Numerical investigations of nonlinear Maxwell fluid flow in the presence of non-Fourier heat flux theory: Keller box-based simulations

Afraz Hussain Majeed; Sadia Irshad; Bagh Ali; Ahmed Kadhim Hussein; Nehad Ali Shah; Thongchai Botmart; Afraz Hussain Majeed; Sadia Irshad; Bagh Ali; Ahmed Kadhim Hussein; Nehad Ali Shah; Thongchai Botmart

doi:10.3934/math.2023631

AIMS Mathematics

2023, Volume 8, Issue 5: 12559-12575. doi: 10.3934/math.2023631

Previous Article Next Article

Research article

Numerical investigations of nonlinear Maxwell fluid flow in the presence of non-Fourier heat flux theory: Keller box-based simulations

1.
Department of Mathematics, Air University, PAF Complex E-9, Islamabad 44000, Pakistan
2.
Institute of Mathematics, Khwaja Fareed University of Engineering and Information Technology, Rahim Yar Khan, Punjab 64200, Pakistan
3.
Faculty of Computer Science and Information Technology, Superior University, Lahore 54000, Pakistan
4.
Mechanical Engineering Department, College of Engineering, University of Babylon, Hilla 00964, Iraq
5.
College of Engineering, University of Warith Al-Anbiyaa, Karbala 56001, Iraq
6.
Department of Mechanical Engineering, Sejong University, Seoul 05006, South Korea
7.
Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand
† These authors contributed equally to this work and are co-first authors

Received: 18 October 2022 Revised: 17 November 2022 Accepted: 18 December 2022 Published: 27 March 2023
MSC : 76-10, 76R10

We investigate the thermal flow of Maxwell fluid in a rotating frame using a numerical approach. The fluid has been considered a temperature-dependent thermal conductivity. A non-Fourier heat flux term that accurately reflects the effects of thermal relaxation is incorporated into the model that is used to simulate the heat transfer process. In order to simplify the governing system of partial differential equations, boundary layer approximations are used. These approximations are then transformed into forms that are self-similar with the help of similarity transformations. The mathematical model includes notable quantities such as the rotation parameter $ \lambda $, Deborah number $ \beta $, Prandtl number Pr, parameter $ ϵ $ and the dimensionless thermal relaxation times $ \gamma $. These are approximately uniformly convergent. The Keller box method is used to find approximate solutions to ODEs. We observed due to the addition of elastic factors, the hydrodynamic boundary layer gets thinner. The thickness of the boundary layer can be reduced with the use of the k rotation parameter as well. When Pr increases, the wall slope of the temperature increases as well and approaches zero, which is an indication that Pr is decreasing. In addition, a comparison of the Cattaneo-Christov (CC) and Fourier models are provided and discussed.
- Maxwell fluid,
- CC model,
- rotating surface,
- heat flux,
- Keller box method
Citation: Afraz Hussain Majeed, Sadia Irshad, Bagh Ali, Ahmed Kadhim Hussein, Nehad Ali Shah, Thongchai Botmart. Numerical investigations of nonlinear Maxwell fluid flow in the presence of non-Fourier heat flux theory: Keller box-based simulations[J]. AIMS Mathematics, 2023, 8(5): 12559-12575. doi: 10.3934/math.2023631

Related Papers:

Abstract

We investigate the thermal flow of Maxwell fluid in a rotating frame using a numerical approach. The fluid has been considered a temperature-dependent thermal conductivity. A non-Fourier heat flux term that accurately reflects the effects of thermal relaxation is incorporated into the model that is used to simulate the heat transfer process. In order to simplify the governing system of partial differential equations, boundary layer approximations are used. These approximations are then transformed into forms that are self-similar with the help of similarity transformations. The mathematical model includes notable quantities such as the rotation parameter $ \lambda $, Deborah number $ \beta $, Prandtl number Pr, parameter $ ϵ $ and the dimensionless thermal relaxation times $ \gamma $. These are approximately uniformly convergent. The Keller box method is used to find approximate solutions to ODEs. We observed due to the addition of elastic factors, the hydrodynamic boundary layer gets thinner. The thickness of the boundary layer can be reduced with the use of the k rotation parameter as well. When Pr increases, the wall slope of the temperature increases as well and approaches zero, which is an indication that Pr is decreasing. In addition, a comparison of the Cattaneo-Christov (CC) and Fourier models are provided and discussed.

