Chebyshev Spectral Collocation Method for Flow and Heat Transfer in Magnetohydrodynamic Dissipative Carreau Nanofluid over a Stretching Sheet with Internal Heat Generation

Document Type : Research Article


1 Mechanical Engineering Department, University of Lagos, Lagos, Nigeria

2 Works and Physical Planning Department, University of Lagos, Lagos, Nigeria

3 Mechanical Engineering Department, Federal University of Agriculture, Abeokuta, Nigeria


In this paper, Chebyshev spectral collocation method is used to solve the unsteady two-dimensional flow and heat transfer of Carreau nanofluid over a stretching sheet subjected to magnetic field, temperature dependent heat source/sink and viscous dissipation. Similarity transformations are used to reduce the systems of the developed governing partial differential equations to nonlinear third and second orders ordinary differential equations which are solved by the numerical method. Good agreements are established between the results of the present numerical solution and the results of Runge- Kutta coupled with shooting method. Using kerosene as the base fluid embedded with the silver (Ag) and copper (Cu) nanoparticles, the effects of pertinent parameters on reduced Nusselt number, flow and heat transfer characteristics of the nanofluid are investigated and discussed. From the results, it is established temperature field and the thermal boundary layers of Ag-Kerosene nanofluid are highly effective when compared with the Cu-Kerosene nanofluid. Heat transfer rate is enhanced by increasing in power-law index and unsteadiness parameter. Skin friction coefficient and local Nusselt number can be reduced by magnetic field parameter and they can be enhanced by increasing the aligned angle. Friction factor is depreciated and the rate of heat transfer increases by increasing the Weissenberg number. It is hope that the present work will enhance the study of the flow and heat transfer processes.


