Document Type : Research Paper


1 Department of Mathematics, Sri Venkateswara Engineering College for Women, Tirupati, India

2 Department of Mathematics, Motilal Nehru National Institute of Technology, Allahabad-211004, India

3 Department of Mathematics, SAS, VIT University, Vellore, T.N

4 Department of Mathematics, Sri Venkateswara University


The present paper focuses on numerical study for an inclined magneto-hydrodynamic effect on free convection flow of a tangent hyperbolic nanofluid embedded with Carbon nanotubes (CNTs) over a stretching surface taking velocity and thermal slip into account. Two types of nanoparticles are considered for the study; they are single and multi-walled nanotubes. The presentation of single-parameter group (Lie group) transformations reduces the independent variable number by one, and hence the partial differential governing equations with the supplementary atmospheres into an ordinary differential equation with the appropriate suitable conditions. The obtained ordinary differential equations are then numerically solved by employing fourth order Runge-Kutta technique along with shooting method. The effects of the various parameters governing the flow field are presented with the help of graphs. The investigation reveals that the non-Newtonian MWWCNTs Tangent hyperbolic nano-liquid reduces the friction near the stretching sheet contrasting SWCNTs. This combination can be used as a friction lessening agent/factor. Usage of CNTs shows an excellent performance in enhancing the thermal conductivity of the nanoliquid and single wall carbon nanotubes (SWCNTs) has higher thermal conductivity than multi wall carbon nanotubes (MWCNTs) even in the presence of radiative heat transfer and heat source. Comparison with existing results available in literature is made and had an excellent coincidence with our numerical method.

Graphical Abstract

Inclined Lorentzian force effect on tangent hyperbolic radiative slip flow imbedded carbon nanotubes: lie group analysis


