MHD flow and heat transfer of fractional nanofluids in a porous medium with ramped wall temperature and heat injection/consumption

i

This article has been updated. A Correction to this article was published on .

i

The Original Article was published on

i

The Original Article was published on

i

This article has been updated

i

This article was retracted on

i

This article has been updated

Abstract

This study focused on the unsteady magnetohydrodynamic (MHD) flow and heat transfer of fractional viscoelastic nanofluids over an infinite vertical plate within a porous medium. Both ramped and isothermal wall temperature conditions were considered, along with the effects of heat injection and consumption. The momentum equation was formulated based on a dual-parameter fractional Maxwell constitutive relation, while the energy equation incorporated a fractional dual-phase-lag (DPL) model. The resulting fractional integrodifferential governing equations were solved numerically using a finite difference method combined with the L1 algorithm and the weighted-shifted Grünwald difference scheme. The accuracy of the proposed numerical scheme was verified through manufactured solutions. Numerical results show that increasing porous medium permeability enhances fluid flow, whereas a stronger magnetic field suppresses it. The effects of the phase-lag parameters on the thermal boundary layer differ under ramped and isothermal wall temperature conditions: the phase lag of the temperature gradient leads to a monotonic thickening in the former but a nonmonotonic variation in the latter, whereas the phase lag of the heat flux exhibits an opposite trend. This study provides valuable insights into the application of fractional integrodifferential models for the design and optimization of thermal systems involving nanofluids.

Keywords

nanofluids / MHD flow / heat transfer / dual-parameter fractional Maxwell model / fractional dual-phase-lag model

CHANGE HISTORY

A Correction to this paper has been published:
https://doi.org

The Retraction Note was updated to reflect reception of statements of disagreement from all authors.

This article has been retracted.Please see the Retraction Notice for more detail:
https://doi.org

References

[1] S. U. S. Choi, J. A. Eastman, Enhancing thermal conductivity of fluids with nanoparticles, in ASME’s International Mechanical Engineering Congress & Exposition , ASME, 66(1995), 99-105. https://doi.org/10.1115/IMECE1995-0926

[2] W. Yu, D. M. France, J. L. Routbort, S. U. S. Choi, Review and comparison of nanofluid thermal conductivity and heat transfer enhancements, Heat Transfer Eng., 29(2008), 432-460. https://doi.org/10.1080/01457630701850851

[3] M. U. Sajid, H. M. Ali, Thermal conductivity of hybrid nanofluids: A critical review, Int. J. Heat Mass Transfer, 126(2018), 211-234. https://doi.org/10.1016/j.ijheatmasstransfer.2018.05.021

[4] J. A. Eastman, S. U. S. Choi, S. Li, W. Yu, L. J. Thompson, Anomalously increased effective thermal conductivities of ethylene glycol-based nanofluids containing copper nanoparticles, Appl. Phys. Lett., 78(2001), 718-720. https://doi.org/10.1063/1.1341218

[5] O. Mahian, L. Kolsi, M. Amani, P. Estellé, G. Ahmadi, C. Kleinstreuer, et al., Recent advances in modeling and simulation of nanofluid flows-Part I: Fundamentals and theory, Phys. Rep., 790(2019), 1-48. https://doi.org/10.1016/j.physrep.2018.11.004

[6] D. D. Kumar, A. V. Arasu, A comprehensive review of preparation, characterization, properties and stability of hybrid nanofluids, Renewable Sustainable Energy Rev., 81(2018), 1669-1689. https://doi.org/10.1016/j.rser.2017.05.257

[7] P. C. Mishra, S. Mukherjee, S. K. Nayak, A. Panda, A brief review on viscosity of nanofluids, Int. Nano Lett., 4(2014), 109-120. https://doi.org/10.1007/s40089-014-0126-3

[8] J. Zhao, L. Zheng, X. Zhang, F. Liu, Unsteady natural convection boundary layer heat transfer of fractional Maxwell viscoelastic fluid over a vertical plate, Int. J. Heat Mass Transfer, 97(2016), 760-766. https://doi.org/10.1016/j.ijheatmasstransfer.2016.02.059

[9] M. Shen, L. Chen, M. Zhang, F. Liu, A renovated Buongiorno’s model for unsteady Sisko nanofluid with fractional Cattaneo heat flux, Int. J. Heat Mass Transfer, 126(2018), 277-286. https://doi.org/10.1016/j.ijheatmasstransfer.2018.05.131