Main Subjects

[1] T. Hayat, I. Ullah, B. Ahmad, A. Alsaedi. Radiative flow of Carreau liquid in presence of Newtonian heating and
chemical reaction. Results in Physics 7, 715–722, 2017.
[2] H. 1. Andersson, J. B. Aarseth, N. Braud and B. S. Dandapat. Flow of a Power-Law Fluid on an Unsteady Stretching Surface. J. Non-Newtonian Fluid Mech. , 62, 1-8, 1996.
[3] H. I. Anderson, K. H. Bech, and B. S. Dandapat. Magneto Hydrodynamics Flow of a Power-Law Fluid Over A Stretching Sheet. Int. J. Non- Linear Mech. , 27, 929-936, 1992.
[4] C. H. Chen. “Heat Transfer in a Power-Law Fluid Film over an Unsteady Stretching Sheet,” Heat Mass Transf. Und Stoffuebertragung. 39, 791–796, 2003.
[5] B. S. Dandapat, B. Santra and H. 1. Andersson. Thermo Capillarity in a Liquid Film on an Unsteady Stretching Surface. Int. J. Heat and Mass transfer, 46, 3009-3015, 2003.
[6] B. S. Dandapat, B. Santra and K. Vejravelu. “The Effects of Variable Fluid Properties and the Thermo Capillarity on The Flow of a Thin Film On Stretching Sheet,” Int. J. Heat and Mass transfer, 50, 991-996, 2007.
[7] C. Wang. Analytic Solutions for a Liquid Film on an Unsteady Stretching Surface. Heat Mass Transf. Und Stoffuebertragung. 42, 759– 766, 2006.
[8] C. H. Chen. Effect of Viscous Dissipation on Heat Transfer in a Non-Newtonian Liquid Film over an Unsteady Stretching Sheet,” J. Nonnewton. Fluid Mech. 135, 128–135, 2006.
[9] M. Sajid, T. Hayat and S. Asghar. Comparison between the HAM and HPM Solutions of Thin Film Flows of Non-Newtonian Fluids on a Moving Belt,” Nonlinear Dyn. 50, 27–35, 2007.
[10] B. S. Dandapat, S. Maity and A. Kitamura. “Liquid Film Flow Due to an Unsteady Stretching Sheet,” Int. J. Non. Linear. Mech. 43, 880–886, 2008.
[11] S. Abbasbandy, M. Yurusoy and M. Pakdemirli. The Analysis Approach of Boundary Layer Equation of Power-Law Fluids of Second Grade. Zzeitschrift fur Naturforschung A, 63, 564-570, 2008.
[12] B. Santra and B. S. Dandapat. Unsteady Thin Film
Flow over A Heated Stretching Sheet,” Int. J. Heat. Mass transfer, 52, 1965-1970, 2008.
[13] M. Sajid, N. Ali and T. Hayat. On Exact Solutions for Thin Film Flows of A Micropolar Fluid. Commun. Nonlinear Sci. Numer. Simul. 14, 451–461, 2009.
[14] N. F. M. Noor and I. Hashim. Thermocapillarity and Magnetic Field Effects in a Thin Liquid Film on an Unsteady Stretching Surface,” Int. J. Heat and mass Transfer, 53, 2044-2051, 2010.
[15] B. S. Dandapat and S. Chakraborty, S. “Effects of Variable Fluid Properties on Unsteady Thin-Film Flow over a Non-Linear Stretching Sheet,” Int. J. Heat Mass Transf., 53, 5757–5763, 2010.
[16] B. S. Dandapat and S. K. Singh. “Thin Film Flow over a Heated Nonlinear Stretching Sheet in Presence of Uniform Transverse Magnetic Field,” Int. Commun. Heat Mass Transf., 38, 324–328, 2011.
[17] G. M. Abdel-Rahman. Effect of Magnetohydrodynamic on Thin Films of Unsteady Micropolar Fluid through A Porous Medium,” J. Mod. Phys., 2, 1290–1304, 2011.
[18] Y. Khan, Q. Wu, N. Faraz and A. Yildirim. The Effects of Variable Viscosity and Thermal Conductivity on a Thin Film Flow over a Shrinking/Stretching Sheet. Comput. Math. with Appl., 61, 3391– 3399, 2011.
[19] I. C. Liu, A. Megahed and H.H. Wang. “Heat Transfer in a Liquid Film due to an Unsteady Stretching Surface with Variable Heat Flux,” J. Appl. Mech. , 80, 1–7, 2013.
[20] K. Vajravelu, K. V. Prasad and C. O. Ng. Unsteady Flow and Heat Transfer in a Thin Film of Ostwald-De Waele Liquid Over A Stretching Surface. Commun. Nonlinear Sci. Numer. Simul. , 17, 4163–4173, 2012.
[21] I.C. Liu and A. M. Megahed. Homotopy Perturbation Method for Thin Film Flow and Heat Transfer over an Unsteady Stretching Sheet with Internal Heating and Variable Heat Flux,” J. Appl. Math., 2012.
[22] R. C. Aziz, I. Hashim and S. Abbasbandy. “Effects of Thermocapillarity and Thermal Radiation on Flow and Heat Transfer In A Thin Liquid Film on an Unsteady Stretching Sheet,” Math. Probl. Eng., 2012.
