**ABSTRACT**

The effect of the induced magnetic field on the motion of Eyring-Powell nanofluid Al2O3, containing gyrotactic microorganisms through the boundary-layer is investigated. The viscoelastic dissipation is taken into consideration. The system is stressed by an external magnetic field. The continuity, momentum, induced magnetic field, temperature, concentration, and microorganisms equations that describe our problem are written in the form of 2-D non-linear differential equations. The system of non-linear PDE is transformed into ODE using appropriate similarity transformations with suitable boundary conditions and solved numerically by applying the NDSolve command in the MATHEMATICA program. The obtained numerical results for velocity, induced magnetic field, temperature, the nanoparticles concentration, and microorganisms are discussed and presented graphically through some figures. The physical parameters of the problem play an important role in the control of the obtained solutions. Moreover, it is obvious that as Grashof numbe increases, both the velocity, f ′, and the induced magnetic field, h’, increase, while, the reciprocal magnetic Prandtl number, A, works on decreasing both f ′ and h′. As Eckert number increases the temperature increases, while it decreases as the velocity ratio B increases.

**KEYWORDS**

PAPER SUBMITTED: 2020-04-08

PAPER REVISED: 2021-04-28

PAPER ACCEPTED: 2021-05-05

PUBLISHED ONLINE: 2021-06-05

**THERMAL SCIENCE** YEAR

**2022**, VOLUME

**26**, ISSUE

**Issue 1**, PAGES [411 - 422]

- W. A. Khan and I. Pop, ‘Boundary-layer flow of a nanofluid past a stretching sheet', Int. J. Heat Mass Transf., vol. 53, no. 11-12, pp. 2477-2483, 2010.
- N. Bachok, A. Ishak, and I. Pop, ‘Boundary-layer flow of nanofluids over a moving surface in a flowing fluid', Int. J. Therm. Sci., vol. 49, no. 9, pp. 1663-1668, 2010.
- M. Hassani, M. Mohammad Tabar, H. Nemati, G. Domairry, and F. Noori, ‘An analytical solution for boundary layer flow of a nanofluid past a stretching sheet', Int. J. Therm. Sci., vol. 50, no. 11, pp. 2256-2263, 2011.
- M. E. Ouaf, M.Y. Abou-zeid, ‘Electromagnetic and non-Darcian effects on a micropolar non-Newtonian fluid boundary-layer flow with heat and mass transfer', Int J Appl Electromagn Mech., DOI: 10.3233/JAE-201623.
- N. T. M. Eldabe, R. R. Rizkallah, M. Y. Abou-zeid and V. M. Ayad, Thermal diffusion and diffusion thermo effects of Eyring ‐ Powell nanofluid flow with gyrotactic microorganisms through the boundary layer, Heat Transfer — Asian Res., vol. 49, pp. 383 - 405, 2020.
- A. V. Kuznetsov and D. A. Nield, ‘Natural convective boundary-layer flow of a nanofluid past a vertical plate: A revised model', Int. J. Therm. Sci., vol. 77, pp. 126-129, 2014.
- D. A. Nield and A. V. Kuznetsov, ‘The Cheng-Minkowycz problem for the double-diffusive natural convective boundary layer flow in a porous medium saturated by a nanofluid', Int. J. Heat Mass Transf., vol. 54, no. 1-3, pp. 374-378, 2011.
- F. Mabood, W. A. Khan, and A. I. M. Ismail, ‘MHD boundary layer flow and heat transfer of nanofluids over a nonlinear stretching sheet: A numerical study', J. Magn. Magn. Mater., vol. 374, pp. 569-576, 2015.
- M. Y. Abou-zeid, ‘Magnetohydrodynamic Boundary Layer Heat Transfer To a Stretching Sheet Including Viscous Dissipation and Internal Heat Generation in a Porous Medium', J. Porous Media, vol. 14, no. 11, pp. 1007-1018, 2011.
