Efficacy of a Modulated Viscosity-dependent Temperature/nanoparticles Concentration Parameter on a ‎Nonlinear Radiative Electromagneto-nanofluid Flow along an Elongated Stretching Sheet

Document Type : Research Paper


1 National Engineering Laboratory for Modern Silk, College of Textile and Clothing Engineering, Soochow University, Suzhou, China

2 School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo, China

3 School of Science, Xi'an University of Architecture and Technology, China

4 Department of Mathematics, Faculty of Education, Ain Shams University,‎ El-Makrizy St, Roxy, Heliopolis: 11566, Cairo, Egypt‎

5 Department of Basic Science, Cairo Higher Institute for Engineering,‎ Computer Science and Management, New Cairo, Cairo, Egypt‎


The purpose of the present communication is to investigate the flow of a radiative electromagnetic-Casson nanofluid past a stretching sheet under the impacts of a chemical reaction and nonlinear thermal radiation. To enrich the blood flow, a modulated viscosity/thermal conductivity dependent temperature/nanoparticles concentration parameter is included in the governing equations. The system of PDEs is transformed to ODEs by invoking similarity transformations and then solved numerically by the well- known fourth-order Runge-Kutta integration scheme based on shooting approach. The main factors affecting the Casson fluid’s temperature profiles are revealed.


Main Subjects

Publisher’s Note Shahid Chamran University of Ahvaz remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