[10] L. Liu, L. Feng, Q. Xu, L. Zheng, F. Liu, Flow and heat transfer of generalized Maxwell fluid over a moving plate with distributed order time fractional constitutive models, Int. Commun. Heat Mass Transfer, 116(2020), 104679. https://doi.org/10.1016/j.icheatmasstransfer.2020.104679

[11] L. Feng, F. Liu, I. Turner, V. V. Anh, Magnetohydrodynamics flow and heat transfer of novel generalized Kelvin-Voigt viscoelastic nanofluids over a moving plate, Phys. Fluids, 36(2024), 063109. https://doi.org/10.1063/5.0213855

[12] Z. Mao, L. Feng, I. Turner, A. Xiao, F. Liu, Transient free convective flow of viscoelastic nanofluids governed by fractional integrodifferential equations under Newtonian heating and thermal radiation, Chin. J. Phys., 93(2025), 584-600. https://doi.org/10.1016/j.cjph.2024.12.025

[13] M. Omama, A. A. Arafa, A. Elsaid, W. K. Zahra, Numerical analysis of double-fractional PDEs in MHD hybrid nanofluid blood flow with slip velocity, heat source, and radiation effects, Phys. Scr., 100(2024), 015288. https://doi.org/10.1088/1402-4896/ada06f

[14] H. Sun, Y. Jiang, Y. Zhang, L. Jiang, A review of constitutive models for non-Newtonian fluids, Fract. Calc. Appl. Anal., 27(2024), 1483-1526. https://doi.org/10.1007/s13540-024-00294-0

[15] L. Feng, I. Turner, V. V. Anh, F. Liu, Review of classical and nonlocal nanofluid models for solar collectors, Renewable Sustainable Energy Rev., 212(2025), 115382. https://doi.org/10.1016/j.rser.2025.115382

[16] E. R. Priest, T. G. Forbes, The magnetic nature of solar flares, Astron. Astrophys. Rev., 10(2002), 313-377. https://doi.org/10.1007/s001590100013

[17] K. Rafique, Z. Mahmood, U. Khan, Mathematical analysis of MHD hybrid nanofluid flow with variable viscosity and slip conditions over a stretching surface, Mater. Today Commun., 36(2023), 106692. https://doi.org/10.1016/j.mtcomm.2023.106692

[18] A. J. Chamkha, A. M. Aly, MHD free convection flow of a nanofluid past a vertical plate in the presence of heat generation or absorption effects, Chem. Eng. Commun., 198(2010), 425-441. https://doi.org/10.1080/00986445.2010.520232

[19] T. Anwar, P. Kumam, D. Baleanu, I. Khan, P. Thounthong, Radiative heat transfer enhancement in MHD porous channel flow of an Oldroyd-B fluid under generalized boundary conditions, Phys. Scr., 95(2020), 115211. https://doi.org/10.1088/1402-4896/abbe50

[20] G. Dharmaiah, O. D. Makinde, K. S. Balamurugan, Influence of magneto hydro dynamics (MHD) nonlinear radiation on micropolar nanofluid flow over a stretching surface: Revised Buongiorno’s nanofluid model, J. Nanofluids, 11(2022), 1009-1022. https://doi.org/10.1166/jon.2022.1890

[21] S. G. Bejawada, M. M. Nandeppanavar, Effect of thermal radiation on magnetohydrodynamics heat transfer micropolar fluid flow over a vertical moving porous plate, Exp. Comput. Multiphase Flow, 5(2023), 149-158. https://doi.org/10.1007/s42757-021-0131-5

[22] S. Saleem, T. Abbas, H. Abutuqayqah, E. U. Haq, S. U. Khan, Numerical simulation accompanied by an intelligent computing system for the chemical reaction of Casson nanofluid and radiative heat flux on a nonlinear stretching surface, Alexandria Eng. J., 79(2023), 629-643. https://doi.org/10.1016/j.aej.2023.08.016

[23] Z. Sheng, Y. Liu, Y. Li, Unsteady magnetohydrodynamic flow, heat, and mass transfer of the fractional Oldroyd-B fluid along a moving infinite vertical plate, Phys. Fluids, 37(2025), 113114. https://doi.org/10.1063/5.0293008

[24] S. Kavya, V. Nagendramma, N. A. Ahammad, S. Ahmad, C. S. K. Raju, N. A. Shah, Magnetic-hybrid nanoparticles with stretching/shrinking cylinder in a suspension of MoS4 and copper nanoparticles, Int. Commun. Heat Mass Transfer, 136(2022), 106150. https://doi.org/10.1016/j.icheatmasstransfer.2022.106150

