Engineering Transactions, 69, 3, pp. 271–292, 2021

Aspects of the Aligned Magnetic Field Past a Stratified Inclined Sheet with Nonlinear Convection and Variable Thermal Conductivity

Muhammad BILAL
The University of Lahore

Fatima Jinnah Women University

Fatima Jinnah Women University

The University of Lahore

An analytical study of a two-dimensional boundary layer flow of an upper-convected Maxwell fluid is examined. In addition, an aligned magnetic field over the inclined shrinking/stretching stratified sheet in a non-Darcian porous medium is considered. The heat transfer effects are employed through nonlinear convection and variable thermal conductivity. The associated higher-order nonlinear equations are transformed to ordinary first-order differential equations by using similarity transformation. The resulting ordinary first-order differential equations are then solved numerically by the shooting method. This paper aims to investigate the special effects of parameters on velocity and temperature profiles. The results are also discussed, graphically and numerically for the skin friction and Nusselt number.
Keywords: inclined sheet; stratification; nonlinear convection; Darcy-Forchheimer flow; aligned magnetic field
Full Text: PDF
Copyright © The Author(s). This is an open-access article distributed under the terms of the Creative Commons Attribution-ShareAlike 4.0 International (CC BY-SA 4.0).


Shankar D.G., Raju C.S.K., Kumar M.S., Makinde O.D., Cattaneo-Christov heat flux on an MHD 3D free convection Casson fluid flow over a stretching sheet, Engineering Transactions, 68(3):223–238, 2020, doi: 10.24423/engtrans.1099.20200720.

Sakidis B.C., Boundary layer behavior on continuous solid surfaces, AIChE Journal, 7(1): 26–28, 1961, doi: 10.1002/aic.690070108.

Chinyoka T., Makinde O.D., Numerical analysis of the transient and non-isothermal channel flow of a third-grade fluid with convective cooling, Engineering Transactions, 68(4): 335–351, 2020, doi: 10.24423/engtrans.1182.20200720.

Hsiao K.-L., Micropolar nanofluid flow with MHD and viscous dissipation effects towards a stretching sheet with multimedia feature, International Journal of Heat and Mass Transfer, 112: 983–990, 2017, doi: 10.1016/j.ijheatmasstransfer.2017.05.042.

Hsiao K-.L., Combined electrical MHD heat transfer thermal extrusion system using Maxwell fluid with radiative and viscous dissipation effects, Applied Thermal Engineering, 112(5): 1281–1289, 2017, doi: 10.1016/j.applthermaleng.2016.08.208.

Bilal M., Mazhar S.Z., Ramzan M., Mehmood Y., Time dependent hydro-magnetic stagnation point flow of a Maxwell nanofluid with melting heat effect and amended Fourier and Fick's laws, Heat Transfer, 50(5): 4417–4434, 2021, doi: 10.1002/htj.22081.

Makinde O.D., Eegunjobi A.S., Entropy analysis of a variable viscosity MHD Couette flow between two concentric pipes with convective cooling, Engineering Transactions, 68(4): 317–334, 2020, doi: 10.24423/EngTrans.1104.20200720.

Jang H.K., Hong S.O., Lee S.B., Kim J.M., Hwang W.R., Viscosity measurement of non-Newtonian fluids in pressure-driven flows of general geometries based on energy dissipation rate, Journal of Non-Newtonian Fluid Mechanics, 274: 104204, 2019, doi :10.1016/j.jnnfm.2019.104204.

Lenci A., Chiapponi L., An experimental setup to investigate non-Newtonian fluid flow in variable aperture channels, Water, 12(5): 1284, 2020, doi: 10.3390/w12051284.

Rajagopal K.R., Gupta A.S., Wineman A.S. On a boundary layer theory for non-Newtonian fluids, International Journal of Engineering Science, 18(6): 875–883, 1980, doi: 10.1016/0020-7225(80)90035-X.

Eldesoky I.M., Abdelsalam S.I., El-Askary W.A., Ahmed M.M., Concurrent development of thermal energy with magnetic field on a particle-fluid suspension through a porous conduit, Bio- NanoScience, 9: 186–202, 2019, doi:10.1007/s12668-018-0585-5.

Abdelsalam S.I., Bhatti M.M., New insight into Au NP applications in tumour treatment and cosmetics through wavy annuli at the nanoscale, Scientific Reports, 9: 260, 2019, doi:10.1038/s41598-018-36459-0.

