, Volume 50, Issue 8, pp 2023–2035 | Cite as

MHD convective stagnation-point flow of nanofluids over a non-isothermal stretching sheet with induced magnetic field



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.


Nanofluids Stagnation-point flow Induced magnetic field Mixed convection Nanoparticles Stretching sheet 

List of symbols


Skin friction coefficient


Grashof number


Magnetic induction vector


Uniform magnetic field at infinity upstream


Magnetic component at x-direction


Magnetic component at y-direction


Current density


Characteristic length


Magnetic parameter


Local Nusselt number


Fluid pressure


Magnetohydrodynamic pressure


Thermal Prandtl number


Magnetic Prandtl number


Reynolds number


Suction/injection parameter


Temperature of the fluid

\(T_{\infty }\)

Free stream temperature


Temperature at the wall


Velocity component in x-direction


Stretching/shrinking sheet velocity


Free stream velocity of the nanofluid


Fluid velocity


Velocity component in y-direction


Direction along and perpendicular to the sheet, respectively

Greek symbols

\({\alpha _{nf}}\)

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










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.


  1. 1.
    Sakiadis BC (1961) Boundary-layer behavior on continuous solid surfaces: I. Boundary-layer equations for two-dimensional and axisymmetric flow. AIChE J 7:26–28CrossRefGoogle Scholar
  2. 2.
    Crane LJ (1970) Flow past a s tretching plate. J Appl Math Phys (ZAMP) 21:645–647CrossRefGoogle Scholar
  3. 3.
    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):892–895CrossRefGoogle Scholar
  4. 4.
    Hayat T, Abbas Z, Sajid M (2009) MHD stagnation-point flow of an upper-convected Maxwell fluid over a stretching surface. Chaos Solitons Fractals 39:840–848ADSCrossRefGoogle Scholar
  5. 5.
    Kumaran V, Banerjee AK, Vanav AK, Vajravelu K (2009) MHD flow past a stretching permeable sheet. Appl Math Comput 210:26–32MathSciNetCrossRefGoogle Scholar
  6. 6.
    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 12:1338–1346MathSciNetCrossRefGoogle Scholar
  7. 7.
    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 56:1–9CrossRefGoogle Scholar
  8. 8.
    Aydin O, Kaya A (2009) MHD mixed convection of a viscous dissipating fluid about a permeable vertical flat plate. Appl Math Model 33:4086–4096CrossRefGoogle Scholar
  9. 9.
    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):109–125Google Scholar
  10. 10.
    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):448–454MathSciNetGoogle Scholar
  11. 11.
    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 38:487–492CrossRefGoogle Scholar
  12. 12.
    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 32:2013MathSciNetGoogle Scholar
  13. 13.
    Ishak A, Nazar R, Bachok A, Pop I (2010) MHD mixed convection flow near the stagnation-point on a vertical permeable surface. Phys A 389:40–46CrossRefGoogle Scholar
  14. 14.
    Kuznetsov AV, Nield DA (2010) Natural convective boundary-layer flow of a nanofluid past a vertical plate. Int J Therm Sci 49(2):243–247MathSciNetCrossRefGoogle Scholar
  15. 15.
    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):843–851CrossRefGoogle Scholar
  16. 16.
    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 32:409–418MathSciNetCrossRefGoogle Scholar
  17. 17.
    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 48:1596–1606CrossRefGoogle Scholar
  18. 18.
    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 42:58–66CrossRefGoogle Scholar
  19. 19.
    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):331–336ADSCrossRefGoogle Scholar
  20. 20.
    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 129:1212–1216CrossRefGoogle Scholar
  21. 21.
    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)CrossRefGoogle Scholar
  22. 22.
    Ghosh SK, Anwar Bg O, Zueco J (2010) Hydromagnetic free convection flow with induced magnetic field effects. Meccanica 45:175–185MathSciNetCrossRefGoogle Scholar
  23. 23.
    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):397–402CrossRefGoogle Scholar
  24. 24.
    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 2013:20MathSciNetCrossRefGoogle Scholar
  25. 25.
    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):74–78CrossRefGoogle Scholar
  26. 26.
    Choi S, Zhang Z, Yu W, Lockwood F, Grulke E (2001) Anomalously thermal conductivity enhancement in nanotube suspensions. Appl Phys Lett 79(14):2252–2254ADSCrossRefGoogle Scholar
  27. 27.
    Karwe MV, Jaluria Y (1991) Numerical simulation of thermal transport associated with a continuous moving flat sheet in materials processing. ASME J Heat Transf 113:612–619CrossRefGoogle Scholar
  28. 28.
    Cowling TG (1957) Magnetohydrodynamics. Interscience, New YorkGoogle Scholar
  29. 29.
    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
  30. 30.
    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 47:155–162ADSCrossRefGoogle Scholar
  31. 31.
    Oztop HF, Abu-Nada E (2008) Numerical study of natural convection in partially heated rectangular enclosures filled with nanofluids. Int J Heat Fluid Flow 29:1326–1336CrossRefGoogle Scholar
  32. 32.
    Ishak A, Nazar R, Pop I (2006) Mixed convection boundary layers in the stagnation-point flow towards a stretching vertical sheet. Meccanica 41:509–518CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.Department of Mathematics, Siksha BhavanaVisva-Bharati UniversitySantiniketanIndia
  2. 2.Siksha SatraVisva-Bharati UniversitySriniketanIndia

Personalised recommendations