Skip to main content
SpringerLink
Log in

MHD mixed convection stagnation-point flow of a nanofluid over a vertical permeable surface: a comprehensive report of dual solutions

  • Original
  • Published:
Heat and Mass Transfer Aims and scope Submit manuscript

Abstract

The steady laminar magnetohydrodynamic mixed convection boundary layer flow of a nanofluid near the stagnation-point on a vertical permeable plate with prescribed external flow and surface temperature is investigated in this study. Here, both assisting and opposing flows are considered and studied. Using appropriate similarity variables, the governing equations are transformed into nonlinear ordinary differential equations in the dimensionless stream function, which is solved numerically using the Runge–Kutta scheme coupled with a conventional shooting procedure. Three different types of nanoparticles, namely copper Cu, alumina Al2O3 and titania TiO2 with water as the base fluid are considered. Numerical results are obtained for the skin-friction coefficient and Nusselt number as well as for the velocity and temperature profiles for some values of the governing parameters, namely, the volume fraction of nanoparticles ϕ, permeability parameter f o , magnetic parameter M and mixed convection parameter λ. It is found that dual solutions exist for both assisting and opposing flows, and the range of the mixed convection parameter for which the solution exists, increases with suction, magnetic field and volume fraction of nanoparticles.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22
Fig. 23
Fig. 24
Fig. 25
Fig. 26
Fig. 27

Similar content being viewed by others

Abbreviations

a, b :

Constant

C f :

Skin friction coefficient

g :

Acceleration due to gravity

k :

Thermal conductivity

\( Nu_{x} \) :

Local Nusselt number

\( Gr_{x} \) :

Local Grashof number

\( Re_{x} \) :

Local Reynolds number

Pr :

Prandtl number

q w :

Surface heat flux

T :

Fluid temperature

T w :

Surface temperature

\( T_{\infty } \) :

Ambient temperature

u, v :

Velocity components

x, y :

Cartesian coordinates

U(x):

Free stream velocity

f(η):

Dimensionless stream function

α :

Thermal diffusivity

β :

Thermal expansion coefficient

ϕ :

Nanoparticle volume fraction

η :

Similarity variable

θ(η):

Dimensionless temperature

λ :

Buoyancy or mixed convection parameter

μ :

Dynamic viscosity

υ :

Kinematic viscosity

ρ :

Fluid density

\( \tau_{w} \) :

Wall shear stress

Ψ :

Stream function

w :

Condition at the surface of the plate

\( \infty \) :

Ambient condition

f :

Fluid

nf :

Nanofluid

s :

Solid

\( ^{'} \) :

Differentiation with respect to η

References

  1. Ece MC (2005) Free convection flow about a cone under mixed thermal boundary conditions and a magnetic field. Appl Math Model 29:1121–1134

    Article  MATH  Google Scholar 

  2. Ramachandran N, Chen TS, Armaly BF (1988) Mixed convection in stagnation flows adjacent to a vertical surface. ASME J Heat Transf 110:373–377

    Article  Google Scholar 

  3. Devi CDS, Takhar HS, Nath G (1991) Unsteady mixed convection flow in stagnation region adjacent to a vertical surface. Heat Mass Transf 26:71–79

    Google Scholar 

  4. Lok YY, Amin N, Campean D, Pop I (2005) Steady mixed convection flow of a micropolar fluid near the stagnation point on a vertical surface. Int J Numer Methods Heat Fluid Flow 15:654–670

    Article  Google Scholar 

  5. Ridha A (1996) Aiding flows non-unique similarity solutions of mixed-convection boundary-layer equations. J Appl Math Phys (ZAMP) 47:341–352

    Article  MATH  MathSciNet  Google Scholar 

  6. Ishak A, Nazar R, Bachok N, Pop I (2010) MHD mixed convection flow near the stagnation-point on a vertical permeable surface. Phys A 389:40–46

    Article  Google Scholar 

  7. Das SK, Choi SUS, Yu W, Pradeep T (2007) Nanofluids: science and technology. Wiley, New Jersey

    Book  Google Scholar 

  8. Choi SUS (1995) Enhancing thermal conductivity of fluids with nanoparticles, In: The Proceedings of the 1995 ASME International Mechanical Engineering Congress, San Francisco, USA, ASME, FED 231/MD 66:99–105

  9. Akoh H, Tsukasaki Y, Yatsuya S, Tasaki A (1978) Magnetic properties of ferromagnetic ultrafine particles prepared by a vacuum evaporation on running oil substrate. J Crystal Growth 45:495–500