Bhatti M.M., Marin M., Zeeshan A., Ellahi R., Abdelsalam S.I., Swimming of motile gyrostactic microorganisms and nanoparticles in blood flow through anisotropically tapered arteries, Frontiers in Physics, 8: 95, 2020, doi: 10.3389/fphy.2020.00095.

Bilal M., Urva Y., Analysis of non-Newtonian fluid flow over fine rotating thin needle for variable viscosity and activation energy, Archive of Applied Mechanics, 91(3):1079–1095, 2020, doi: 10.1007/s00419-020-01811-2.

Abdelsalam S.I., Bhatti, M.M., Anomalous reactivity of thermo-bio convective nanofluid towards oxytactic microorganisms, Applied Mathematics and Mechanics, English Edition, 41(5): 711–724, 2020, doi: 10.1007/s10483-020-2609-6.

Bilal M., Ashbar S., Flow and heat transfer analysis of Eyring-Powell fluid over stratified sheet with mixed convection, Journal of the Egyptian Mathematical Society, 28: 40, 2020, doi: 10.1186/s42787-020-00103-6.

Abdelsalamab S.I., Vafai K., Combined effects of magnetic field and rheological properties on the peristaltic flow of a compressible fluid in a microfluidic channel, European Journal of Mechanics – B/Fluids, 65: 398–411, 2017, doi: 10.1016/j.euromechflu.2017.02.002.

Akbar N.S., Khan Z.H., Effect of variable thermal conductivity and thermal radiation with CNTS suspended nanofluid over a stretching sheet with convective slip boundary conditions: Numerical study, Journal of Molecular Liquids, 222, 279–286, 2016, doi: 10.1016/j.molliq.2016.06.102.

Rana S., Mehmood R., Akbar N.S., Mixed convective oblique ow of a Casson fluid with partial slip, internal heating and homogeneous heterogeneous reactions, Journal of Molecular Liquids, 222: 1010–1019, 2016, doi: 10.1016/j.molliq.2016.07.137.

Lu D.-C., Ramzan M., Bilal M., Chung J.D., Farooq U., A numerical investigation of 3D MHD rotating flow with binary chemical reaction, activation energy and non-Fourier heat flux, Communication in Theoretical Physics, 70(1): 89–96, 2018, doi: 10.1088/0253-6102/70/1/89.

Bilal M., Sagheer M., Hussain S., On MHD 3D upper convected Maxwell fluid flow with thermophoretic effect using non-linear radiative heat flux, Canadian Journal of Physics, 96(1): 1–10, 2018, doi: 10.1139/cjp-2017-0250.

Lu D.-C, Ramzan M., Bilal M., Chung J.D., Farooq U., Upshot of chemical species and nonlinear thermal radiation on Oldroyd-B nanofluid flow past a bidirectional stretched surface with heat generation/absorption in a porous media, Communication in Theoretical Physics, 70(1): 71–80, 2018, doi: 10.1088/0253-6102/70/1/71.

Forchheimer P.H., Wasserbewegun Durch Boden Zeitschrift des Vereines Deutscher Ingenieure [in German], 49: 1736–1749, 1901.

Izbash, S.V., O Filtracii v Kropnozernstom Materiale [in Russian], Leningrad, USSR 1931.

Alsabery A.I., Tayebi T., Chamkha A.J., Hashim I., Natural convection of Al2O3–water nanofluid in a non-Darcian wavy porous cavity under the local thermal non-equilibrium condition, Scientific Reports, 10: 18048, 2020, doi: 10.1038/s41598-020-75095-5.

Rasool G., Shafiq A., Baleanu D., Consequences of Soret-Dufour effects, thermal radiation, and binary chemical reaction on Darcy-Forchheimer flow of nanofluids, Symmetry, 12(9): 1421, 2020, doi: 10.3390/sym12091421.

Sadiq M. A., Haider F., Hayat T., , Alsaedi A., Partial slip in Darcy-Forchheimer carbon nanotubes flow by rotating disk, International Communications in Heat and Mass Transfer, 116: 104641, 2020, doi:10.1016/j.icheatmasstransfer.2020.104641.

Pavlov K.B., Magnetohydrodynamic flow of an incompressible viscous fluid caused by deformation of a plane surface, Magnetohydrodynamics, 10(4): 507–510, 1974, (or [in Russian], Magnitnaya Gidrodinamika, 4(1): 146–147, 1974).

Andersson H.I., MHD flow of a viscoelastic fluid past a stretching surface, Acta Mechanica, 95(1): 227–230, 1992, doi: 10.1007/BF01170814.

