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Mixed convection flow of second grade fluid along a vertical stretching flat surface with variable surface temperature

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Abstract

In the present study we have explored the effects of thermal buoyancy on flow of a viscoelastic second grade fluid past a vertical, continuous stretching sheet of which the velocity and temperature distributions are assumed to vary according to a power-law form. The governing differential equations are transformed into dimensionless form using appropriate transformations and then solved numerically. The methods here employed are (1) the perturbation method together with the Shanks transformation, (2) the local non-similarity method with second level of truncation and (3) the implicit finite difference method for values of ξ ( = Gr x /Re 2 x , defined as local mixed convection parameter) ranging in [0, 10]. The comparison between the solutions obtained by the aforementioned methods found in excellent agreement. Effects of the elasticity parameter λ on the skin-friction and heat transfer coefficients have been shown graphically for the fluids having the values of the Prandtl number equal to 0.72, 7.03 and 15.0. Effects of the viscoelastic parameter and the mixed convection parameter, ξ, on the temperature and velocity fields have also been studied. We notice that with the increase in visco-elastic parameter λ, velocity decreases whereas temperature increases and that velocity gradient is higher than that of temperature.

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Abbreviations

ξ:

local mixed convection parameter

Gr x :

local Grashof number

Re x :

local Reynolds number

λ:

elasticity parameter

u :

velocity in x-direction (m/s)

v :

velocity in y-direction (m/s)

w :

velocity in z-direction (m/s)

x, y :

Cartesian coordinates (m)

T :

temperature (°C)

P :

fluid pressure (Pa)

ρ:

fluid density at reference temperature (T 0)

ν:

effective kinematic viscosity (μ/ρ)

∇2 :

∂2/∂x 2 + ∂2/∂y 2, Laplacian operator

κ:

second grade parameter

K :

kinematic elasticity (=κ/ρ)

g :

gravitational acceleration (m/s2)

β:

coefficient of thermal expansion of fluid (K −1)

T ∞ :

ambient temperature (°C)

T w :

surface temperature (°C)

α:

thermal diffusivity of the ambient fluid (k/ρ c p )

δ:

momentum boundary layer thickness (m)

δT :

thermal boundary layer thickness (m)

U :

velocity of the moving surface (m/s)

ψ:

stream function (m2/s)

η:

similarity variable (m)

θ:

dimensionless temperature

Pr :

Prandtl number

C f :

local skin friction coefficient

τw :

shear-stress at the surface

Nu :

local Nusselt number

q w :

heat transfer per unit area at the surface

μ:

effective dynamic viscosity (Pa/s)

M(a,b,z) :

Kummer’s function

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Authors and Affiliations

  1. Department of Mathematics, COMSATS Institute of Information Technology, Islamabad, Pakistan

    M. Mushtaq, S. Asghar & M. A. Hossain

Authors
  1. M. Mushtaq
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  2. S. Asghar
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  3. M. A. Hossain
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Corresponding author

Correspondence to M. A. Hossain.

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On leave of absence from the Department of Mathematics, University of Dhaka, Bangladesh.

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Mushtaq, M., Asghar, S. & Hossain, M.A. Mixed convection flow of second grade fluid along a vertical stretching flat surface with variable surface temperature. Heat Mass Transfer 43, 1049–1061 (2007). https://doi.org/10.1007/s00231-006-0177-8

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  • DOI: https://doi.org/10.1007/s00231-006-0177-8

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