References

[1]	S. H. Han, L. C. Zheng, C. R. Li, X. X. Zhang, Coupled flow and heat transfer in viscoelastic fluid with Cattaneo-Christov heat flux model, Appl. Math. Lett., 38 (2014), 87–93. https://doi.org/10.1016/j.aml.2014.07.013 doi: 10.1016/j.aml.2014.07.013
[2]	M. Mustafa, Cattaneo-Christov heat flux model for rotating flow and heat transfer of upper-convected Maxwell fluid, AIP Adv., 5 (2015), 047109. https://doi.org/10.1063/1.4917306 doi: 10.1063/1.4917306
[3]	B. Krishnendu, H. Krishnendu, A. Ahmed, Dual solutions in boundary layer flow of Maxwell fluid over a porous shrinking sheet, Chinese Phys. B, 23 (2014), 124701. https://doi.org/10.1088/1674-1056/23/12/124701 doi: 10.1088/1674-1056/23/12/124701
[4]	C. Fetecau, M. Jamil, C. Fetecau, I. Siddique, A note on the second problem of Stokes for Maxwell fluids, Int. J. Non-Linear Mech., 44 (2009), 1085–1090. https://doi.org/10.1016/j.ijnonlinmec.2009.08.003 doi: 10.1016/j.ijnonlinmec.2009.08.003
[5]	C. Fetecau, M. Athar, C. Fetecau, Unsteady flow of generalized Maxwell fluid with fractional derivative due to a constantly accelerating plate, Comput. Math. Appl., 57 (2009), 596–603. https://doi.org/10.1016/j.camwa.2008.09.052 doi: 10.1016/j.camwa.2008.09.052
[6]	Y. Mahsud, N. A. Shah, D. Vieru, Influence of time-fractional derivatives on the boundary layer flow of Maxwell fluids, Chinese J. Phys., 55 (2017), 1340–1351. https://doi.org/10.1016/j.cjph.2017.07.006 doi: 10.1016/j.cjph.2017.07.006
[7]	I. Khan, N. A. Shah, L. C. C. Dennis, A scientific report on heat transfer analysis in mixed convection flow of Maxwell fluid over an oscillating vertical plate, Sci. Rep., 7 (2017), 40147. https://doi.org/10.1038/srep40147 doi: 10.1038/srep40147
[8]	S. K. Nandy, Unsteady flow of Maxwell fluid in the presence of nanoparticles toward a permeable shrinking surface with Navier slip, J. Taiwan Inst. Chem. Eng., 52 (2015) 22–30. https://doi.org/10.1016/j.jtice.2015.01.025 doi: 10.1016/j.jtice.2015.01.025
[9]	W. Na, N. A. Shah, I. Tlili, I. Siddique, Maxwell fluid flow between vertical plates with damped shear and thermal flux: free convection, Chinese J. Phys., 65 (2020), 367–376. https://doi.org/10.1016/j.cjph.2020.03.005 doi: 10.1016/j.cjph.2020.03.005
[10]	L. M. Cao, X. H. Si, L. C. Zheng, Convection of Maxwell fluid over stretching porous surface with heat source/sink in presence of nanoparticles: Lie group analysis, Appl. Math. Mech., 37 (2016), 433–442. https://doi.org/10.1007/s10483-016-2052-9 doi: 10.1007/s10483-016-2052-9
[11]	M. E. Karim, M. A. Samad, Effect of Brownian diffusion on squeezing elastico-viscous nanofluid flow with Cattaneo-Christov heat flux model in a channel with double slip effect, Appl. Math., 11 (2020), 277–291. https://doi.org/10.4236/am.2020.114021 doi: 10.4236/am.2020.114021
[12]	S. Shateyi, S. S. Motsa, Thermal radiation effects on heat and mass transfer over an unsteady stretching surface, Math. Probl. Eng., 2009 (2009), 1–13. https://doi.org/10.1155/2009/965603 doi: 10.1155/2009/965603
[13]	M. S. Abel, J. V. Tawade, J. N. Shinde, The effects of MHD flow and heat transfer for the UCM fluid over a stretching surface in presence of thermal radiation, Adv. Math. Phys., 2012 (2012), 1–21. https://doi.org/10.1155/2012/702681 doi: 10.1155/2012/702681
[14]	C. I. Christov, On frame indifferent formulation of the Maxwell-Cattaneo model of finite-speed heat conduction, Mech. Research Commun., 36 (2009), 481–486. https://doi.org/10.1016/j.mechrescom.2008.11.003 doi: 10.1016/j.mechrescom.2008.11.003
[15]	B. Straughan, Thermal convection with the Cattaneo-Christov model, Int. J. Heat Mass Transfer, 53 (2010), 95–98. https://doi.org/10.1016/j.ijheatmasstransfer.2009.10.001 doi: 10.1016/j.ijheatmasstransfer.2009.10.001
[16]	V. Tibullo, V. Zampoli, A uniqueness result for the Cattaneo-Christov heat conduction model applied to incompressible fluids, Mech. Research Commun., 38 (2011), 77–79. https://doi.org/10.1016/j.mechrescom.2010.10.008 doi: 10.1016/j.mechrescom.2010.10.008
[17]	S. A. M. Haddad, Thermal instability in Brinkman porous media with Cattaneo-Christov heat flux, Int. J. Heat Mass Transfer, 68 (2014), 659–668. https://doi.org/10.1016/j.ijheatmasstransfer.2013.09.039 doi: 10.1016/j.ijheatmasstransfer.2013.09.039
[18]	R. M. Khan, N. Imran, Z. Mehmood, M. Sohail, A Petrov-Galerkin finite element approach for the unsteady boundary layer upper-convected rotating Maxwell fluid flow and heat transfer analysis, Waves Random Complex Media, 2022 (2022), 1–18. https://doi.org/10.1080/17455030.2022.2055201 doi: 10.1080/17455030.2022.2055201
[19]	S. Bilal, A. H. Majeed, R. Mahmood, I. Khan, A. H. Seikh, E. S. M. Sherif, Heat and mass transfer in hydromagnetic second-grade fluid past a porous inclined cylinder under the effects of thermal dissipation, diffusion and radiative heat flux, Energies, 13 (2020), 1–17. https://doi.org/10.3390/en13010278 doi: 10.3390/en13010278
[20]	A. H. Majeed, S. Bilal, R. Mahmood, M. Y. Malik, Heat transfer analysis of viscous fluid flow between two coaxially rotated disks embedded in permeable media by capitalizing non-Fourier heat flux model, Phys. A, 540 (2020), 1231182 https://doi.org/10.1016/j.physa.2019.123182 doi: 10.1016/j.physa.2019.123182
[21]	S. Bilal, A. Tassaddiq, A. H. Majeed, K. S. Nisar, F. Ali, M. Y. Malik, Computational and physical examination about the aspects of fluid flow between two coaxially rotated disks by capitalizing non-Fourier heat flux theory: finite difference approach, Front. Phys., 7 (2019), 209. https://doi.org/10.3389/fphy.2019.00209 doi: 10.3389/fphy.2019.00209

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)