Thermal boundary layer flow over a stretching sheet in a micropolar fluid with radiation effect Authors Anuar Ishak School of Mathematical Sciences, Faculty of Science and Technology Universiti Kebangsaan Malaysia Article

First Online: 16 October 2009 Received: 09 December 2008 Accepted: 02 October 2009 DOI :
10.1007/s11012-009-9257-4

Cite this article as: Ishak, A. Meccanica (2010) 45: 367. doi:10.1007/s11012-009-9257-4
Abstract In the present paper, we study the effects of radiation on the thermal boundary layer flow induced by a linearly stretching sheet immersed in an incompressible micropolar fluid with constant surface temperature. Similarity transformation is employed to transform the governing partial differential equations into ordinary ones, which are then solved numerically using the Runge-Kutta-Fehlberg method. Results for the local Nusselt number as well as the temperature profiles are presented for different values of the governing parameters. It is found that the heat transfer rate at the surface decreases in the presence of radiation. Comparison with known results for certain particular cases is excellent.

Keywords Boundary layer Heat transfer Micropolar fluid Radiation Stretching sheet Fluids mechanics Nomenclature a ,b constants

c _{p} specific heat at constant pressure

f dimensionless stream function

h dimensionless microrotation

j microinertia density

k thermal conductivity

k ^{*} mean absorption coefficient

K material parameter

m boundary parameter

N microrotation or angular velocity

N _{R} radiation parameter

Pr Prandtl number

q _{r} radiative heat flux

T fluid temperature

T _{w} surface temperature

T _{∞} ambient temperature

u ,v velocity components in the x - and y -directions, respectively

U _{w} velocity of the stretching sheet

x ,y Cartesian coordinates along the sheet and normal to it, respectively

Greek Letters α thermal diffusivity

β thermal expansion coefficient

γ spin gradient viscosity

η similarity variable

θ dimensionless temperature

κ vortex viscosity

ν kinematic viscosity

μ dynamic viscosity

ρ fluid density

σ ^{*} Stefan-Boltzmann constant

ψ stream function

Subscripts w condition at the solid surface

∞ ambient condition

Superscript ′ differentiation with respect to η

References 1.

Crane LJ (1970) Flow past a stretching plate. Z Angew Math Phys 21:645–647

CrossRef 2.

Gupta PS, Gupta AS (1977) Heat and mass transfer on a stretching sheet with suction or blowing. Can J Chem Eng 55:744–746

CrossRef 3.

Grubka LJ, Bobba KM (1985) Heat transfer characteristics of a continuous, stretching surface with variable temperature. ASME J Heat Transf 107:248–250

CrossRef 4.

Chen CK, Char MI (1988) Heat transfer of a continuous stretching surface with suction or blowing. J Math Anal Appl 135:568–580

MATH CrossRef MathSciNet 5.

Dutta BK, Roy P, Gupta AS (1985) Temperature field in the flow over a stretching sheet with uniform heat flux. Int Commun Heat Mass Transfer 12:89–94

CrossRef 6.

Ali ME (1994) Heat transfer characteristics of a continuous stretching surface. Heat Mass Transfer 29:227–234

7.

Ali ME (1995) On thermal boundary layer on a power-law stretched surface with suction or injection. Int J Heat Fluid Flow 16:280–290

CrossRef 8.

Afzal N, Varshney IS (1980) The cooling of a low heat resistance stretching sheet moving through a fluid. Heat Mass Transfer 14:289–293

9.

Afzal N (1993) Heat transfer from a stretching surface. Int J Heat Mass Transfer 36:1128–1131

MATH CrossRef 10.

Chen CH (1998) Laminar mixed convection adjacent to vertical, continuously stretching sheets. Heat Mass Transfer 33:471–476

CrossRef ADS 11.

Ali M, Al-Yousef F (1998) Laminar mixed convection from a continuously moving vertical surface with suction or injection. Heat Mass Transfer 33:301–306

CrossRef ADS 12.

Daskalakis JE (1993) Free convection effects in the boundary layer along a vertically stretching flat surface. Can J Phys 70:1253–1260

ADS 13.

Partha MK, Murthy PVSN, Rajasekhar GP (2005) Effect of viscous dissipation on the mixed convection heat transfer from an exponentially stretching surface. Heat Mass Transfer 41:360–366

CrossRef ADS 14.