- N. T. M. Eldabe, M. Y. Abou-zeid and O. S. Ahmed, Motion of a thin film of a fourth grade nanofluid with heat transfer down a vertical cylinder: homotopy perturbation method application, J. Adv. Res. Fluid Mech. Thermal Sci. 66 (2) 101-113, (2020).
- T. M. Agbaje, S. Mondal, S. S. Motsa, and P. Sibanda, ‘A numerical study of unsteady non-Newtonian Powell-Eyring nanofluid flow over a shrinking sheet with heat generation and thermal radiation', Alexandria Eng. J., vol. 56, no. 1, pp. 81-91, 2017.
- M. Y. Abou-zeid, Implicit homotopy perturabation method for mhd non-Newtonian nanofluid flow with Cattaneo-Christov heat flux due to parallel rotating disks, J. Nanofluids, 8 (8), 1648-1653, 2019.
- N. T. M. Eldabe, and M. Y. Abou-zeid, Radially varying magnetic field effect on peristaltic motion with heat and mass transfer of a non-Newtonian fluid between two co-axial tubes, Thermal Sci., vol. 22, pp. 2069-2080, 2018.
- N. T. Eldabe, G. M. Moatimid, M. Y. Abouzeid, A. A. ElShekhipy, N. F. Abdallah, A semianalytical technique for MHD peristalsis of pseudoplastic nanofluid with temperature-dependent viscosity: Application in drug delivery system, Heat Transfer - Asian Research, 49(1), 424-440, (2020)
- Mansour, H.M., Abou-zeid, M.Y., Heat and mass transfer effect on non-Newtonian fluid flow in a non-uniform vertical tube with peristalsis, Journal of Advanced Research in Fluid Mechanics and Thermal Sciences, 61(1), 44-62, 2019
- N. T. Eldabe, M. Y. Abou-Zeid, O. H. El-Kalaawy, S. M. Moawad, O. S. Ahmed, Electromagnetic Steady motion of Casson fluid with heat and mass transfer through porous medium past a shrinking surface, Thermal Science, 2021, 25(1), pp. 257-265
- M. M. Bhatti, S.Z. Alamri, R. Ellahi, S. I. Abdelsalam, ‘Intra‑uterine particle-fluid motion through a compliant asymmetrictapered channel with heat transfer', J. Therm. Anal. Calorim., published online, doi.org/10.1007/s10973-020-10233-9, 2020.
- N. T. M. Eldabe, B. M. Agoor, and H. Alame, ‘Peristaltic motion of non-Newtonian fluid with heat and mass transfer through a porous medium in channel under uniform magnetic field', J. Fluids, vol. 2014, pp. 1-12, 2014.
- L. Zhang , M. M. Bhatti, M. Marin, and K. S. Mekheimer, ‘Entropy analysis on the blood flow through anisotropically tapered ateries filled with Magnetic Zinc-Oxide (ZnO) Nanoparticles', Entropy. vol. 22, 2020.
- N. T. Eldabe, M. Y. Abou-zeid, H. A. Shawky, MHD peristaltic transport of Bingham blood fluid with heat and mass transfer through a non-uniform channel, Journal of advanced research in fluid mechanics and thermal science, 77(2): 145-159, 2021.
- N. T. M. Eldabe, M. A. Hassan and M. Y. Abou-zeid, Wall Properties effect on the peristaltic motion of a coupled stress fluid with heat and mass transfer through a porous media, J. Eng. Mech. 142 (3): 4015102, (2015).
- J. Rahimi, D. D. Ganji, M. Khaki, and K. Hosseinzadeh, ‘Solution of the boundary layer flow of an Eyring-Powell non-Newtonian fluid over a linear stretching sheet by collocation method', Alexandria Eng. J., vol. 56, no. 4, pp. 621-627, 2017.
- B. J. Gireesha, B. Mahanthesh, I. S. Shivakumara, and K. M. Eshwarappa, ‘Melting heat transfer in boundary layer stagnation-point flow of nanofluid toward a stretching sheet with induced magnetic field', Eng. Sci. Technol. an Int. J., vol. 19, no. 1, pp. 313-321, 2016.
- M. Sheikholeslami and H. B. Rokni, ‘Nanofluid two-phase model analysis in the existence of induced magnetic field', Int. J. Heat Mass Transf., vol. 107, pp. 288-299, 2017.
