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Nonlinear Analysis: Real World Applications 10 (2009) 2909–2913 Contents lists available at ScienceDirect Nonlinear Analysis: Real World Applications journal homepage: www.elsevier.com/locate/nonrwa Heat transfer over an unsteady stretching permeable surface with prescribed wall temperature Anuar Ishak a , Roslinda Nazar a,* , Ioan Pop b a School of Mathematical Sciences, Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor, Malaysia b Faculty of Mathematics, University of Cluj, R-3400 Cluj, CP 253, Romania article info Article history: Received 18 November 2007 Accepted 26 September 2008 Keywords: Unsteady flow Heat transfer Similarity solutions Stretching permeable surface abstract The unsteady laminar boundary layer flow over a continuously stretching permeable surface is investigated. The unsteadiness in the flow and temperature fields is caused by the time-dependence of the stretching velocity and the surface temperature. Effects of the unsteadiness parameter, suction/injection parameter and Prandtl number on the heat transfer characteristics are thoroughly examined. © 2008 Elsevier Ltd. All rights reserved. 1. Introduction Fluid dynamics due to a stretching surface is important since it has many practical applications in manufacturing processes which include both metal and polymer sheets, for example the cooling of an infinite metallic plate in a cooling bath, the boundary layer along material handling conveyers, the aerodynamic extrusion of plastic sheets, the boundary layer along a liquid film in the condensation processes, paper production, glass blowing, metal spinning and drawing of plastic films. The quality of the final product depends on the rate of heat transfer at the stretching surface. Since the pioneering study by Crane [1], who presented an exact analytical solution for the steady two-dimensional flow due to a stretching surface in a quiescent fluid, many authors have considered various aspects of this problem and obtained similarity solutions (cf. [2–15]). All of the above mentioned studies deal with stretching surfaces where the flows were assumed to be steady. Unsteady flows due to stretching surfaces have received less attention; a few of them are those considered by Devi et al. [16], Andersson et al. [17], Nazar et al. [18], and very recently by Ali and Mehmood [19]. In Refs. [18] and [19], the similarity transformation introduced by Williams and Rhyne [20] was used, which transforms the governing partial differential equations with three independent variables to two independent variables, which are more convenient for numerical computations. Motivated by the above investigations, in this paper we present the characteristics of the heat transfer caused by a stretching permeable surface. The governing partial differential equations with three independent variables are transformed to ordinary differential equations using the similarity transformation, before being solved numerically by the Keller-box method. The results obtained are then compared with those of Grubka and Bobba [3], Ali [5] and the exact solution for the steady-state flow case to support their validity. 2. Analysis Consider the unsteady laminar boundary layer flow due to a stretching permeable surface in a quiescent viscous and incompressible fluid, as shown in Fig. 1. At time t = 0, the sheet is impulsively stretched with the velocity U w (x, t ) along * Corresponding author. Tel.: +60 3 8921 3371; fax: +60 3 8925 4519. E-mail address: [email protected] (R. Nazar). 1468-1218/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.nonrwa.2008.09.010

Heat transfer over an unsteady stretching permeable surface with prescribed wall temperature

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Nonlinear Analysis: Real World Applications 10 (2009) 2909–2913

Contents lists available at ScienceDirect

Nonlinear Analysis: Real World Applications

journal homepage: www.elsevier.com/locate/nonrwa

Heat transfer over an unsteady stretching permeable surface withprescribed wall temperatureAnuar Ishak a, Roslinda Nazar a,∗, Ioan Pop ba School of Mathematical Sciences, Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor, Malaysiab Faculty of Mathematics, University of Cluj, R-3400 Cluj, CP 253, Romania

a r t i c l e i n f o

Article history:Received 18 November 2007Accepted 26 September 2008

Keywords:Unsteady flowHeat transferSimilarity solutionsStretching permeable surface

a b s t r a c t

The unsteady laminar boundary layer flow over a continuously stretching permeablesurface is investigated. The unsteadiness in the flow and temperature fields is caused bythe time-dependence of the stretching velocity and the surface temperature. Effects ofthe unsteadiness parameter, suction/injection parameter and Prandtl number on the heattransfer characteristics are thoroughly examined.

© 2008 Elsevier Ltd. All rights reserved.

1. Introduction

Fluid dynamics due to a stretching surface is important since it has many practical applications in manufacturingprocesses which include both metal and polymer sheets, for example the cooling of an infinite metallic plate in a coolingbath, the boundary layer alongmaterial handling conveyers, the aerodynamic extrusion of plastic sheets, the boundary layeralong a liquid film in the condensation processes, paper production, glass blowing, metal spinning and drawing of plasticfilms. The quality of the final product depends on the rate of heat transfer at the stretching surface. Since the pioneeringstudy by Crane [1], who presented an exact analytical solution for the steady two-dimensional flow due to a stretchingsurface in a quiescent fluid, many authors have considered various aspects of this problem and obtained similarity solutions(cf. [2–15]).All of the above mentioned studies deal with stretching surfaces where the flows were assumed to be steady. Unsteady

