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Numerical solutions of free convection boundary layer flow on a solid sphere with Newtonian heating in a micropolar fluid

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Abstract

In this paper, the problem of free convection boundary layer flow on a solid sphere in a micropolar fluid with Newtonian heating, in which the heat transfer from the surface is proportional to the local surface temperature, is considered. The transformed boundary layer equations in the form of partial differential equations are solved numerically using an implicit finite-difference scheme. Numerical solutions are obtained for the local wall temperature, the local skin friction coefficient, as well as the velocity, angular velocity and temperature profiles. The features of the flow and heat transfer characteristics for different values of the material or micropolar parameter K, the Prandtl number Pr and the conjugate parameter γ are analyzed and discussed.

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Abbreviations

a :

radius of the sphere

h s :

heat transfer parameter for Newtonian heating

C f :

skin friction coefficient

f :

dimensionless stream function

g :

acceleration due to gravity

Gr :

Grashof number

H :

angular velocity of micropolar fluid

j :

microinertia density

K :

material parameter of micropolar fluid

k :

thermal conductivity

Pr :

Prandtl number

Re :

Reynolds number

T :

fluid temperature

T ∞ :

ambient temperature

U ∞ :

free stream velocity

u,v :

velocity components along the x and y directions, respectively

x,y :

Cartesian coordinates along the sphere and normal to it, respectively

β :

thermal expansion coefficient

γ :

conjugate parameter for Newtonian heating

μ :

dynamic viscosity

ν :

kinematic viscosity

θ :

dimensionless temperature

κ :

vortex viscosity

φ :

spin gradient viscosity

ρ :

fluid density

ψ :

stream function

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Acknowledgement

The authors gratefully acknowledge the financial supports received from the Ministry of Higher Education, Malaysia (UKM-ST-07-FRGSS0036-2009) and the Universiti Malaysia Pahang (RDU110108). They also wish to express their very sincere thanks to the reviewers for the valuable comments and suggestions.

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Authors and Affiliations

  1. Faculty of Industrial Science and Technology, Universiti Malaysia Pahang, 26300, UMP Kuantan, Pahang, Malaysia

    M. Z. Salleh

  2. School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600, UKM Bangi, Selangor, Malaysia

    R. Nazar

  3. Solar Energy Research Institute, Universiti Kebangsaan Malaysia, 43600, UKM Bangi, Selangor, Malaysia

    R. Nazar

  4. Faculty of Mathematics, University of Cluj, 3400, Cluj, CP 253, Romania

    I. Pop

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  1. M. Z. Salleh
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  2. R. Nazar
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  3. I. Pop
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Correspondence to M. Z. Salleh.

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Salleh, M.Z., Nazar, R. & Pop, I. Numerical solutions of free convection boundary layer flow on a solid sphere with Newtonian heating in a micropolar fluid. Meccanica 47, 1261–1269 (2012). https://doi.org/10.1007/s11012-011-9510-5

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  • DOI: https://doi.org/10.1007/s11012-011-9510-5

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