Physics Letters A 372 (2008) 559–561

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Heat transfer over a stretching surface with variable heat fluxin micropolar fluids

Anuar Ishak a, Roslinda Nazar a,∗, Ioan Pop b

a School of Mathematical Sciences, Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor, Malaysiab Faculty of Mathematics, University of Cluj, R-3400 Cluj, CP 253, Romania

Received 14 March 2007; accepted 2 August 2007

Available online 9 August 2007

Communicated by A.P. Fordy

Abstract

Heat transfer over a stretching surface with uniform or variable heat flux in micropolar fluids is investigated in this Letter. The boundarylayer equations are transformed into ordinary differential equations, and then they are solved numerically by a finite-difference method. Theeffects of the material parameter K , Prandtl number Pr, velocity exponent parameter m, and heat flux exponent parameter n on the heat transfercharacteristics are studied. It is found that the local Nusselt number is higher for micropolar fluids compared to Newtonian fluids.© 2007 Elsevier B.V. All rights reserved.

Keywords: Heat transfer; Micropolar fluid; Stretching surface

1. Introduction

The fluid dynamics over a stretching surface is important inmany practical applications such as extrusion of plastic sheets,paper production, glass blowing, metal spinning and drawingplastic films, to name just a few. The quality of the final prod-uct depends on the rate of heat transfer at the stretching sur-face.

Elbashbeshy [1] studied the heat transfer over a stretchingsurface immersed in an incompressible Newtonian fluid whenthe surface is subjected to a variable heat flux. The purposeof the present study is therefore, to extend the work of El-bashbeshy [1] to micropolar fluids, which display the effectsof local rotary inertia and couple stresses. The theory of mi-cropolar fluids can explain the flow behavior in which the clas-sical Newtonian fluids theory is inadequate. Since introducedby Eringen [2,3], many researchers have considered variousproblems in micropolar fluids, see for example Seddeek [4] andIshak et al. [5,6].

* Corresponding author. Tel.: +603 89213371; fax: +603 89254519.E-mail address: rmn72my@yahoo.com (R. Nazar).

0375-9601/$ – see front matter © 2007 Elsevier B.V. All rights reserved.doi:10.1016/j.physleta.2007.08.003

2. Formulation of the problem

Consider a steady, two-dimensional laminar flow of an in-compressible micropolar fluid on a continuous, stretching sur-face with velocity Uw(x) = axm and variable surface heat fluxqw(x) = bxn (where a, b, m and n are constants) moving axi-ally through a stationary fluid, which is assumed to have con-stant physical properties. The x-axis runs along the continuoussurface in the direction of the motion and the y-axis is per-pendicular to it. Under the boundary layer approximations, thebasic equations are

(1)∂u

∂x+ ∂v

∂y= 0,

(2)u∂u

∂x+ v

∂u

∂y=

(μ + κ

ρ

)∂2u

∂y2+ κ

ρ

∂N

∂y,

(3)ρj

(u

∂N

∂x+ v

∂N

∂y

)= ∂

∂y

(γ

∂N

∂y

)− κ

(2N + ∂u

∂y

),

(4)u∂j

∂x+ v

∂j

∂y= 0,

(5)u∂T

∂x+ v

∂T

∂y= α

∂2T

∂y2,

560 A. Ishak et al. / Physics Letters A 372 (2008) 559–561

Table 1Values of Nux/Re1/2

x for various values of m, n and Pr when K = 0 (Newtonian fluids)

m n Elbashbeshy [1] Ali [10] Present results

Pr = 1 Pr = 10 Pr = 1 Pr = 10 Pr = 1 Pr = 10

−0.2 0.0 0.81990 2.79970 0.81476 2.80052 0.8147 2.80081.0 1.2606 4.188 1.2606 4.1882

1.0 0.0 0.58210 2.30730 0.5820 2.30801.0 1.0 3.7202 1.0000 3.7205

3.0 0.0 0.44036 1.87528 0.4373 1.87535.0 0.0 0.3670 1.6191 0.3653 1.6195

1.0 0.6849 2.8857 0.6847 2.8857

Fig. 1. Temperature profiles θ(η) for various values of K when Pr = 1, m = 1and n = 1.

subject to the boundary conditions

u = Uw, v = 0, j = 0, N = −1

2

∂u

∂y,

∂T

∂y= −qw

kat y = 0,

(6)u → 0, N → 0, T → T∞ as y → ∞,

where u and v are the velocity components along the x- andy-axes, respectively, T is the fluid temperature, N is the com-ponent of the microrotation vector normal to the x–y plane, ρ

is the density, j is the micro-inertia density, μ is the dynamicviscosity, κ is the gyro-viscosity (or vortex viscosity), γ is thespin-gradient viscosity and α is the thermal diffusivity. We as-sume that j and γ are functions of the coordinates x and y, andare not constants as in many other papers.

