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


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.


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


dimensionless needle size


skin friction coefficient


dimensionless stream function


acceleration due to gravity


Grashof number


thermal conductivity


characteristic length of the needle


power index


local Nusselt number


Prandtl number


characteristic heat flux

qw (x)

dimensionless heat flux from the surface of the needle


Reynolds number


local Reynolds number


dimensionless needle radius


dimensionless local fluid temperature


dimensionless temperature for the surface of the needle


ambient temperature


dimensionless velocity components along the x and r directions, respectively


characteristic velocity of the moving needle

Uw (x)

dimensionless velocity of the moving needle

x, r

dimensionless axial and radial coordinates, respectively

Greek symbols


thermal diffusivity


thermal expansion coefficient


similarity variable


mixed convection parameter


dimensionless temperature


dynamic viscosity


kinematic viscosity


fluid density


skin friction from the surface of the needle


stream function



condition at the surface of the needle

condition at infinity


differentiation with respect to η

dimensional variables



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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