupled with Crank Nicolson method is developed and successfully applied. The beauty of Crank Nicolson finite difference scheme is, this scheme is unconditionally stable. A very careful survey of literate witnesses that this scheme has never been reported in the literary for fluid problems. The physical changes are discussed with the help of graphics. The expression for both velocity field and temperature distribution has been made via said scheme. A comprehensive discussion about the influence of various related dimensionless parameters upon the flow properties disclosed our work. It is observed that velocity field decreases as enhancing the magnetic field effects. Heat transfer enhanced as enhancing the nanoparticle volume fraction parameter. Velocity field and heat transfer shows the dominant behavior for the case of Cu-based nanofluid as compare to Al₂O₃ based nanofluid. Comparative study also included to show the accuracy of the proposed finite difference scheme. It is to be highlighted that the proposed scheme is very efficient and well-matched to investigate the solutions of modeled problem and can be extended to diversify problems of physical nature.

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