[25] P. Jalili, A. A. Azar, B. Jalili, D. D. Ganji, Study of nonlinear radiative heat transfer with magnetic field for non-Newtonian Casson fluid flow in a porous medium, Results Phys., 48(2023), 106371. https://doi.org/10.1016/j.rinp.2023.106371

[26] C. P. Malhotra, R. L. Mahajan, W. S. Sampath, K. L. Barth, R. A. Enzenroth, Control of temperature uniformity during the manufacture of stable thin-film photovoltaic devices, In ASME International Mechanical Engineering Congress and Exposition, 4711(2004), 547-555. https://doi.org/10.1115/IMECE2004-61331

[27] N. Ahmed, M. Dutta, Transient mass transfer flow past an impulsively started infinite vertical plate with ramped plate velocity and ramped temperature, Int. J. Phys. Sci., 8(2013), 254-263. https://doi.org/10.5897/IJPS12.713

[28] P. Chandran, N. C. Sacheti, A. K. Singh, Natural convection near a vertical plate with ramped wall temperature, Heat Mass Transfer, 41(2005), 459-464. https://doi.org/10.1007/s00231-004-0568-7

[29] G. S. Seth, M. S. Ansari, R. Nandkeolyar, MHD natural convection flow with radiative heat transfer past an impulsively moving plate with ramped wall temperature, Heat Mass Transfer, 47(2011), 551-561. https://doi.org/10.1007/s00231-010-0740-1

[30] S. Rao, P. N. Deka, Numerical analysis of MHD hybrid nanofluid flow a porous stretching sheet with thermal radiation, Int. J. Appl. Comput. Math., 10(2024), 95. https://doi.org/10.1007/s40819-024-01734-4

[31] M. VeeraKrishna, A. J. Chamkha, Hall effects on unsteady MHD flow of second grade fluid through porous medium with ramped wall temperature and ramped surface concentration, Phys. Fluids, 30(2018), 053101. https://doi.org/10.1063/1.5025542

[32] Y. D. Reddy, B. S. Goud, M. A. Kumar, Radiation and heat absorption effects on an unsteady MHD boundary layer flow along an accelerated infinite vertical plate with ramped plate temperature in the existence of slip condition, Partial Differ. Equ. Appl. Math., 4(2021), 100166. https://doi.org/10.1016/j.padiff.2021.100166

[33] Z. Mao, L. Feng, A. Xiao, F. Liu, I. Turner, Magnetohydrodynamic transient flow of dual-parameter fractional Maxwell nanofluids past a vertical plate with generalised dual-phase-lagging heat conduction under ramped wall temperature conditions, Eng. Comput., 42(2026), 51. https://doi.org/10.1007/s00366-025-02267-0

[34] B. Jin, Fractional Differential Equations , Springer International Publishing, 2021. https://doi.org/10.1007/978-3-030-76043-4

[35] F. Mainardi, G. Spada, Creep, relaxation and viscosity properties for basic fractional models in rheology, Eur. Phys. J. Spec. Top., 193(2011), 133-160. https://doi.org/10.1140/epjst/e2011-01387-1

[36] Applications of Fractional Calculus in Physics , (ed. R. Hilfer), World Scientific, 2000. https://doi.org/10.1142/3779

[37] H. Schiessel, R. Metzler, A. Blumen, T. F. Nonnenmacher, Generalized viscoelastic models: Their fractional equations with solutions, J. Phys. A: Math. Gen., 28(1995), 6567-6584. https://doi.org/10.1088/0305-4470/28/23/012

[38] C. Friedrich, Relaxation and retardation functions of the Maxwell model with fractional derivatives, Rheol. Acta, 30(1991), 151-158. https://doi.org/10.1007/BF01134604

[39] W. Tan, W. Pan, M. Xu, A note on unsteady flows of a viscoelastic fluid with the fractional Maxwell model between two parallel plates, Int. J. Non-Linear Mech., 38(2003), 645-650. https://doi.org/10.1016/S0020-7462(01)00121-4

[40] S. Rosseland, Astrophysics: On an Atomic-Theoretical Basis , Springer-Verlag, Berlin, 1931.

