Magnetohydrodynamic free convection boundary layer flow of non-Newtonian tangent hyperbolic fluid from a vertical permeable cone with variable surface temperature

  • S. Abdul Gaffar
  • V. Ramachandra Prasad
  • S. Keshava Reddy
  • O. Anwar Bég
Technical Paper


The nonlinear, non-isothermal steady-state boundary layer flow and heat transfer of an incompressible tangent hyperbolic non-Newtonian (viscoelastic) fluid from a vertical permeable cone with magnetic field are studied. The transformed conservation equations are solved numerically subject to physically appropriate boundary conditions using the second-order accurate implicit finite difference Keller-box technique. The numerical code is validated with previous studies. The influence of a number of emerging non-dimensional parameters, namely a Weissenberg number (We), rheological power law index (m), surface temperature exponent (n), Prandtl number (Pr), magnetic parameter (M) suction/injection parameter (f w ) and dimensionless tangential coordinate (ξ) on velocity and temperature evolution in the boundary layer regime, is examined in detail. Furthermore, the effects of these parameters on surface heat transfer rate and local skin friction are also investigated. It is observed that velocity, surface heat transfer rate and local skin friction are reduced with increasing Weissenberg number, but temperature is increased. Increasing m enhances velocity and surface heat transfer rate but reduces temperature and local skin friction. An increase in non-isothermal power law index (n) is observed to decrease the velocity and temperature. Increasing magnetic parameter (M) is found to decrease the velocity and increase the temperature. Overall, the primary influence on free convection is sustained through the magnetic body force parameter, M, and also the surface mass flux (injection/suction) parameter, f w . The rheological effects, while still prominent, are not as dramatic. Boundary layers (both hydrodynamic and thermal) are, therefore, most strongly modified by the applied magnetic field and wall mass flux effect. The study is pertinent to smart coatings, e.g., durable paints, aerosol deposition processing and water-based solvent thermal treatment in chemical engineering.


Thermo-magnetohydrodynamics Free convection Non-Newtonian tangent hyperbolic fluid Finite difference numerical method Weissenberg number Non-isothermal cone Magnetic field 

List of symbols



Half angle of the cone


Material parameter


Constant imposed magnetic field


Skin friction coefficient


Non-dimensional steam function


Suction (wall transpiration) parameter


Local Grashof number


Acceleration due to gravity


Thermal conductivity of fluid


Local magnetic parameter


Rheological power-law index


Surface temperature exponent


Heat transfer rate (Local Nusselt number)


Prandtl number


Local radius of the cone


Temperature of the fluid

u, v

Non-dimensional velocity components along the x and y directions, respectively


Velocity vector


Local Weissenberg number


Stream wise coordinate


Transverse coordinate



Thermal diffusivity


Coefficient of thermal expansion


Non-dimensional concentration


Dimensionless radial coordinate


Dynamic viscosity


Kinematic viscosity


Non-dimensional temperature


Density of non-Newtonian fluid


Dimensionless tangential coordinate


Dimensionless stream function

\(\dot{\gamma }\)

Shear rate


Time-dependent non-Newtonian material constant


Second invariant strain tensor



Conditions at the wall (cone surface)

Free stream condition



The authors are grateful to both reviewers and their comments which have served to significantly improve the interpretative and other aspects of the present article.


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Copyright information

© The Brazilian Society of Mechanical Sciences and Engineering 2016

Authors and Affiliations

  • S. Abdul Gaffar
    • 1
  • V. Ramachandra Prasad
    • 2
  • S. Keshava Reddy
    • 1
  • O. Anwar Bég
    • 3
  1. 1.Department of MathematicsJawaharlal Nehru Technological UniversityAnantapuramuIndia
  2. 2.Department of MathematicsMadanapalle Institute of Technology and ScienceMadanapalleIndia
  3. 3.Gort (Aerospace and Chemical Engineering)Gabriel’s Wing HouseBradfordUK

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