‎[1]‎ Al-Khaled, K., Khan, S.U., Thermal aspects of casson nanoliquid with gyrotactic microorganisms, ‎temperature-dependent viscosity, and variable thermal conductivity: Bio-technology and ‎thermal applications, Inventions, 5(3), 2020, 1–14.‎
‎[2]‎ Gbadeyan, J.A., Titiloye, E.O., Adeosun, A.T., Effect of variable thermal conductivity and viscosity on ‎Casson nanofluid flow with convective heating and velocity slip, Heliyon, 6(1), 2020, e03076.‎
‎[3]‎ Fung, Y., Biodynamics: Circulation, Springer-Verlag, New York, 1984.‎
‎[4]‎ Nakamura, M., Sawada, T., Numerical study on the flow of a non-Newtonian fluid through an ‎axisymmetric stenosis, Journal of Biomechanical Engineering, 110(2), 1988, 137-143.‎
‎[5]‎ Haemmerich, D., Wright, A.W., Mahvi, D.M., Lee, F.T., Webster, J.G., Hepatic bipolar radiofrequency ‎ablation creates coagulation zones close to blood vessels: a finite element study, Medical & Biological Engineering & Computing, 41(3), 2003, 317–323.‎
‎[6]‎ Casson, N., A flow equation for pigment-oil suspensions of the printing ink type, In: Mill, C.C., Ed., ‎Rheology of Disperse Systems, Pergamon Press, Oxford, 1959.‎
‎[7]‎ Gireesha, B.J., Kumar, P.B.S., Mahanthesh, B.S., Shehzad, A., Rauf, A., Nonlinear 3D flow of Casson-‎Carreau fluids with homogeneous – heterogeneous reactions : A comparative study, Results in  Physics, ‎‎7, 2017, 2762–2770.‎
‎[8]‎ Walawender, W.P., Chen, T.Y., Cala, D.F., An approximate Casson fluid model for tube flow of blood, ‎Biorheology, 12(2), 1975, 111–119.‎
‎[9]‎ Srivastava, L.M., Srivastava, V.P., Peristaltic transport of blood: Casson model-II, Journal of Biomechanics, 17(11), ‎‎1984, 821–829.‎
‎[10]‎ Dash, R.K., Mehta, K.N., Jayaraman, G., Casson fluid flow in a pipe filled with a homogeneous porous ‎medium, International Journal of Engineering Science, 34(10), 199, 1145–1156.‎
‎[11]‎ Eldabe, N.T.M., Saddeck, G., El-Sayed, A.F., Heat transfer of MHD non-Newtonian Casson fluid flow ‎between two rotating cylinders, Mechanics and Mechanical Engineering, 5(2), 2001, 237–251.‎
‎[12]‎ Mernone, A.V., Mazumdar, J.N., Lucas, S.K., A mathematical study of peristaltic transport of a Casson ‎fluid, Mathematical and Computer Modelling, 35(7-8), 2002, 895–912.‎
‎[13]‎ Mustafa, M., Hayat, T., Pop, I., Aziz, A., Unsteady boundary layer flow of a Casson fluid due to an ‎impulsively started moving flat plate, Heat Transfer Research, 40(6), 2011, 563–576.‎
‎[14]‎ Shehzad, S.A., Hayat, T., Qasim, M., Asghar, S., Effects of mass transfer on MHD flow of Casson fluid ‎with chemical reaction and suction, Brazilian Journal of Chemical Engineering, 30, 2013, 187–195.‎
‎[15]‎ Hayat, T., Asad, S., Alsaedi, A., Flow of Casson fluid with nanoparticles, Applied Mathematics and Mechanics, 37(4), ‎‎2001, 459–470.‎
‎[16]‎ Ghadikolaei, S.S., Hosseinzadeh, K., Ganji, D.D., Jafari, B., Nonlinear thermal radiation effect on ‎magneto Casson nanofluid flow with Joule heating effect over an inclined porous stretching sheet, ‎Case Studies in Thermal Engineering, 12, 2018, 176–187.‎
‎[17]‎ Aman, S., Zokri, S.M., Ismail, Z., Salleh, M.Z., Khan, I., Effect of MHD and porosity on exact solutions ‎and flow of a hybrid casson-nanofluid, Journal of Advanced Research in Fluid Mechanics and Thermal Sciences, 44(1), 2018, 131–139.‎
‎[18]‎ Siddiqa, S., Begum, N., Hossain, M.A., Shoaib, M., Reddy Gorla, R.S., Radiative heat transfer analysis ‎of non-Newtonian dusty Casson fluid flow along a complex wavy surface, Numer. Heat Transf. Part ‎A Appl., 73 (4), 2018, 209–221.‎
‎[19]‎ Elgazery, N.S., Flow of non-Newtonian magneto-fluid with gold and alumina nanoparticles ‎through a non-Darcian porous medium, Journal of the Egyptian Mathematical Society, 27(1), 2019, 1–25.‎
‎[20]‎ Hady, F.M., Mahdy, A., Mohamed, R.A., Ahmed, S.E., Abo-zaid, O.A., Unsteady natural convection ‎flow of a dusty non-Newtonian Casson fluid along a vertical wavy plate: numerical approach, Journal of the Brazilian Society of Mechanical Sciences and Engineering, 41(11), 2019, 1–20.‎
‎[21]‎ Akolade, M.T., Tijani, Y.O., A comparative study of three dimensional flow of Casson–Williamson ‎nanofluids past a riga plate : Spectral quasi-linearization approach, Partial Differential Equations in Applied Mathematics, 4, 2021, 100108.‎
‎[22]‎ Elelamy, A.F., Elgazery, N.S., Ellahi, R., Blood flow of MHD non-Newtonian nanofluid with heat ‎transfer and slip effects: Application of bacterial growth in heart valve, International Journal of Numerical Methods for Heat & Fluid Flow, 30(11), 2020, 4883–4908.‎
‎[23]‎ Usman, M., Gul, T., A., Khan, Alsubie, A., Zaka, M., Electromagnetic couple stress film flow of hybrid ‎nanofluid over an unsteady rotating disc, International Communications in Heat and Mass Transfer, 127, 2021, 105562.‎
‎[24]‎ Varun Kumar, R.S., Gunderi Dhananjaya, P., Naveen Kumar, R.R., Punith Gowda, J., ‎Prasannakumara, B.C., Modeling and theoretical investigation on Casson nanofluid flow over a ‎curved stretching surface with the influence of magnetic field and chemical reaction, International Journal for Computational Methods in Engineering Science and Mechanics, 23(1), 2022, 12–19.‎
‎[25]‎ Venkata Ramudu, A.C., Anantha Kumar, K., Sugunamma, V., Sandeep, N., Impact of Soret and ‎Dufour on MHD Casson fluid flow past a stretching surface with convective–diffusive conditions, Journal of Thermal Analysis and Calorimetry, 147(3), 2022, 2653–2663.‎
‎[26]‎ Elgazery, N.S., Hassan, M.A., The effects of variable fluid properties and magnetic field on the flow ‎of non-Newtonian fluid film on an unsteady stretching sheet through a porous medium, Communications in Numerical Methods in Engineering, 24(12), 2008, 2113-2129. ‎