Main Subjects

[1] S.U.S. Choi, “Nanofluids: from vision to reality through research”, J. Heat Transfer, Vol. 131, 033106-1–033106-9, (2009).
[2] M.Chandrasekar, and M.S.Kasiviswa- nathan, “Analysis of heat and mass transfer on MHD flow of a nanofluid past a stretching sheet”, Procedia Engineering, Vol. 127,pp.493-500, (2015).
[3] T. Hayat, M. Rashid, M. Imtiaz, and A.Alsaedi, “Magnetohydrodynamic (MHD) stretched flow of nanofluid with power-law velocity and chemical reaction”, AIP Advances, Vol. 5, 117121, (2015).
[4] R. S. R. Gorla and B. Vasu, “Unsteady convective heat transfer to a stretching surface in a non-Newtonian nanofluid”, Journal of Nanofluids, Vol. 5(4), pp. 581-594 (2016)
[5] P.K.Kameswaran, B. Vasu, P.V.S.N. Murthy, and Rama S R Gorla, “Mixed Convection from a Wavy Surface Embedded in a Thermally Stratified Nanofluid saturated Porous Medium with non-Linear Boussinesq Approximation”, International Communications in Heat and Mass Transfer, Vol. 77, pp.78-86 (2016)
[6] R. UlHaq,Z. Hayat Khan, and W. A. Khan, “Thermophysical effects of carbon nanotubes on MHD flow over a stretching surface”, Physica E: Low-dimensional Systems and Nanostructures”, Vol. 63, pp.215-222, (2014).
[7] A. Waqar, I. Khan, I. Richard Culham, and R. UlHaq, “Heat transfer analysis of MHD water functionalized carbon nanotube flow over a static/moving wedge”, J. Nanomater,ID 934367, 13 pages, (2015),
[8] W. Khan, Z. Khan, and M. Rahi, “Fluid flow and heat transfer of carbon nanotubes along a flat plate with Navier slip boundary”, Appl. Nano Sci., Vol. 4,pp.633–641, (2014).
[9] R. UlHaq, S. Nadeem, Z.H. Khan, and N.F.M. Noor, “Convective heat transfer in MHD slip flow over a stretching surface in the presence of carbon nanotubes”, Physica B, Vol. 457,pp.40–47, (2015).
[10] R. U. I. Haq , Z. H. Khan, and W. A. Khan , “Thermo physical effects of carbon nanotubes on MHD flow over a stretching surface”,Physica E: Low-Dimensional Systems and Nanostructures, Vol. 63, pp. 215–222, (2014).
[11] P. Sreenivasulu, and T. Poornima, “Magnetohydrodynamic boundary layer flow of a dissipating nanofluid past an exponential stretching sheet with convective boundary condition, Mathematical Sciences International Research Journal, Vol. 4, No. 2, pp.288-292, (2015).
[12] T. Poornima, N. Bhaskar reddy and P. Sreenivasulu, “Temperature dependent viscosity effect on MHD mixed convective dissipating flow of cylinder shaped Cu-water nanofluid past a vertical moving surface”, Proc. of Int. Conference on “Frontiers in Mathematics”, Gauhati University, Guwahati, Assam, India., pp.235-242. (2015).
[13] T. Hayat , I. Ullah, A. Alsaedi, and M. Farooq, “MHD flow of Powell-Eyringnanofluid over a non-linear stretching sheet with variable thickness”, Results in Physics, Vol. 7, pp. 189–196, (2017).
[14] O. A. Bég, V. R. Prasad and B. Vasu, “Numerical study of mixed bioconvection in porous media saturated with nanofluid containing oxytactic microorganisms”, “Journal of Mechanics in Medicine and Biology”, Vol. 13(04), pp. 1350067 (2013)
[15] S. C. Reddy, K. Naikoti, and M. M., Rashidi, “MHD flow and heat transfer characteristics of Williamson nanofluid over a stretching sheet with variable thickness and variable thermal conductivity”, Transactions of A. Razmadze Mathematical Institute, Vol. 171, pp. 195–211, (2017).
[16] R. E. Ergun, C. W. Carlson, J. P. McFadden, F. S. Mozer, L. Muschietti, I. Roth, and R. J. Strangeway, “Debye-Scale plasma structures associated with magnetic-field-aligned electric fields, Phys. Rev. Lett., Vol. 81, N0.4, 826, (1998).
[17] M.M. Rashidi, N. Vishnu Ganesh, A.K. Abdul Hakeem, and B. Ganga, “Buoyancy effect on MHD flow of nanofluid over a stretching sheet in the presence of thermal radiation”, Journal of Molecular Liquids, Vol. 198, pp.234-238, (2014).
[18] M. A. Wahed and M. Akl, “Effect of hall current on MHD flow of a nanofluid with variable properties due to a rotating disk with viscous dissipation and nonlinear thermal radiation”,AIP Advances, Vol. 6, 095308-1-14, (2016).
[19] D. Lu, M. Ramzan, N.U. L. Huda, J. D. Chung and U. Farooq, “Nonlinear radiation effect on MHD Carreau nanofluid flow over a radially stretching surface with zero mass flux at the surface”, Scientific Reports, Vol. 8 , 3709, pp.1-17, (2018).
[20] T. Poornima, N. Bhaskar Reddy and P. Sreenivasulu, “Slip Flow of Casson Rheological Fluid under Variable Thermal Conductivity with Radiation”, Heat Transfer Asian Research Journal, Vol. 44, No. 8, pp.718-737, (2015).
