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Heat and Mass Transfer

, 44:473 | Cite as

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

  • S. Ahmad
  • N. M. Arifin
  • R. NazarEmail author
  • I. Pop
Original

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.

Keywords

Mixed Convection Boundary Layer Flow Thermal Boundary Layer Skin Friction Coefficient Needle Size 
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

dimensionless needle size

Cf

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

Nux

local Nusselt number

Pr

Prandtl number

q0

characteristic heat flux

qw (x)

dimensionless heat flux from the surface of the needle

Re

Reynolds number

Rex

local Reynolds number

R(x)

dimensionless needle radius

T

dimensionless local fluid temperature

Tw

dimensionless temperature for the surface of the needle

T

ambient temperature

u,v

dimensionless velocity components along the x and r directions, respectively

U0

characteristic velocity of the moving needle

Uw (x)

dimensionless velocity of the moving needle

x, r

dimensionless axial and radial coordinates, respectively

Greek symbols

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

τw

skin friction from the surface of the needle

ψ

stream function

Subscripts

w

condition at the surface of the needle

condition at infinity

Superscripts

differentiation with respect to η

dimensional variables

Notes

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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Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Institute for Mathematical Research & Department of MathematicsUniversiti Putra MalaysiaUPM SerdangMalaysia
  2. 2.School of Mathematical SciencesNational University of MalaysiaUKM BangiMalaysia
  3. 3.Faculty of MathematicsUniversity of ClujClujRomania

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