Following Ahmadi [7], Kline [8] or Gorla [9], we assumethat the spin gradient viscosity γ is given by

(7)γ = (μ + κ/2)j = μ(1 + K/2)j,

where K = κ/μ is the material parameter. This assumptionis invoked to allow the field of equations predicts the correctbehavior in the limiting case when the microstructure effectsbecome negligible and the total spin N reduces to the angularvelocity [7].

The continuity equation (1) is satisfied by introducing astream function ψ such that u = ∂ψ/∂y and v = −∂ψ/∂x. Fur-

Fig. 2. Temperature profiles θ(η) for various values of n when K = 1, Pr = 1and m = 0.

Fig. 3. Temperature profiles θ(η) for various values of n when K = 1, Pr = 1and m = 1.

ther, we introduce the similarity transformation

η =[(m + 1)Uw

2νx

]1/2

y, ψ =[

2νxUw

m + 1

]1/2

f (η),

N = Uw

[(m + 1)Uw

2νx

]1/2

h(η), j = 2νx

(m + 1)Uw

i(η),

(8)T = T∞ + qw

k

[2νx

(m + 1)Uw

]1/2

θ(η).

A. Ishak et al. / Physics Letters A 372 (2008) 559–561 561

Fig. 4. Variation of the local Nusselt number Nux/Re1/2x with K for various

values of n when Pr = 0.72 (air) and m = 1.

Fig. 5. Temperature profiles θ(η) for various values of Pr when K = 1, m = 1and n = 1.

Thus, Eqs. (2)–(5) reduce to the following ordinary differentialequations:

(9)(1 + K)f ′′′ + ff ′′ − 2m

m + 1f ′2 + Kh′ = 0,

(1 + K

2

)(ih′)′ + i

[f h′ − 3m − 1

m + 1f ′h

]

(10)− K(2h + f ′′) = 0,

(11)2(1 − m)if ′ − (m + 1)f i′ = 0,

(12)1

Prθ ′′ + f θ ′ − 2n + 1 − m

m + 1f ′θ = 0,

where Pr is the Prandtl number and prime denotes differentia-tion with respect to η. The boundary conditions (6) become

f (0) = 0, f ′(0) = 1, i(0) = 0,

h(0) = −1

2f ′′(0), θ ′(0) = −1,

(13)f ′(η) → 0, h(η) → 0, θ(η) → 0, as η → ∞.

We notice that when K = 0 (Newtonian fluids), the prob-lem reduces to those considered by Elbashbeshy [1], for animpermeable surface. The physical quantity of interest is the lo-cal Nusselt number, which can be easily shown that it is givenby

(14)Nux√

Rex

=√

m + 1

2

1

θ(0),

where Rex = Uwx/ν is the local Reynolds number.

3. Results and discussion

The non-linear ordinary differential equations (9)–(12), sat-isfying the boundary conditions (13) have been solved numer-ically using the Keller box-method for several values of theinvolved parameters, namely the material parameter K , Prandtlnumber Pr, velocity exponent m, and heat flux exponent n. Itis found that the values of the local Nusselt number Nux/Re1/2

x

compare well with the results reported by Elbashbeshy [1] andAli [10], as shown in Table 1.

The temperature profiles presented in Figs. 1–3 show thatthe surface temperature decreases with K and n but increaseswith m. This fact is in agreement with the variation of thelocal Nusselt number presented in Fig. 4, for fixed valuesof Pr and m. The surface temperature is higher for fluids withsmaller Pr, as shown in Fig. 5. Thus, the local Nusselt numberincreases as Pr increases.

Acknowledgement

The financial support received from the Academy of Sci-ences Malaysia under the SAGA grant no. STGL-013-2006 isgratefully acknowledged.

References

[1] E.M.A. Elbashbeshy, J. Phys. D: Appl. Phys. 31 (1998) 1951.[2] A.C. Eringen, J. Math. Mech. 16 (1966) 1.[3] A.C. Eringen, J. Math. Anal. Appl. 38 (1972) 480.[4] M.A. Seddeek, Phys. Lett. A 306 (2003) 255.[5] A. Ishak, R. Nazar, I. Pop, Int. J. Eng. Sci. 44 (2006) 1225.[6] A. Ishak, R. Nazar, I. Pop, Fluid Dyn. Res. 38 (2006) 489.[7] G. Ahmadi, Int. J. Eng. Sci. 14 (1976) 639.[8] K.A. Kline, Int. J. Eng. Sci. 15 (1977) 131.[9] R.S.R. Gorla, Int. J. Eng. Sci. 26 (1988) 385.

[10] M.E. Ali, Int. J. Heat Fluid Flow 16 (1995) 280.