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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

Abstract

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.

Keywords

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

List of symbols

Nomenclature

A

Half angle of the cone

B

Material parameter

B0

Constant imposed magnetic field

Cf

Skin friction coefficient

f

Non-dimensional steam function

fw

Suction (wall transpiration) parameter

Grx

Local Grashof number

g

Acceleration due to gravity

k

Thermal conductivity of fluid

M

Local magnetic parameter

m

Rheological power-law index

n

Surface temperature exponent

Nu

Heat transfer rate (Local Nusselt number)

Pr

Prandtl number

r

Local radius of the cone

T

Temperature of the fluid

u, v

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

V

Velocity vector

We

Local Weissenberg number

x

Stream wise coordinate

y

Transverse coordinate

Greek

\(\alpha\)

Thermal diffusivity

\(\beta\)

Coefficient of thermal expansion

ϕ

Non-dimensional concentration

η

Dimensionless radial coordinate

μ

Dynamic viscosity

υ

Kinematic viscosity

θ

Non-dimensional temperature

ρ

Density of non-Newtonian fluid

\(\xi\)

Dimensionless tangential coordinate

ψ

Dimensionless stream function

\(\dot{\gamma }\)

Shear rate

\(\varGamma\)

Time-dependent non-Newtonian material constant

\(\varPi\)

Second invariant strain tensor

Subscripts

w

Conditions at the wall (cone surface)

Free stream condition

Notes

Acknowledgments

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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