Mabood F., Ibrahim S.M., Rashidi M.M., Shadloo M.S., Lorenzini G., Non-uniform heat source/sink and Soret effects on MHD non-Darcian convective flow past a stretching sheet in a micropolar fluid with radiation, International Journal of Heat and Mass Transfer, 93: 674–682, 2016, doi: 10.1016/j.ijheatmasstransfer.2015.10.014.

Khan M.I., Nasir T., Hayat T., Khan N.B., Alsaedi A., Binary chemical reaction with activation energy in rotating flow subject to nonlinear heat flux and heat source/sink, Journal of Computational Design and Engineering, 7(3): 279–286, 2020, doi: 10.1093/jcde/qwaa023.

Kumar K.A., Reddy J.V.R., Sugunamma V., Sandeep N., MHD flow of chemically reacting Williamson fluid over a curved/flat surface with variable heat source/sink, International Journal of Fluid Mechanics Research, 46(5): 407–425, 2019, doi: 10.1615/InterJFluidMechRes.2018025940.

Khan A.A., Bukhari S.R., Marin M., Ellahi R., Effects of chemical reaction on third grade MHD fluid flow under the influence of heat and mass transfer with variable reactive index, Heat Transfer Research, 50(11): 1061–1080, 2019, doi: 10.1615/HeatTransRes.2018028397.

Ramzan M., Bilal M., Chung J.D., Influence of homogeneous heterogeneous reactions on MHD 3D Maxwell fluid flow with Cattaneo-Christov heat flux and convective boundary condition, Journal of Molecular Liquids, 230: 415–422, 2017, doi: 10.1016/j.molliq.2017.01.061.

Majeed A., Zeeshan A., Noori F.M., Analysis of chemically reactive species with mixed convection and Darcy-Forchheimer flow under activation energy: a novel application for geothermal reservoirs, Journal of Thermal Analysis and Calorimetry, 140: 2357–2367, 2020, doi: 10.1007/s10973-019-08978-z.

Majeed A., Zeeshan A., Noori F.M., Numerical study of Darcy-Forchheimer model with activation energy subject to chemically reactive species and momentum slip of order two, AIP Advances, 9(4): 045035, 2019, doi: 10.1063/1.5095546.

Raju C.S.K., Sandeep N., Sulochana C., Sugunamma V., Babu M.J.. Radiation, inclined magnetic field and cross-diffusion effects on flow over a stretching surface, Journal of the Nigerian Mathematical Society, 34(2): 169–180, 2015, doi: 10.1016/j.jnnms.2015.02.003.

Sajid T., Sagheer M., Hussain S., Bilal M., Darcy-Forchheimer flow of Maxwell nanofluid flow with nonlinear thermal radiation and activation energy, AIP Advances, 8: 035102, 2018, doi: 10.1063/1.5019218.

Ilias M.R., Ismail N.S., AbRaji N.H., Rawi N.A., Shafie S., Unsteady aligned MHD boundary layer flow and heat transfer of a magnetic nanofluids past an inclined plate, International Journal of Mechanical Engineering and Robotics Research, 9(2): 197–206, 2020, doi: 10.18178/ijmerr.9.2.197-206.

Ramzan M., Bilal M., Kanwal S., Chung J.D., Effects of variable thermal conductivity and non-linear thermal radiation past an Eyring Powell nanofluid flow with chemical reaction, Communication in Theoretical Physics, 67(6): 723–731, 2017, doi: 10.1088/0253-6102/67/6/723.

Abel M.S., Tawade J.V., Nandeppanavar M.M., MHD flow and heat transfer for the upper-convected Maxwell fluid over a stretching sheet, Meccanica, 47: 385–393, 2012, doi: 10.1007/s11012-011-9448-7.

Waini I., Zainal N.A., Khashi’ie N.S., Aligned magnetic field effects on flow and heat transfer of the upper-convected Maxwell fluid over a stretching/shrinking sheet, MATEC Web Conf., 97: 01078, 2017, doi: 10.1051/matecconf/20179701078.

Bhattacharyya K., Effects of heat source/sink on MHD flow and heat transfer over a shrinking sheet with mass suction, Chemical Engineering Research Bulletin, 15(1): 12–17, 2011, doi: 10.3329/cerb.v15i1.6524.

Bilal M., Nazeer M., Numerical analysis for the non-Newtonian flow over stratified stretching/shrinking inclined sheet with the aligned magnetic field and nonlinear convection, Archive of Applied Mechanics, 91: 949–964, 2021, doi: 10.1007/s00419-020-01798-w.

DOI: 10.24423/EngTrans.1239.20210826