[23] M. M. Khader and A. M. Megahed. Numerical Simulation Using The Finite Difference Method for the Flow and Heat Transfer in a Thin Liquid Film over an Unsteady Stretching Sheet in a Saturated Porous Medium in the Presence of Thermal Radiation. J. King Saud Univ. - Eng. Sci., 25, 29–34, 2013.
[24] Y. Lin, L. Zheng, X. Zhang, L. Ma and G. Chen. MHD Pseudo-Plastic Nanofluid Unsteady Flow And Heat Transfer In A Finite Thin Film Over Stretching Surface With Internal Heat Generation. Int. J. Heat Mass Transf., 84, 903–911, 2015.
[25] N. Sandeep, C. Sulochana and I. L. Animasaun. Stagnation Point Flow of a Jeffrey Nano Fluid over a Stretching Surface with Induced Magnetic Field and Chemical Reaction. Int. J. Eng. Research in Afrika, 20, 93-111, 2016.
[26] J. Tawade, M. S. Abel, P. G. Metri and A. Koti, A.. “Thin Film Flow and Heat Transfer Over an Unsteady Stretching Sheet with Thermal Radiation, Internal Heating In Presence of External Magnetic Field,” Int. Adv. Appl. Math. And Mech.,3, 29-40, 2016.
[27] C. S. K. Raju and N. Sandeep. Unsteady three-dimensional flow of Casson-Carreau fluids past a stretching surface. Alex Eng J. 55:1115–26, 2016.
[28] C. S. K. Raju and N. Sandeep. Falkner-Skan flow of a magnetic-Carreau fluid past a wedge in the presence of cross diffusion effects. Eur Phys J Plus. 131:267, 2016
[29] C. S. K. Raju, N. Sandeep and V. Sugunamma. “Dual Solutions For Three-Dimensional MHD Flow Of A Nanofluid Over A Nonlinearly Permeable Stretching Sheet ,” Alexandria Engineering Journal, 55, 151162, 2016.
[30] N. Sandeep, O. K. Koriko and I. L. Animasaun, I.L. Modified Kinematic Viscosity Model for 3D-Casson Fluid Flow within Boundary Layer Formed On A Surface At Absolute Zero,” Journal of Molecular Liquids, 221, 2016.
[31] M. J. Babu, N. Sandeep and C. S. K. Raju. Heat and Mass Transfer in MHD Eyring-Powell Nanofluid Flow Due To Cone in Porous Medium,” International Journal of Engineering Research in Africa, 19,57–74, 2016.
[32] I. L. Animasun, C. S. K. Raju and N. Sandeep, N. Unequal Diffusivities Case of Homogeneous-Heterogeneous Reactions within Viscoelastic Fluid Flow in the Presence of Induced Magnetic Field and Nonlinear Thermal Radiation,” Alexandria Engineering Journal, 55(2),1595-1606, 2016.
[33] O. D. Makinde and I. L. Animasaun, I. L. Thermophoresis and Brownian Motion Effect on MHD Bioconvection of Nanofluid with Nonlinear Thermal Radiation and Quartic Chemical Reaction Past an Upper Horizontal Surface of A Paraboloid of Revolution. J. of Mole. Liquids, 221, 733-743, 2016.
[34] O. D. Makinde and I. L. Animasaun. Bioconvection in MHD Nanofluid Flow with Nonlinear Thermal Radiation and Quartic Autocatalysis Chemical Reaction Past an Upper Surface of A Paraboloid Of Revolution,” J. of Ther. Sci., 109, 159-171, 2016.
[35] N. Sandeep, N. Effect of Aligned Magnetic Field on Liquid Thin Film Flow of Magnetic-Nanofluid Embedded With Graphene Nanoparticles,” Advanced Powder Technology, (in Press).
[36] J. V. Ramana Reddy, V. Sugunamma, N. Sandeep. “Effect Of Frictional Heating on Radiative Ferrofluid Flow over a Slendering Stretching Sheet With Aligned Magnetic Field,” Europen Physical Journal Plus 132:7, 2017.
[37] M. E. Ali and N. Sandeep. Cattaneo-Christov Model for Radiative Heat Transfer of Magnetohydrodynamic Casson-Ferrofluid: A Numerical Study,” Results in Physics, 7, 21-30, 2017.
[38] P. J. Carreau. Rheological equations from molecular network theories. Trans Soc Rheol., 116:99–127, 1972.
[39] M. S. Kumar, N. Sandeep, B. R. Kumar. Free convection Heat transfer of MHD dissipative Carreau Nanofluid Flow Over a Stretching Sheet. Frontiers in Heat and Mass Transfer, 813, 2017.
[40] T. Hayat, N. Saleem, S. Asghar, M. S. Alhothuali, A. Alhomaidan A.. Influence of induced magnetic field and
heat transfer on peristaltic transport of a Carreau fluid. Commun Nonlinear Sci Numer Simul., 16:3559–77, 2011.
[41] B. I. Olajuwon B. I. Convective heat and mass transfer in a hydromagnetic Carreau fluid past a vertical porous plated in presence of thermal radiation and thermal diffusion. Therm Sci., 15:241–52, 2011.
[42] T. Hayat , S. Asad, M. Mustafa, A. Alsaedi. Boundary layer flow of Carreau fluid over a convectively heated stretching sheet. Appl Math Comput.,246:12–22, 2014.