‎[27]‎ Elgazery, N.S., El-Sayed, M.F., Effects of magneto-Marangoni convection with variable properties ‎on non-Newtonian biviscosity fluid over stretching sheet in porous medium, Journal of Porous Media, 17‎‎(10), 2014, 901-912.‎
‎[28]‎ Elgazery, N.S., Numerical simulation for biviscosity fluid flow through a porous medium under the ‎effects of variable properties, Special Topics and Reviews in Porous Media, 3(1), 2012, 1–11.‎
‎[29]‎ Animasaun, I.L., Effects of thermophoresis, variable viscosity and thermal conductivity on free ‎convective heat and mass transfer of non-darcian MHD dissipative Casson fluid flow with suction ‎and nth order of chemical reaction, Journal of the Nigerian Mathematical Society, 34(1), 2015, 11–31.‎
‎[30]‎ El-Aziz, M.A., Afify, A.A., Effects of Variable Thermal Conductivity with Thermal Radiation on MHD ‎Flow and Heat Transfer of Casson Liquid Film Over an Unsteady Stretching Surface, Brazilian Journal of Physics, 46(5), 2001, 516–525.‎
‎[31]‎ Nawaz, M., Naz, R., Awais, M., Magnetohydrodynamic axisymmetric flow of Casson fluid with ‎variable thermal conductivity and free stream, Alexandria Engineering Journal, 57(3), 2018, 2043–2050.‎
‎[32]‎ Sohail, M., Shah, Z., Tassaddiq, A., Kumam, P., Roy, P., Entropy generation in MHD Casson fluid flow ‎with variable heat conductance and thermal conductivity over non-linear bi-directional ‎stretching surface, Scientific Reports, 10(1), 2020, 1–16.‎
‎[33]‎ Bisht, A., Sharma, R., Non-similar solution of Casson nanofluid with variable viscosity and variable ‎thermal conductivity, International Journal of Numerical Methods for Heat & Fluid Flow, 22(7), 2020, 3919–3938.‎
‎[34]‎ Elgazery, N.S., Elelamy, A.F., Multiple solutions for non-Newtonian nanofluid flow over a stretching ‎sheet with nonlinear thermal radiation: Application in transdermal drug delivery, Pramana, 94(1), 2020, 8-94.‎
‎[35]‎ Kumar, K., Chauhan, P.R., Kumar, R., Bharj, R.S., Irreversibility analysis in Al2O3-water ‎nanofluid flow with variable property, Facta Universitatis, Series: Mechanical Engineering, 20(3), 2022, 503–518.‎
‎[36]‎ Feijó, B.C., Pavlovic, A., Rocha, L.A.O., Isoldi, L.A., Lorente, S., dos Santos, E.D., Geometrical ‎investigation of microchannel with two trapezoidal blocks subjected to laminar convective flows ‎with and without boiling, Reports in Mechanical Engineering, 3(1), 2022, 20–36.‎
‎[37]‎ Aliy, G., Kishan, N., Effect of Electric Field on MHD Flow and Heat Transfer Characteristics of ‎Williamson Nanofluid over a Heated Surface with Variable Thickness, OHAM Solution, 30(1), ‎‎2019, 1–23.‎
‎[38]‎ Bhargava, R., Chandra, H., Numerical simulation of MHD boundary layer flow and heat transfer ‎over a nonlinear stretching sheet in the porous medium with viscous dissipation using hybrid ‎approach, arXiv:1711.03579 [physics.flu-dyn].‎
‎[39]‎ Ahmed, S.E., Mansour, M.A., Mahdy, A. Mohamed, S.S., Entropy generation due to double diffusive ‎convective flow of Casson fluids over nonlinearity stretching sheets with slip conditions, Engineering Science and Technology, an International Journal, 20, 2017, 1553–1562.‎
‎[40]‎ Crane, L.J., Flow past a stretching plate, Zeitschrift für Angewandte Mathematik und Physik, 21(4), 1970, ‎‎645–647.‎
‎[41]‎ Elgazery, N.S., Nanofluids flow over a permeable unsteady stretching surface with non-uniform ‎heat source/sink in the presence of inclined magnetic field, Journal of the Egyptian Mathematical Society, 27(1), 2019, 1–26.‎
‎[42]‎ Afify, A.A., Elgazery, N.S., Lie group analysis for the effects of chemical reaction on MHD ‎stagnation-point flow of heat and mass transfer towards a heated porous stretching sheet with ‎suction or injection, Nonlinear Analysis: Modelling and Control, 17(1), 2012, 1–15.‎
‎[43] Qian, M.Y., He, J.H., Two-scale thermal science for modern life: Making the Impossible Possible, ‎Thermal Science, 26(3), 2022, 2409-2412‎.
‎[44] He, J.H., Seeing with a single scale is always unbelieving: From magic to two-scale fractal, Thermal Science, 25(2), 2021, 1217-1219 ‎
‎[45] Liu, F.J., Zhang, T., He, C.H., Tian, D., Thermal oscillation arising in a heat shock of a porous ‎hierarchy and its application, Facta Universitatis Series Mechanical Engineering, ‎‎20(3), 2022, 633-645.‎
‎[46] Xue, R.J., Liu, F.J., A Fractional model and its application to heat prevention coating with ‎cocoon-like hierarchy, Thermal Science, 26(3), 2022, 2493-2498‎.
‎[47] He, C.H., A variational principle for a fractal nano/microelectromechanical (N/MEMS) ‎system, International Journal of Numerical Methods for Heat & Fluid Flow, 2022, DOI: ‎‎10.1108/HFF-03-2022-0191‎.
‎[48] Wang, K.L., Novel analytical approach to modified fractal gas dynamics model with the ‎variable coefficients, ZAMM Zeitschrift fur Angewandte Mathematik und Mechanik, 2023, DOI: 10.1002/zamm.202100391.‎
‎[49] Wang, K.L., Wei, C.F., Fractal soliton solutions for the fractal-fractional shallow water wave ‎equation arising in ocean engineering, Alexandria Engineering Journal, 65, 2023, 859-865.‎
‎[50] Li, Z.Y., Wang, M.C., Wang, Y.L., Solving a class of variable order nonlinear fractional ‎integral differential equations by using reproducing kernel function, AIMS Mathematics, 7(7), 2022, 12935-12951 ‎
‎[51] Li, Z.Y., Chen, Q.T., Wang, Y.L., Li, X.Y., Solving two-sided fractional super-diffusive partial ‎differential equations with variable coefficients in a class of new reproducing kernel ‎spaces, Fractal and Fractional, 6(9), 2022, 492.