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Heat and Mass Transfer

, Volume 50, Issue 5, pp 639–650 | Cite as

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

  • Hossein Tamim
  • Saeed DinarvandEmail author
  • Reza Hosseini
  • Ioan Pop
Original

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.

Keywords

Mixed Convection Base Fluid Thermal Boundary Layer Heat Transfer Characteristic Dual Solution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

List of symbols

a, b

Constant

Cf

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

qw

Surface heat flux

T

Fluid temperature

Tw

Surface temperature

\( T_{\infty } \)

Ambient temperature

u, v

Velocity components

x, y

Cartesian coordinates

U(x)

Free stream velocity

f(η)

Dimensionless stream function

Greek symbols

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

Subscripts

w

Condition at the surface of the plate

\( \infty \)

Ambient condition

f

Fluid

nf

Nanofluid

s

Solid

Superscript

\( ^{'} \)

Differentiation with respect to η

References

  1. 1.
    Ece MC (2005) Free convection flow about a cone under mixed thermal boundary conditions and a magnetic field. Appl Math Model 29:1121–1134CrossRefzbMATHGoogle Scholar
  2. 2.
    Ramachandran N, Chen TS, Armaly BF (1988) Mixed convection in stagnation flows adjacent to a vertical surface. ASME J Heat Transf 110:373–377CrossRefGoogle Scholar
  3. 3.
    Devi CDS, Takhar HS, Nath G (1991) Unsteady mixed convection flow in stagnation region adjacent to a vertical surface. Heat Mass Transf 26:71–79Google Scholar
  4. 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 15:654–670CrossRefGoogle Scholar
  5. 5.
    Ridha A (1996) Aiding flows non-unique similarity solutions of mixed-convection boundary-layer equations. J Appl Math Phys (ZAMP) 47:341–352CrossRefzbMATHMathSciNetGoogle Scholar
  6. 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 389:40–46CrossRefGoogle Scholar
  7. 7.
    Das SK, Choi SUS, Yu W, Pradeep T (2007) Nanofluids: science and technology. Wiley, New JerseyCrossRefGoogle Scholar
  8. 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 66:99–105Google Scholar
  9. 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 45:495–500CrossRefGoogle Scholar
  10. 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: 3–11Google Scholar
  11. 11.
    Wang XQ, Mujumdar AS (2007) Heat transfer characteristics of nanofluids: a review. Int J Thermal Sci 46:1–19CrossRefzbMATHGoogle Scholar
  12. 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 29:242–249CrossRefGoogle Scholar
  13. 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 50:2002–2018CrossRefzbMATHGoogle Scholar
  14. 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 29:1326–1336CrossRefGoogle Scholar
  15. 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 52:5792–5795CrossRefzbMATHGoogle Scholar
  16. 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 55:1888–1895CrossRefGoogle Scholar
  17. 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 56:48–57CrossRefGoogle Scholar
  18. 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 55:642–648CrossRefzbMATHGoogle Scholar
  19. 19.
    Chen TS, Mucoglu A (1975) Buoyancy effects on forced convection along a vertical cylinder. ASME J Heat Transf 97:198–203CrossRefGoogle Scholar
  20. 20.
    Mahmood T, Merkin JH (1988) Mixed convection on a vertical circular cylinder. Appl Math Phys (ZAMP) 39:186–203CrossRefzbMATHGoogle Scholar
  21. 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 42:1010–1017CrossRefzbMATHGoogle Scholar
  22. 22.
    Grosan T, Pop I (2011) Axisymmetric mixed convection boundary layer flow past a vertical cylinder in a nanofluid. Int J Heat Mass Transf 54:3139–3145CrossRefzbMATHGoogle Scholar
  23. 23.
    Merkin JH, Mahmood T (1990) On the free convection boundary layer on a vertical plate with prescribed surface heat flux. J Eng Math 24:95–107CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Wilks G, Bramley JS (1981) Dual solutions in mixed convection. Proc R Soc Edinburgh 87A:349–358CrossRefMathSciNetGoogle Scholar
  25. 25.
    Merkin JH, Pop I (1996) Conjugate free convection on a vertical surface. Int J Heat Mass Transf 39:1527–1534CrossRefzbMATHGoogle Scholar
  26. 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 19:459–483CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Hossein Tamim
    • 1
  • Saeed Dinarvand
    • 1
    Email author
  • Reza Hosseini
    • 1
  • Ioan Pop
    • 2
  1. 1.Mechanical Engineering DepartmentAmirkabir University of TechnologyTehranIran
  2. 2.Mathematics DepartmentUniversity of ClujClujRomania

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