Abstract
The flow and heat transfer characteristics of a second-grade hybrid nanofluid over a permeable stretching/shrinking sheet are investigated. Using similarity transformations, the governing equations are reduced to a set of differential equations. These equations have been solved utilizing nonlinear shooting method. It is found that the region of the existence of the dual solutions becomes wider for higher values of the suction parameter, non-Newtonian parameter, magnetic field parameter and volume fraction of Cu nanoparticles. Due to the increase of the suction parameter, non-Newtonian viscosity parameter and volume fraction of Cu nanoparticles, the velocity of the fluid significantly increases, whereas the temperature decreases. The results elucidated with streamlines show that the flow field can be consisted of three layers depending on the velocity ratio parameter which is defined as the ratio of velocities of the sheet and the free stream. In addition, stability analysis is carried out to determine the stable and unstable solutions of the problem. It is found that the upper solutions are stable and physically acceptable.
Similar content being viewed by others
References
S.K. Das, S.U.S. Choi, W. Yu, T. Pradeep, Nanofluids: Science and Technology (Wiley-Interscience, New Jersey, 2007)
V. Trisaksri, S. Wongwises, Critical review of heat transfer characteristics of nanofluids. Renew. Sustain. Energy Rev. 11, 512–523 (2007)
S. Kakaç, A. Pramuanjaroenkij, Review of convective heat transfer enhancement with nanofluids. Int. J. Heat Mass Transf. 52, 3187–3196 (2009)
S. Akilu, K.V. Sharma, A.T. Baheta, R. Mamat, A review of thermophysical properties of water based composite nanofluids. Renew. Sustain. Energy Rev. 66, 654–678 (2016)
J. Sarkar, P. Ghosh, A. Adil, A review on hybrid nanofluids: recent research, development and applications. Renew. Sustain. Energy Rev. 43, 164–177 (2015)
K.Y. Leong, K.Z.K. Ahmad, H.C. Ong, M.J. Ghazali, A. Baharum, Synthesis and thermal conductivity characteristic of hybrid nanofluids—a review. Renew. Sustain. Energy Rev. 75, 868–878 (2017)
L.S. Sundar, K.V. Sharma, M.K. Singh, A.C.M. Sousa, Hybrid nanofluids preparation, thermal properties, heat transfer and friction factor—a review. Renew. Sustain. Energy Rev. 68, 185–198 (2017)
J.A.R. Babu, K.K. Kumar, S.S. Rao, State-of-art review on hybrid nanofluids. Renew. Sustain. Energy Rev. 77, 551–565 (2017)
N.A.C. Sidik, I.M. Adamu, M.M. Jamil, G.H.R. Kefayati, R. Mamat, G. Najafi, Recent progress on hybrid nanofluids in heat transfer applications: a comprehensive review. Int. Commun. Heat Mass Transf. 78, 68–79 (2016)
G. Huminic, A. Huminic, The influence of hybrid nanofluids on the performances of elliptical tube: recent research and numerical study. Int. J. Heat Mass Transf. 129, 132–143 (2019)
G. Huminic, A. Huminic, Hybrid nanofluids for heat transfer applications–a state-of-the-art review. Int. J. Heat Mass Transf. 125, 82–103 (2018)
C.Y. Wang, Stagnation flow towards a shrinking sheet. Int. J. Non Linear Mech. 43, 377–382 (2008)
Y.Y. Lok, N. Amin, I. Pop, Non-orthogonal stagnation point flow towards a stretching sheet. Int. J. Nonlin. Mech. 41, 622–627 (2006)
K. Bhattacharyya, Dual solutions in unsteady stagnation-point flow over a shrinking sheet. Chin. Phys. Lett. 28, 084702 (2011)
K. Bhattacharyya, Dual solutions in boundary layer stagnation-point flow and mass transfer with chemical reaction past a stretching/shrinking sheet. Int. Comm. Heat Mass Transf. 38, 917–922 (2011)
I. Waini, A. Ishak, I. Pop, Hybrid nanofluid flow induced by an exponentially shrinking sheet. Chin. J. Phys. (2019). https://doi.org/10.1016/j.cjph.2019.12.015
S.S.U. Devi, S.P.A. Devi, Heat transfer enhancement of Cu-Al2O3/water hybrid nanofluid flow over a stretching sheet. J. Nigerian Mathem. Soc. 36, 419–433 (2017)
S.S.U. Devi, S.P.A. Devi, Numerical investigation of three-dimensional hybrid Cu-Al2O3/water nanofluid flow over a stretching sheet with effecting Lorentz force subject to Newtonian heating. Can. J. Phys. 94, 490–496 (2016)
T. Hayat, S. Nadeem, Heat transfer enhancement with Ag–CuO/water hybrid nanofluid. Res. Phys. 7, 2317–2324 (2017)
A.V. Khan, I. Pop, Boundary-layer flow of a nanofluid past a stretching sheet. Int. J. Heat Mass Transf. 53, 2477–2483 (2010)
