Abstract
This paper is concern with the investigation of buoyancy-driven hydromagnetic mixed convection stagnation-point flow of nanofluids over a stretching sheet in the presence of induced magnetic field. Three types of nanofluids namely, Cu–water, \(\hbox {Al}_2\hbox {O}_3\)–water, \(\hbox {TiO}_2\)–water are considered. The resulting system is solved numerically by a fifth-order Runge–Kutta–Fehlberg scheme with shooting technique. Numerical results are validated by comparing the present results with previously published results. Investigations predict that the effects of magnetic field is to increase induced magnetic field, whereas reverse effects is seen by increasing the volume fraction of the nanoparticles.
Similar content being viewed by others
Abbreviations
- \(C_f\) :
-
Skin friction coefficient
- \(Gr\) :
-
Grashof number
- \(\overrightarrow{H}\) :
-
Magnetic induction vector
- \(H_0\) :
-
Uniform magnetic field at infinity upstream
- \(H_1\) :
-
Magnetic component at x-direction
- \(H_2\) :
-
Magnetic component at y-direction
- \(\overrightarrow{J}\) :
-
Current density
- \(L\) :
-
Characteristic length
- \(M\) :
-
Magnetic parameter
- \({Nu_x}\) :
-
Local Nusselt number
- \(p\) :
-
Fluid pressure
- \(P\) :
-
Magnetohydrodynamic pressure
- \(Pr\) :
-
Thermal Prandtl number
- \(Pr_m\) :
-
Magnetic Prandtl number
- \({Re}\) :
-
Reynolds number
- \(S\) :
-
Suction/injection parameter
- \(T\) :
-
Temperature of the fluid
- \(T_{\infty }\) :
-
Free stream temperature
- \(T_w\) :
-
Temperature at the wall
- \(u\) :
-
Velocity component in x-direction
- \({u_w}\) :
-
Stretching/shrinking sheet velocity
- \({u_e}\) :
-
Free stream velocity of the nanofluid
- \(\overrightarrow{V}\) :
-
Fluid velocity
- \(v\) :
-
Velocity component in y-direction
- \(x,y\) :
-
Direction along and perpendicular to the sheet, respectively
- \({\alpha _{nf}}\) :
-
Effective thermal diffusivity of the nanofluid
- \({\alpha _f}\) :
-
Fluid thermal diffusivity
- \(\alpha _1\) :
-
Magnetic diffusivity
- \({\beta }\) :
-
Coefficient of thermal expansion
- \({{\beta }_{nf}}\) :
-
Thermal expansion of nanofluid
- \({\beta _f}\) :
-
Thermal expansion coefficient of the fluid
- \({\beta _s}\) :
-
Thermal expansion coefficient of the nanoparticles
- \({\phi } \) :
-
Solid volume fraction of the nanoparticles
- \({\eta }\) :
-
Similarity variable
- \({\varLambda }\) :
-
Buoyancy parameter
- \(\mu _0\) :
-
Magnetic permeability
- \({\mu _{nf}}\) :
-
Effective dynamic viscosity of the nanofluid
- \({\mu _f}\) :
-
Dynamic viscosity of the fluid
- \({\nu _f}\) :
-
Kinematic viscosity of the fluid
- \({\rho _{nf}}\) :
-
Effective density of the nanofluid
- \(\theta \) :
-
Dimensionless temperature of the fluid
- \({\psi }\) :
-
Stream function
- \(\sigma \) :
-
Electric conductivity
- \(\kappa _{nf}\) :
-
Effective thermal conductivity of the nanofluid
- \(\kappa _f\) :
-
Thermal conductivity of the fluid
- \('\) :
-
Differentiation with respect to y
- \({nf}\) :
-
Nanofluid
- \({f}\) :
-
Fluid
- \({s}\) :
-
Solid
References
Sakiadis BC (1961) Boundary-layer behavior on continuous solid surfaces: I. Boundary-layer equations for two-dimensional and axisymmetric flow. AIChE J 7:26–28
Crane LJ (1970) Flow past a s tretching plate. J Appl Math Phys (ZAMP) 21:645–647
Fang T, Zhang J (2008) Flow between two stretchable disks—an exact solution of the Navier–Stokes equations. Int Commun Heat Mass Transf 35(8):892–895
Hayat T, Abbas Z, Sajid M (2009) MHD stagnation-point flow of an upper-convected Maxwell fluid over a stretching surface. Chaos Solitons Fractals 39:840–848
Kumaran V, Banerjee AK, Vanav AK, Vajravelu K (2009) MHD flow past a stretching permeable sheet. Appl Math Comput 210:26–32
Hamad MAA, Pop I, Md. Ismailc AI (2011) Magnetic field effects on free convection flow of a nanofluid past a vertical semi-infinite flat plate. Nonlinear Anal Real World Appl 12:1338–1346
Wubshet I, Bandari S, Mahantesh MN (2013) MHD stagnation point flow and heat transfer due to nanofluid towards a stretching sheet. Int J Heat Mass Transf 56:1–9
Aydin O, Kaya A (2009) MHD mixed convection of a viscous dissipating fluid about a permeable vertical flat plate. Appl Math Model 33:4086–4096
Ahmed S, Anwar OB, Vedad S, Zeinalkhani M, Heidari A (2012) Mathematical modelling of magnetohydrodynamic transient free and forced convective flow with induced magnetic field effects. Int J Pure Appl Sci Technol 11(1):109–125