    Article  Google Scholar 

  10. Eastman JA, Choi US, Li S, Thompson LJ, Lee S (1997) Enhanced thermal conductivity through the development of nanofluids, In: Materials Research Society Symposium e Proceedings, Materials Research Society, Pittsburgh 457: 3–11

  11. Wang XQ, Mujumdar AS (2007) Heat transfer characteristics of nanofluids: a review. Int J Thermal Sci 46:1–19

    Article  MATH  Google Scholar 

  12. Abu-Nada E (2008) Application of nanofluids for heat transfer enhancement of separated flows encountered in a backward facing step. Int J Heat Fluid Flow 29:242–249

    Article  Google Scholar 

  13. Tiwari RJ, Das MK (2007) Heat transfer augmentation in a two-sided lid-driven differentially heated square cavity utilizing nanofluids. Int J Heat Mass Transf 50:2002–2018

    Article  MATH  Google Scholar 

  14. 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

    Article  Google Scholar 

  15. Nield DA, Kuznetsov AV (2009) The ChengeMinkowycz problem for natural convective boundary-layer flow in a porous medium saturated by a nanofluid. Int J Heat Mass Transf 52:5792–5795

    Article  MATH  Google Scholar 

  16. Rohni AM, Ahmad S, Pop I (2012) Flow and heat transfer over an unsteady shrinking sheet with suction in nanofluids. Int J Heat Mass Transf 55:1888–1895

    Article  Google Scholar 

  17. Aziz A, Khan WA, Pop I (2012) Free convection boundary layer flow past a horizontal flat plate embedded in porous medium filled by nanofluid containing gyrotactic microorganisms. Int J Thermal Sci 56:48–57

    Article  Google Scholar 

  18. Bachok N, Ishak A, Pop I (2012) Flow and heat transfer characteristics on a moving plate in a nanofluid. Int J Heat Mass Transf 55:642–648

    Article  MATH  Google Scholar 

  19. Chen TS, Mucoglu A (1975) Buoyancy effects on forced convection along a vertical cylinder. ASME J Heat Transf 97:198–203

    Article  Google Scholar 

  20. Mahmood T, Merkin JH (1988) Mixed convection on a vertical circular cylinder. Appl Math Phys (ZAMP) 39:186–203

    Article  MATH  Google Scholar 

  21. Ishak A, Nazar R, Pop I (2007) The effects of transpiration on the boundary layer flow and heat transfer over a vertical slender cylinder. Int J Non-Linear Mech 42:1010–1017

    Article  MATH  Google Scholar 

  22. Grosan T, Pop I (2011) Axisymmetric mixed convection boundary layer flow past a vertical cylinder in a nanofluid. Int J Heat Mass Transf 54:3139–3145

    Article  MATH  Google Scholar 

  23. Merkin JH, Mahmood T (1990) On the free convection boundary layer on a vertical plate with prescribed surface heat flux. J Eng Math 24:95–107

    Article  MATH  MathSciNet  Google Scholar 

  24. Wilks G, Bramley JS (1981) Dual solutions in mixed convection. Proc R Soc Edinburgh 87A:349–358

    Article  MathSciNet  Google Scholar 

  25. Merkin JH, Pop I (1996) Conjugate free convection on a vertical surface. Int J Heat Mass Transf 39:1527–1534

    Article  MATH  Google Scholar 

  26. Lok YY, Pop I, Ingham DB, Amin N (2009) Mixed convection flow of a micropolar fluid near a non-orthogonal stagnation-point on a stretching vertical sheet. Int J Numer Methods Heat Fluid Flow 19:459–483

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mechanical Engineering Department, Amirkabir University of Technology, 424 Hafez Avenue, Tehran, Iran

    Hossein Tamim, Saeed Dinarvand & Reza Hosseini

  2. Mathematics Department, University of Cluj, CP 253, 3400, Cluj, Romania

    Ioan Pop

Authors
  1. Hossein Tamim
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Saeed Dinarvand
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Reza Hosseini
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Ioan Pop
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Saeed Dinarvand.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Tamim, H., Dinarvand, S., Hosseini, R. et al. MHD mixed convection stagnation-point flow of a nanofluid over a vertical permeable surface: a comprehensive report of dual solutions. Heat Mass Transfer 50, 639–650 (2014). https://doi.org/10.1007/s00231-013-1264-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00231-013-1264-2

Keywords

Search

Navigation