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Mixed convection boundary layer flow along vertical moving thin needles with variable heat flux

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Abstract

The problem of steady laminar mixed convection boundary layer flow of an incompressible viscous fluid along vertical moving thin needles with variable heat flux for both assisting and opposing flow cases is theoretically considered in this paper. The governing boundary layer equations are first transformed into non-dimensional forms. The curvature effects are incorporated into the analysis whereas the pressure variation in the axial direction has been neglected. These equations are then transformed into similarity equations using the similarity variables, which are solved numerically using an implicit finite-difference scheme known as the Keller-box method. The solutions are obtained for a blunt-nosed needle (m = 0). Numerical calculations are carried out for various values of the dimensionless parameters of the problem, which include the mixed convection parameter λ, the Prandtl number Pr and the parameter a representing the needle size. It is shown from the numerical results that the skin friction coefficient, the surface (wall) temperature and the velocity and temperature profiles are significantly influenced by these parameters. The results are presented in graphical form and are discussed in detail.

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Abbreviations

a :

dimensionless needle size

C f :

skin friction coefficient

f :

dimensionless stream function

g :

acceleration due to gravity

Gr :

Grashof number

k :

thermal conductivity

L :

characteristic length of the needle

m :

power index

Nu x :

local Nusselt number

Pr :

Prandtl number

q 0 :

characteristic heat flux

q w (x):

dimensionless heat flux from the surface of the needle

Re :

Reynolds number

Re x :

local Reynolds number

R(x):

dimensionless needle radius

T :

dimensionless local fluid temperature

T w :

dimensionless temperature for the surface of the needle

T ∞ :

ambient temperature

u,v :

dimensionless velocity components along the x and r directions, respectively

U 0 :

characteristic velocity of the moving needle

U w (x):

dimensionless velocity of the moving needle

x, r :

dimensionless axial and radial coordinates, respectively

α:

thermal diffusivity

β:

thermal expansion coefficient

η:

similarity variable

λ:

mixed convection parameter

θ:

dimensionless temperature

μ:

dynamic viscosity

υ:

kinematic viscosity

ρ:

fluid density

τ w :

skin friction from the surface of the needle

ψ:

stream function

w :

condition at the surface of the needle

∞:

condition at infinity

′:

differentiation with respect to η

−:

dimensional variables

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Acknowledgments

The authors gratefully acknowledge the financial support received in the form of a fundamental research grant (SAGA Fund) from the Academy of Sciences Malaysia and the Ministry of Science, Technology and Innovation (MOSTI), Malaysia. One of the authors (I. Pop) also wishes to thank the Royal Society (London) for partial financial support to enable collaboration on this research. The authors also wish to express their sincere thanks to the reviewers for the valuable comments and suggestions.

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Authors and Affiliations

  1. Institute for Mathematical Research & Department of Mathematics, Universiti Putra Malaysia, 43400, UPM Serdang, Selangor, Malaysia

    S. Ahmad & N. M. Arifin

  2. School of Mathematical Sciences, National University of Malaysia, 43600, UKM Bangi, Selangor, Malaysia

    R. Nazar

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

    I. Pop

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  1. S. Ahmad
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  2. N. M. Arifin
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  3. R. Nazar
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  4. I. Pop
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Correspondence to R. Nazar.

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Ahmad, S., Arifin, N.M., Nazar, R. et al. Mixed convection boundary layer flow along vertical moving thin needles with variable heat flux. Heat Mass Transfer 44, 473–479 (2008). https://doi.org/10.1007/s00231-007-0263-6

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  • DOI: https://doi.org/10.1007/s00231-007-0263-6

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