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


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.


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



Skin friction coefficient


Acceleration due to gravity


Thermal conductivity

\( Nu_{x} \)

Local Nusselt number

\( Gr_{x} \)

Local Grashof number

\( Re_{x} \)

Local Reynolds number


Prandtl number


Surface heat flux


Fluid temperature


Surface temperature

\( T_{\infty } \)

Ambient temperature

u, v

Velocity components

x, y

Cartesian coordinates


Free stream velocity


Dimensionless stream function

Greek symbols


Thermal diffusivity


Thermal expansion coefficient


Nanoparticle volume fraction


Similarity variable


Dimensionless temperature


Buoyancy or mixed convection parameter


Dynamic viscosity


Kinematic viscosity


Fluid density

\( \tau_{w} \)

Wall shear stress


Stream function



Condition at the surface of the plate

\( \infty \)

Ambient condition








\( ^{'} \)

Differentiation with respect to η


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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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