[21] N. Bhaskar Reddy, T. Poornima and P. Sreenivasulu, “Radiative heat transfer effect on MHD slip flow of Dissipating Nanofluid past an exponential stretching porous sheet”, International Journal of Pure and Applied Mathematics, Vol. 109, No. 9, pp.134 – 142, (2016).
[22] Y. B.Kho, A.Hussanan, M. K. A. Mohamed, N. M. Sarif, Z. Ismail and M. Z.Salleh, “Thermal radiation effect on MHD flow and heat transfer analysis of Williamson nanofluid past over a stretching sheet with constant wall temperature”, Journal of Physics: Conf. Ser., Vol. 890, No. 1, pp.1-6, (2017).
[23] T. Poornima and P. Sreenivasulu, “Radiation effects on MHD convective boundary layer flow of Eg-based Cu nanotubes due to an extensible moving surface”, Mathematical Sciences International Research Journal, Vol. 4, No. 2, pp. 293-297, (2015).
[24] T. Hayat, M. Ijaz Khan, M. Waqas, A. Alsaedi, and M. Farooq, “Numerical simulation for melting heat transfer and radiation effects in stagnation point flow of carbon–water nanofluid” ,Comput. Methods Appl. Mech. Engrg,Vol. 315,pp. 1011–1024, (2017).
[25] I. Rashid, R. Ul. Haq, and Q. M.AlMdallal, “Aligned magnetic field effects on water based metallic nanoparticles over a stretching sheet with PST and thermal radiation effects”, Physica E: Low-dimensional Systems and Nanostructures, Vol. 89, pp. 33-42, (2017).
[26] A. Hussain, M.Y. Malik, T. Salahuddin, A. Rubab, and M. Khan, “Effects of viscous dissipation on MHD tangent hyperbolic fluid over a nonlinear stretching sheet with convective boundary conditions”, Results in Physics, Vol. 7 pp. 3502–3509, (2017).
[27] Z. Ullah, and G. Zaman, “Lie group analysis of magnetohydrodynamic tangent hyperbolic fluid flow towards a stretching sheet with slip conditions”, Heliyon, Vol. 3, No. 111, e00443, pp. 1-15, (2017).
[28] N. S. Akbar , S.Nadeem , R. U.Haq and Z. H. Khan, “Numerical solutions of magnetohydrodynamic boundary layer flow of tangent hyperbolic fluid towards a stretching sheet”, Indian J. Phys, V0l. 87, No. 11, pp. 1121-1124.
[29] G.T. Thammanna, B.J. Gireesha, andB. Mahanthesh, “Partial slip and Joule heating on magnetohydrodynamic radiated flow of nanoliquid with dissipation and convective condition”, Results in Physics, Vol. 7, pp. 2728–2735, (2017).
[30] M.Naseer, M. Y. Malik, S. Nadeem, and A.Rehman, “The boundary layer flow of hyperbolic tangent fluid over a vertical exponentially stretching cylinder”, Alexandria Engineering Journal, Vol. 53, pp.747–750, (2014).
[31] N. Sandeep, “Effect of aligned magnetic field on liquid thin film flow of magnetic-nanofluids embedded with graphene nanoparticles”, Advanced Powder Technology, Vol. 28, No. 3, pp. 865-875, (2017).
[32] M. Sailaja, R.Hemadri Reddy, R.Saravana and K.Avinash, “Aligned magnetic field effect on unsteady liquid film flow of Casson fluid over a stretching surface”, Materials Science and Engineering, Vol. 263, No. 6, pp.1-8, (2017).
[33] N. S.Arifin, S. M.Zokri, A. R. M.Kasim, M. Z.Salleh, N. F. Mohammad and W. N. S. W. Yusoff, “Aligned magnetic field of two-phase mixed convection flow in dusty Casson fluid over a stretching sheet with Newtonian heating”, Journal of Physics: Conference Series, Vol. 890, No. 1, pp. 1-6, (2017).
[34] M. Rashidi,M. Babu, M. Jayachandra, N. Sandeep, M. Ali, “MHD squeezing flow of nanofluid between parallel plates in the presence of aligned magnetic field, Journal of Computational and Theoretical Nanoscience, Vol. 13, No. 11, pp. 8700-8708(9), (2016).
[35] I. Waini, N. A. Zainal, and N. S. Khashi'ie, “Aligned magnetic field effects on flow and heat transfer of the upper-convected Maxwell fluid over a stretching/shrinking sheet, MATEC Web of Conferences Vol. 97, 01078 , 10 pages, (2017).
[36] G.P. Ashwinkumar, and C. Sulochana, S. P. Samrat, “Effect of the aligned magnetic field on the boundary layer analysis of magnetic-nanofluid over a semi-infinite vertical plate with ferrous nanoparticles”, Multidiscipline Modelling in Materials and Structures, Vol. 14, No. 3 pp.437-515, (2017).
[37] Q.Z. Xue, “Model for thermal conductivity of carbon nanotube-based composites”, Physica B,Vol. 368 pp. 302–307, (2005).
[38] S. Mukhopadyaya, G.C. Layek and S.A. Samad, “Study of MHD boundary layer flow over a heated stretching sheet with variable viscosity, International Journal of Numerical Methods for Heat & Fluid Flow, Vol. 48, No. 21-22, pp.4460-4466, (2005).
[39] K. Uma Maheswararao, G. and Koteswararao, “Boundary layer flow of MHD tangent hyperbolic fluid past a vertical plate in the presence of thermal dispersion using spectral relaxation method”, International Journal of Computational Engineering Research, Vol. 07, No. 06, (2017).