[43] N. S. Akbar, S. Nadeem, U. I. Haq Rizwan, Ye Shiwei. MHD stagnation point flow of Carreau fluid toward a permeable shrinking sheet: Dual solutions. Ain Shams Eng J., 5:1233–9, 2014.
[44] N. S. Akbar, N. S. Blood flow of Carreau fluid in a tapered artery with mixed convection, Int. J. Biomath. 7 (6) 1450066–1450087, 2014.
[45] S. Mekheimer Kh,, F. Salama, M. A. El Kot. The unsteady flow of a carreau fluid through inclined catheterized arteries haveing a balloon with Time-Variant Overlapping Stenosis, Walailak Journal of Science and Technology (WJST) 12, 863-883, 2015.
[46] Y. A. Elmaboud, K. S. Mekheimer, M. S. Mohamed M. S. Series solution of a natural convection flow for a Carreau fluid in a vertical channel with peristalsis. J Hydrodyn Ser B, 27:969–79, 2015.
[47] M. Hashim and Khan. A revised model to analyze the heat and mass transfer mechanisms in the flow of Carreau nanofluids. Int J Heat Mass Transfer, 103:291–7, 2016.
[48] G. R. Machireddy and S. Naramgari. Heat and mass transfer in radiative MHD Carreau fluid with cross diffusion. Ain Shams Eng J 2016.
[49] Sulochana C, Ashwinkumar GP, Sandeep N. 2016. Transpiration effect on stagnationpoint flow of a Carreau nanofluid in the presence of thermophoresis and Brownian motion. Alex Eng J. 55:1151–7
[50] D. Gottlieb, S.A. Orszag, Numerical analysis of spectral methods: Theory and applications, in: Regional Conference Series in Applied Mathematics, vol. 28, SIAM, Philadelphia, 1977, pp. 1–168.
[51] C. Canuto, M.Y. Hussaini, A. Quarteroni, T.A. Zang, Spectral Methods in Fluid Dynamics, Springer-Verlag, New York, 1988.
[52] R. Peyret, Spectral Methods for Incompressible Viscous Flow, SpringerVerlag, New York, 2002.
[53] F.B. Belgacem, M. Grundmann, Approximation of the wave and electromagnetic diffusion equations by spectral methods, SIAM Journal on Scientific Computing 20 (1), (1998), 13–32.
[54] X.W. Shan, D. Montgomery, H.D. Chen, Nonlinear magnetohydrodynamics by Galerkin-method computation, Physical Review A 44 (10) (1991) 6800–6818.
[55] X.W. Shan, Magnetohydrodynamic stabilization through rotation, Physical Review Letters 73 (12) (1994) 1624–1627.
[56] J.P. Wang, Fundamental problems in spectral methods and finite spectral method, Sinica Acta Aerodynamica 19 (2) (2001) 161–171.
[57] E.M.E. Elbarbary, M. El-kady, Chebyshev finite difference approximation for the boundary value problems, Applied Mathematics and Computation 139 (2003) 513–523.
[58] Z.J. Huang, and Z.J. Zhu, Chebyshev spectral collocation method for solution of Burgers’ equation and laminar natural convection in two-dimensional cavities,Bachelor Thesis, University of Science and Technology of China, Hefei, 2009.
[59] N.T. Eldabe, M.E.M. Ouaf, Chebyshev finite difference method for heat and mass transfer in a hydromagnetic flow of a micropolar fluid past a stretching surface with Ohmic heating and viscous dissipation, Applied Mathematics and Computation 177 (2006) 561–571.
[60] A.H. Khater, R.S. Temsah, M.M. Hassan, A Chebyshev spectral collocation method for solving Burgers’-type equations, Journal of Computational and Applied Mathematics 222 (2008) 333–350.
[61] C. Canuto, M.Y. Hussaini, A. Quarteroni, T.A. Zang, Spectral Methods in Fluid Dynamics, Springer, New York, 1988.
[62] E.H. Doha, A.H. Bhrawy, Efficient spectral-Galerkin algorithms for direct solution of fourth-order differential equations using Jacobi polynomials, Appl. Numer. Math. 58 (2008) 1224–1244.
[63] E.H. Doha, A.H. Bhrawy, Jacobi spectral Galerkin method for the integrated forms of fourth-order elliptic differential equations, Numer. Methods Partial Differential Equations 25 (2009) 712–739.
[64] E.H. Doha, A.H. Bhrawy, R.M. Hafez, A Jacobi–Jacobi dual-Petrov–Galerkin method for third- and fifth-order differential equations, Math. Computer Modelling 53 (2011) 1820–1832.
[65] E.H. Doha, A.H. Bhrawy, S.S. Ezzeldeen, Efficient Chebyshev spectral methods for solving multi-term fractional orders differential equations, Appl. Math. Model. (2011) doi:10.1016/j.apm.2011.05.011.