- M. Sheikholeslami, Q. M. Z. Zia, and R. Ellahi, ‘Influence of Induced Magnetic Field on Free Convection of Nanofluid Considering Koo-Kleinstreuer-Li (KKL) Correlation', Appl. Sci., vol. 6, no. 11, p. 324, 2016.
- M. A. Mohamed and . Y. Abou-Zeid, MHD peristaltic flow of micropolar Casson nanofluid through a porous medium between two co-axial tubes, J. Porous Media, vol. 22, no. (9), pp. 1079-1093, (2019).
- M. Y. Abou-zeid, Homotopy perturbation method for couple stresses effect on MHD peristaltic flow of a non-Newtonian nanofluid, Microsys. Technol., vol. 24, pp. 4839-4849, 2018.
- N. T.Eldabe, G. M. Moatimid, M. Y. Abouzeid, A. A. ElShekhipy, N. F. Abdallah, Instantaneous thermal-diffusion and diffusion-thermo effects on carreau nanofluid flow over a stretching porous sheet, Journal of Advanced Research in Fluid Mechanics and Thermal Sciences, Volume 72, Issue 2, 2020, Pages 142-157.
- T. Hayat, I. Ullah, A. Alsaedi, and M. Farooq, ‘MHD flow of Powell-Eyring nanofluid over a non-linear stretching sheet with variable thickness', Results Phys., vol. 7, pp. 189-196, 2017.
- N. T. Eldabe, M. E. Gabr, and S. A. Zaher, ‘Two-dimensional boundary layer flow with heat and mass transfer of magnetohydrodynamic non-Newtonian nanofluid through porous medium over a semi-infinite moving plate', Microsyst. Technol., vol. 24, no. 7, pp. 2919-2928, 2018.
- M. Y.Abou-zeid, ‘Homotopy perturbation method for MHD non-Newtonian nanofluid flow through a porous medium in eccentric annuli with peristalsis', Thermal Sci., vol. 21, no. 5, pp. 2069-2080, 2017.
- F. M. Ali, R. Nazar, N. M. Arifin, and I. Pop, ‘MHD stagnation-point flow and heat transfer towards stretching sheet with an induced magnetic field', Appl. Math. Mech. (English Ed., vol. 32, no. 4, pp. 409-418, 2011.
- W. A. Khan and O. D. Makinde, ‘MHD nanofluid bioconvection due to gyrotactic microorganisms over a convectively heat stretching sheet', Int. J. Therm. Sci., vol. 81, no. 1, pp. 118-124, 2014.
- N. T. M. Eldabe, M. Y. Abou-zeid, M. A. A. Mohamed and M. M. Abd-Elmoneim, MHD peristaltic flow of non-Newtonian power-law nanofluid through a non-Darcy porous medium inside a non-uniform inclined channel, Arch Appl Mech. 91, 1067-1077, (2021).
- M. Waqas, T. Hayat, S. A. Shehzad, and A. Alsaedi, ‘Transport of magnetohydrodynamic nanomaterial in a stratified medium considering gyrotactic microorganisms', Phys. B Condens. Matter, vol. 529, pp. 33-40, 2018.
- M. Y. Abou-zeid, ‘Homotopy Perturbation Method to Gliding Motion of Bacteria on a Layer of Power-Law Nanoslime with Heat Transfer', J. Comput. Theor. Nanosci., vol. 12, pp. 3605-3614, 2015.
- Z. Mehmood and Z. Iqbal, ‘Interaction of induced magnetic field and stagnation point flow on bioconvection nanofluid submerged in gyrotactic microorganisms', J. Mol. Liq., vol. 224, pp. 1083-1091, 2016.
- M. M. Bhatti, A. Zeeshan, and R. Ellahi, ‘Simultaneous effects of coagulation and variable magnetic field on peristaltically induced motion of Jeffrey nanofluid containing gyrotactic microorganism', Microvasc. Res., vol. 110, pp. 32-42, 2017.
- W. Ibrahim, ‘The effect of the induced magnetic field and convective boundary condition on MHD stagnation point flow and heat transfer of upper-convected Maxwell fluid in the presence of nanoparticle past a stretching sheet', Propuls. Power Res., vol. 5, no. 2, pp. 164-175, 2016.
- N. T. El-dabe, M.Y. Abou-zeid, M. A. Mohamed, and M.Maged, ‘Peristaltic flow of Herschel Bulkley nanofluid through a non-Darcy porous medium with heat transfer under slip condition', Int J Appl Electromagn Mech. Doi: 10.3233/JAE-201600, 2021.