[41] E. Magyari, A. Pantokratoras, Note on the effect of thermal radiation in the linearized Rosseland approximation on the heat transfer characteristics of various boundary layer flows, Int. Commun. Heat Mass Transfer, 38(2011), 554-556. https://doi.org/10.1016/j.icheatmasstransfer.2011.03.006

[42] D. Y. Tzou, The generalized lagging response in small-scale and high-rate heating, Int. J. Heat Mass Transfer, 38(1995), 3231-3240. https://doi.org/10.1016/0017-9310(95)00052-B

[43] T. N. Mishra, K. N. Rai, Numerical solution of FSPL heat conduction equation for analysis of thermal propagation, Appl. Math. Comput., 273(2016), 1006-1017. https://doi.org/10.1016/j.amc.2015.10.082

[44] Z. M. Odibat, N. T. Shawagfeh, Generalized Taylor’s formula, Appl. Math. Comput., 186(2007), 286-293. https://doi.org/10.1016/j.amc.2006.07.102

[45] C. Ji, W. Dai, Z. Sun, Numerical method for solving the time-fractional dual-phase-lagging heat conduction equation with the temperature-jump boundary condition, J. Sci. Comput., 75(2018), 1307-1336. https://doi.org/10.1007/s10915-017-0588-3

[46] H. Xu, X. Jiang, Time fractional dual-phase-lag heat conduction equation, Chin. Phys. B, 24(2015), 034401. https://doi.org/10.1088/1674-1056/24/3/034401

[47] Y. D. Reddy, B. S. Goud, Comprehensive analysis of thermal radiation impact on an unsteady MHD nanofluid flow across an infinite vertical flat plate with ramped temperature with heat consumption, Results Eng., 17(2023), 100796. https://doi.org/10.1016/j.rineng.2022.100796

[48] M. Chandrasekar, S. Suresh, R. Srinivasan, A. C. Bose, New analytical models to investigate thermal conductivity of nanofluids, J. Nanosci. Nanotechnol., 9(2009), 533-538. https://doi.org/10.1166/jnn.2009.j025

[49] L. Godson, B. Raja, D. M. Lal, S. Wongwises, Experimental investigation on the thermal conductivity and viscosity of silver-deionized water nanofluid, Exp. Heat Transfer, 23(2010), 317-332. https://doi.org/10.1080/08916150903564796

[50] E. Abu-Nada, Application of nanofluids for heat transfer enhancement of separated flows encountered in a backward facing step, Int. J. Heat Fluid Flow, 29(2008), 242-249. https://doi.org/10.1016/j.ijheatfluidflow.2007.07.001

[51] H. Schlichting, K. Gersten, Boundary-Layer Theory , Springer, Berlin, 2016. https://doi.org/10.1007/978-3-662-52919-5

[52] S. Das, A. Sensharma, R. N. Jana, R. P. Sharma, Slip flow of nanofluid past a vertical plate with ramped wall temperature considering thermal radiation, J. Nanofluids, 6(2017), 1054-1064. https://doi.org/10.1166/jon.2017.1392

[53] B. Jin, Z. Zhou, Numerical Treatment and Analysis of Time-Fractional Evolution Equations , Springer Cham, 2023. https://doi.org/10.1007/978-3-031-21050-1

[54] Z. Wang, S. Vong, Compact difference schemes for the modified anomalous fractional sub-diffusion equation and the fractional diffusion-wave equation, J. Comput. Phys., 277(2014), 1-15. https://doi.org/10.1016/j.jcp.2014.08.012

[55] M. Khan, A. Rasheed, M.S. Anwar, S. T. Hussain Shah, Application of fractional derivatives in a Darcy medium natural convection flow of MHD nanofluid, Ain Shams Eng. J., 14(2023), 102093. https://doi.org/10.1016/j.asej.2022.102093

[56] L. Liu, Y. Liang, X. Bao, H. Fang, An efficient adaptive grid method for a system of singularly perturbed convection-diffusion problems with Robin boundary conditions, Adv. Differ. Equ., 2021(2021), 6. https://doi.org/10.1186/s13662-020-03166-y

[57] Z. Mao, D. Luo, A robust adaptive grid method for first-order nonlinear singularly perturbed Fredholm integro-differential equations, Networks Heterogen. Media, 18(2023), 1006-1023. https://doi.org/10.3934/nhm.2023044

[58] S. Jiang, J. Zhang, Q. Zhang, Z. Zhang, Fast evaluation of the Caputo fractional derivative and its applications to fractional diffusion equations, Commun. Comput. Phys., 21(2015), 650-678. https://doi.org/10.4208/cicp.OA-2016-0136

[59] X. Gu, S. Wu, A parallel-in-time iterative algorithm for Volterra partial integro-differential problems with weakly singular kernel, J. Comput. Phys., 417(2020), 109576. https://doi.org/10.1016/j.jcp.2020.109576

[60] J. Buongiorno, Convective transport in nanofluids, J. Heat Transfer, 128(2006), 240-250. https://doi.org/10.1115/1.2150834