El-Aziz MA (2007) Temperature dependent viscosity and thermal conductivity effects on combined heat and mass transfer in MHD three-dimensional flow over a stretching surface with Ohmic heating. Meccanica 42:375–386

MATH CrossRef 15.

Mahapatra TR, Dholey S, Gupta AS (2007) Momentum and heat transfer in the magnetohydrodynamic stagnation-point flow of a viscoelastic fluid toward a stretching surface. Meccanica 42:263–272

MATH CrossRef 16.

Ishak A, Nazar R, Pop I (2006) Unsteady mixed convection boundary layer flow due to a stretching vertical surface. Arab J Sci Eng 31:165–182

MathSciNet 17.

Ishak A, Nazar R, Pop I (2006) Magnetohydrodynamic stagnation point flow towards a stretching vertical sheet. Magnetohydrodynamics 42:17–30

18.

Ishak A, Nazar R, Pop I (2006) Mixed convection boundary layers in the stagnation-point flow towards a stretching vertical sheet. Meccanica 41:509–518

MATH CrossRef 19.

Ishak A, Nazar R, Pop I (2007) Mixed convection on the stagnation point flow toward a vertical continuously stretching sheet. ASME J Heat Transfer 129:1087–1090

CrossRef 20.

Ishak A, Nazar R, Pop I (2008) Hydromagnetic flow and heat transfer adjacent to a stretching vertical sheet. Heat Mass Transfer 44:921–927

CrossRef ADS 21.

Chiam TC (1982) Micropolar fluid flow over a stretching sheet. Z Angew Math Mech 62:565–568

CrossRef 22.

Heruska MW, Watson LT, Sankara KK (1986) Micropolar flow past a porous stretching sheet. Comput Fluids 14:117–129

MATH CrossRef MathSciNet 23.

Agarwal RS, Bhargava R, Balaji AVS (1989) Finite element solution of flow and heat transfer of a micropolar fluid over a stretching sheet. Int J Eng Sci 27:1421–1428

MATH CrossRef 24.

Hassanien IA, Gorla RSR (1990) Heat transfer to a micropolar fluid from a non-isothermal stretching sheet with suction and blowing. Acta Mech 84:191–199

CrossRef 25.

Nelson NA, Desseaux A (2001) Effect of surface conditions on flow of a micropolar fluid driven by a porous stretching sheet. Int J Eng Sci 39:1881–1897

26.

Kelson NA, Farrell TW (2001) Micropolar flow over a porous stretching sheet with strong suction or injection. Int Commun Heat Mass Transfer 28:479–488

CrossRef 27.

Nazar R, Amin N, Pop I (2004) Stagnation point flow of a micropolar fluid towards a stretching sheet. Int J Non-Linear Mech 39:1227–1235

MATH CrossRef 28.

Hayat T, Abbas Z, Javed T (2008) Mixed convection flow of a micropolar fluid over a non-linearly stretching sheet. Phys Lett A 372:637–647

CrossRef ADS 29.

Ishak A, Nazar R, Pop I (2008) Mixed convection stagnation point flow of a micropolar fluid towards a stretching sheet. Meccanica 43:411–418

MATH CrossRef 30.

Ishak A, Nazar R, Pop I (2008) Heat transfer over a stretching surface with variable surface heat flux in micropolar fluids. Phys Lett A 372:559–561

CrossRef ADS 31.

Bataller RC (2008) Radiation effects in the Blasius flow. Appl Math Comput 198:333–338

MATH CrossRef MathSciNet 32.

Ahmadi G (1976) Self-similar solution of incompressible micropolar boundary layer flow over a semi-infinite plate. Int J Eng Sci 14:639–646

MATH CrossRef 33.

Yücel A (1989) Mixed convection in micropolar fluid flow over a horizontal plate with surface mass transfer. Int J Eng Sci 27:1593–1602

MATH CrossRef 34.

Brewster MQ (1992) Thermal radiative transfer properties. Wiley, New York

35.

Datti PS, Prasad KV, Abel MS, Joshi A (2004) MHD visco-elastic fluid flow over a non-isothermal stretching sheet. Int J Eng Sci 42:935–946

CrossRef © Springer Science+Business Media B.V. 2009