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Constant heat flux solution for mixed convection boundary layer viscoelastic fluid

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Abstract

The mixed convection boundary layer of a viscoelastic fluid past a circular cylinder with constant heat flux is discussed. The boundary layer equations are an order higher than those for the Newtonian (viscous) fluid and the adherence boundary conditions are insufficient to determine the solution of these equations completely. The governing non-similar partial differential equations are transformed into dimensionless forms and then solved numerically using the Keller-box method by augmenting an extra boundary condition at infinity. Numerical results obtained in the form of velocity distributions and temperature profiles are presented for a range of values of the dimensionless viscoelastic fluid parameter. It is found that for some values of the viscoelastic parameter and some negative values of the mixed convection parameter (opposing flow) the momentum boundary layer separates from the cylinder. Heating the cylinder delays separation and can, if the cylinder is warm enough, suppress the separation completely. Similar to the case of a Newtonian fluid, cooling the cylinder brings the separation point nearer to the lower stagnation point.

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Abbreviations

\( C_{f} \) :

Skin friction coefficient

\( f(\eta ),F(x,y) \) :

Dimensionless stream functions

\( f^{\prime } (\eta ) \) :

Velocity profiles

\( g \) :

Acceleration due to gravity

\( k_{0} \) :

Constant material

\( K \) :

Viscoelastic parameter

\( Pr \) :

Prandtl number

\( q_{w} \) :

Constant wall heat flux

\( Re \) :

Reynolds number

\( T \) :

Fluid temperature

\( T_{w} \) :

Temperature at the surface of the cylinder

\( T_{\infty } \) :

Temperature of the ambient fluid

\( \bar{u},\bar{v} \) :

Velocity components along \( \bar{x} \) and \( \bar{y} \) directions

\( u,v \) :

Dimensionless velocity components

\( \bar{u}_{e} (\bar{x}) \) :

Velocity outside boundary layer

\( u_{e} (x) \) :

Dimensionless velocity outside boundary layer

\( \bar{x},\bar{y} \) :

Dimensional Cartesian coordinates

\( x,y \) :

Dimensionless Cartesian coordinates

\( x_{s} \) :

Separation point

\( \alpha_{{}} \) :

Thermal diffusivity of the fluid

\( \beta \) :

Volumetric thermal expansion coefficient of the fluid

\( \lambda \) :

Mixed convection parameter

\( \lambda_{c} (K) \) :

Critical value of \( \lambda \)

\( \lambda_{0} \) :

Value of \( \lambda \) below which the boundary layer separation is not possible

\( \eta \) :

Similarity variable

\( \mu \) :

Dynamic viscosity of the fluid

\( \nu \) :

Kinematic viscosity of the fluid

\( \rho \) :

Fluid density

\( \theta (x,y), \theta (\eta ) \) :

Dimensionless temperatures

\( \theta_{w} (x) \) :

Local wall temperature distribution

\( \psi \) :

Stream function

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Acknowledgments

This work is supported by a research grant (Vot FRGS No. 4F109 and 02H80) from Universiti Teknologi Malaysia, UTM, MOHE and would like to express many thanks to an anonymous reviewer for the valuable comment and suggestions.

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Authors and Affiliations

  1. Department of Mathematics, University Teknologi Malaysia, Johor Bahru, Johor, Malaysia

    Abdul Rahman Mohd Kasim, Nurul Farahain Mohammad & Sharidan Shafie

  2. Department of Mathematics, Babeş-Bolyai University, 400084, Cluj-Napoca, Romania

    Ioan Pop

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  1. Abdul Rahman Mohd Kasim
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  2. Nurul Farahain Mohammad
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  3. Sharidan Shafie
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  4. Ioan Pop
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Correspondence to Abdul Rahman Mohd Kasim.

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Mohd Kasim, A.R., Mohammad, N.F., Shafie, S. et al. Constant heat flux solution for mixed convection boundary layer viscoelastic fluid. Heat Mass Transfer 49, 163–171 (2013). https://doi.org/10.1007/s00231-012-1075-x

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