L.A. Lund, Z. Omara, I. Khan, A.H. Seikh, E.-S.M. Sherif, K.S. Nisar, Stability analysis and multiple solution of Cu–Al2O3/H2O nanofluid contains hybrid nanomaterials over a shrinking surface in the presence of viscous dissipation. J. Mater. Res. Technol. 9(1), 421–432 (2020)
T. Hayat, S. Nadeem, A.U. Khan, Rotating flow of Ag-CuO/H2O hybrid nanofluid with radiation and partial slip boundary effects. Eur. Phys. J. E 41, 75 (2018)
I. Waini, A. Ishak, I. Pop, Hybrid nanofluid flow and heat transfer over a nonlinear permeable stretching/shrinking surface. Int. J. Num. Meth. Heat Fluid Flow 29(9), 3110–3127 (2019)
N. Bachok, A. Ishak, I. Pop, Flow and heat transfer characteristics on a moving plate in a nanofluid. Int. J. Heat Mass Transf. 55, 642–648 (2012)
S. Ahmad, A.M. Rohni, I. Pop, Blasius and Sakiadis problems in nanofluids. Acta Mech. 218, 195–204 (2011)
A.M. Rohni, S. Ahmad, I. Pop, Boundary layer flow over a moving surface in a nanofluid beneath a uniform free stream. Int. J. Numer. Meth. Heat Fluid Flow 21, 828–846 (2011)
A.M. Rohni, S. Ahmad, A.I. Ismail, I. Pop, Flow and heat transfer over an unsteady shrinking sheet with suction in a nanofluid using Buongiorno’s model. Int. Commun. Heat Mass Transf. 43, 75–80 (2013)
A.M. Rohni, S. Ahmad, I. Pop, Flow and heat transfer over an unsteady shrinking sheet with suction in nanofluids. Int. J. Heat Mass Transfer 55, 1888–1895 (2012)
I. Waini, A. Ishak, I. Pop, Unsteady flow and heat transfer past a stretching/shrinking sheet in a hybrid nanofluid. Int. J. Heat Mass Transf. 136, 288–297 (2019)
K. Vajravelu, D. Rollins, Hydromagnetic flow of a second grade fluid over a stretching sheet. Appl. Math. Comput. 148, 783–791 (2004)
M. Sajid, T. Hayat, S. Asghar, Non-similar analytic solution for MHD flow and heat transfer in a third-order fluid over a stretching sheet. Int. J. Heat Mass Transf. 50, 1723–1736 (2007)
A. Keçebaş, M. Yürüsoy, Similarity solutions of unsteady boundary layer equations of a special third grade fluid. Int. J. Eng. Sci. 44, 721–729 (2006)
S. Abbasbandy, T. Hayat, On series solution for unsteady boundary layer equations in a special third grade fluid. Commun. Nonlinar Sci. Numer. Simul. 16, 3140–3146 (2011)
K. Naganthran, R. Nazar, I. Pop, Unsteady stagnation-point flow and heat transfer of a special third grade fluid past a permeable stretching/shrinking sheet. Sci. Rep. 6, 24632 (2016)
K. Vajravelu, T. Roper, Flow and heat transfer in a second grade fluid over a stretching sheet. Int. J. Nonlinear Mech. 34, 1031–1036 (1999)
T. Hayat, I. Ullah, T. Muhammad, A. Alsaedi, Magnetohydrodynamic (MHD) three-dimensional flow of second grade nanofluid by a convectively heated exponentially stretching surface. J. Mol. Liquids 220, 1004–1012 (2016)
T. Hayat, A. Aziz, T. Muhammad, B. Ahmad, On magnetohydrodynamic flow of second grade nanofluid over a nonlinear stretching sheet. J. Magn. Magn. Mater. 408, 99–106 (2016)
M. Ramzan, M. Bilal, U. Farooq, J.D. Chung, Mixed convective radiative flow of second grade nanofluid with convective boundary conditions: an optimal solution. Res. Phys. 6, 796–804 (2016)
M. Khan, M.U. Rahman, Flow and heat transfer to modified second grade fluid over a non-linear stretching sheet. AIP Adv. 5, 087157 (2015)
A.U. Awan, S. Abid, N. Ullah, S. Nadeem, Magnetohydrodynamic oblique stagnation point flow of second grade fluid over an oscillatory stretching surface. Res. Phys. 18, 103233 (2020)
M. Imtiaz, F. Mabood, T. Hayat, A. Alsaedi, Homogeneous-heterogeneous reactions in MHD radiative flow of second grade fluid due to a curved stretching surface. Int. J. Heat Mass Transf. 145, 118781 (2019)
R.L. Fosdick, K.R. Rajagopal, Anomalous features in the model of “second order fluids”. Arch. Rational Mech. Anal. 70, 145–152 (1979)
J.H. Merkin, On dual solutions occurring in mixed convection in a porous medium. J. Eng. Math. 20, 171–179 (1985)
P.D. Weidman, D.G. Kubittschek, A.M.J. Davis, The effect of transpiration on self-similar boundary layer flow over moving surfaces. Int. J. Eng. Sci. 44, 730–737 (2006)
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Rights and permissions
About this article
Cite this article
Roy, N.C., Pop, I. Flow and heat transfer of a second-grade hybrid nanofluid over a permeable stretching/shrinking sheet. Eur. Phys. J. Plus 135, 768 (2020). https://doi.org/10.1140/epjp/s13360-020-00788-9
Received:
Accepted:
Published:
DOI: https://doi.org/10.1140/epjp/s13360-020-00788-9