Elbashbeshy EMA, Aldawody DA (2010) Effects of thermal radiation and magnetic field on unsteady mixed convection flow and heat transfer over a porous stretching surface. Int J Nonlinear Sci 9(4):448–454
Hamad MAA (2011) Analytic solution of natural convection flow of a nanofluid over a linearly stretching sheet in the presence of magnatic field. Int Commun Heat Mass Transf 38:487–492
Ali FM, Nazar R, Arifin NM, Pop I (2013) Dual solutions in MHD flow on a nonlinear porous shrinking sheet in a viscous fluid. Bound Value Probl 32:2013
Ishak A, Nazar R, Bachok A, Pop I (2010) MHD mixed convection flow near the stagnation-point on a vertical permeable surface. Phys A 389:40–46
Kuznetsov AV, Nield DA (2010) Natural convective boundary-layer flow of a nanofluid past a vertical plate. Int J Therm Sci 49(2):243–247
Vajravelu K, Prasad KV, Lee J, Lee C (2011) Convective heat transfer in the flow of viscous Ag–water and Cu–water nanofluids over a stretching surface. Int J Therm Sci 50(5):843–851
Ali FM, Nazar R, Arifin NM, Pop I (2011) MHD stagnation-point flow and heat transfer towards stretching sheet with induced magnetic field. Appl Math Mech Engl Ed 32:409–418
Anwar O, Bakier AY, Prasade VR, Zueco J, Ghosh SK (2009) Nonsimilar laminar steady electrically conducting forced convection liquid metal boundary layer flow with induced magnetic field effects. Int J Therm Sci 48:1596–1606
Nadeem S, Akbar NS (2011) Influence of heat and mass transfer on the peristaltic flow of a Johnson Segalman fluid in a vertical asymmetric channel with induced MHD. J Taiwan Inst Chem Eng 42:58–66
Kumari M, Takhar HS, Nath G (1990) MHD flow and heat transfer over a stretching surface with prescribed wall temperature or heat flux. Warme und Stoff Ubertragung 25(6):331–336
Ishak A, Nazar R, Arifin NM, Pop I (2007) Dual solution in magnetodyrodynamic mixed convective fow near a stagnation-point on a vertical surface. ASME J Heat Transf 129:1212–1216
Ali FM, Nazar R, Arifin NM, Pop I (2010) MHD mixed convective boundary layer flow towards a stagnation point on a vertical surface with induced magnetic field. ASME J Heat Transf 133(2):022502 (6 pages)
Ghosh SK, Anwar Bg O, Zueco J (2010) Hydromagnetic free convection flow with induced magnetic field effects. Meccanica 45:175–185
Jafar K, Nazar R, Ishak A, Pop I (2010) Magnetohydrodynamic flow over amoving plate in a parallel stream with induced magnetic field. Int J Miner Mater 17(4):397–402
Jafar K, Nazar R, Ishak A, Pop I (2013) MHD boundary layer flow due to a moving wedge with induced magnetic field. Bound Value Probl 2013:20
Lotfi R, Saboohi Y, Rashidi A (2010) Numerical study of forced convective heat transfer of nanofluids: comparison of different approaches. Int Commun Heat Mass Transf 37(1):74–78
Choi S, Zhang Z, Yu W, Lockwood F, Grulke E (2001) Anomalously thermal conductivity enhancement in nanotube suspensions. Appl Phys Lett 79(14):2252–2254
Karwe MV, Jaluria Y (1991) Numerical simulation of thermal transport associated with a continuous moving flat sheet in materials processing. ASME J Heat Transf 113:612–619
Cowling TG (1957) Magnetohydrodynamics. Interscience, New York
Ahmed S (2010) Induced magnetic field with radiating fluid over a porous vertical plate: analytical study. J Nav Archit Mar Eng. doi:10.3329/jname.v7i2.5662
Ali FM, Nazar R, Arifin NM, Pop I (2011) MHD boundary layer flow and heat transfer over a stretching sheet with induced mahnetic field. Heat Mass Transf 47:155–162
Oztop HF, Abu-Nada E (2008) Numerical study of natural convection in partially heated rectangular enclosures filled with nanofluids. Int J Heat Fluid Flow 29:1326–1336
Ishak A, Nazar R, Pop I (2006) Mixed convection boundary layers in the stagnation-point flow towards a stretching vertical sheet. Meccanica 41:509–518
Acknowledgments
We are very thankful to the Editor and the referees for their valuable comments and suggestions, which have definitely improved the quality of the paper considerably.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Pal, D., Mandal, G. MHD convective stagnation-point flow of nanofluids over a non-isothermal stretching sheet with induced magnetic field. Meccanica 50, 2023–2035 (2015). https://doi.org/10.1007/s11012-015-0153-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11012-015-0153-9