flowsdue to stretching surfaces have received less attention; a fewof themare those consideredbyDevi et al. [16], Anderssonet al. [17], Nazar et al. [18], and very recently by Ali and Mehmood [19]. In Refs. [18] and [19], the similarity transformationintroduced by Williams and Rhyne [20] was used, which transforms the governing partial differential equations with threeindependent variables to two independent variables, which are more convenient for numerical computations.Motivated by the above investigations, in this paper we present the characteristics of the heat transfer caused by a

stretching permeable surface. The governing partial differential equationswith three independent variables are transformedto ordinary differential equations using the similarity transformation, before being solved numerically by the Keller-boxmethod. The results obtained are then compared with those of Grubka and Bobba [3], Ali [5] and the exact solution for thesteady-state flow case to support their validity.

2. Analysis

Consider the unsteady laminar boundary layer flow due to a stretching permeable surface in a quiescent viscous andincompressible fluid, as shown in Fig. 1. At time t = 0, the sheet is impulsively stretched with the velocity Uw(x, t) along

∗ Corresponding author. Tel.: +60 3 8921 3371; fax: +60 3 8925 4519.E-mail address: [email protected] (R. Nazar).

1468-1218/$ – see front matter© 2008 Elsevier Ltd. All rights reserved.doi:10.1016/j.nonrwa.2008.09.010

2910 A. Ishak et al. / Nonlinear Analysis: Real World Applications 10 (2009) 2909–2913

Fig. 1. Physical model and coordinate system.

the x-axis, keeping the origin fixed in the fluid of ambient temperature T∞. The stationary Cartesian coordinate system hasits origin located at the leading edge of the sheet with the positive x-axis extending along the sheet, while the y-axis ismeasured normal to the surface of the sheet. Under these assumptions along with the boundary layer approximations andneglecting the viscous dissipation, the governing unsteady two-dimensional Navier–Stokes equations and energy equationmay be written as

∂u∂x+∂v

∂y= 0, (1)

∂u∂t+ u

∂u∂x+ v

∂u∂y= ν

∂2u∂y2

, (2)

∂T∂t+ u

∂T∂x+ v

∂T∂y= α

∂2T∂y2

, (3)

subject to the boundary conditionsu = Uw, v = Vw, T = Tw at y = 0,u→ 0, T → T∞ as y→∞, (4)

where u and v are the velocity components in the x and y directions, respectively, T is the fluid temperature insidethe boundary layer, t is time, α and ν are the thermal diffusivity and the kinematic viscosity, respectively, and Vw =− (νUw/x)1/2 f (0) represents the mass transfer at the surface with Vw > 0 for injection and Vw < 0 for suction. We assumethat the stretching velocity Uw(x, t) and the surface temperature Tw(x, t) are of the form

Uw(x, t) =ax1− ct

, Tw(x, t) = T∞ +bx1− ct

, (5)

where a, b and c are constants with a > 0, b ≥ 0 and c ≥ 0 (with ct < 1), and both a and c have dimension time−1. It shouldbe noticed that at t = 0 (initial motion), Eqs. (1)–(3) describe the steady flow over a stretching surface. This particular formof Uw(x, t) and Tw(x, t) has been chosen in order to be able to devise a new similarity transformation, which transformsthe governing partial differential equations (1)–(3) into a set of ordinary differential equations, thereby facilitating theexploration of the effects of the controlling parameters (see Andersson et al. [17]).We now introduce the following dimensionless functions f and θ , and similarity variable η (see Ishak et al. [9,10], Devi

et al. [16] and Andersson et al. [17]):

η =

(Uwνx

)1/2y, ψ = (νxUw)1/2 f (η), θ(η) =

T − T∞Tw − T∞

, (6)

where ψ(x, y, t) is a stream function defined as u = ∂ψ/∂y and v = −∂ψ/∂x, which identically satisfies the massconservation equation (1). Substituting (6) into Eqs. (2) and (3) we obtain

f ′′′ + ff ′′ − f ′2 − A(f ′ +

12ηf ′′

)= 0, (7)

1Prθ ′′ + f θ ′ − f ′θ − A

(θ +

12ηθ ′)= 0, (8)

where primes denote differentiationwith respect to η, A = c/a is a parameter thatmeasures the unsteadiness and Pr = ν/αis the Prandtl number. The boundary conditions (4) now become

f (0) = f0, f ′(0) = 1, θ(0) = 1,

f ′(η)→ 0, θ(η)→ 0 as η→∞, (9)where f (0) = f0, with f0 < 0 and f0 > 0 corresponding to injection and suction, respectively.

A. Ishak et al. / Nonlinear Analysis: Real World Applications 10 (2009) 2909–2913 2911

The quantities of physical interest are the skin friction coefficient Cf and the local Nusselt number Nux, which are definedas

Cf =τw

ρU2w/2, Nux =

xqwk(Tw − T∞)

, (10)

where ρ is the fluid density, and the wall shear stress τw and the surface heat flux qw are given by

τw = µ

(∂u∂y

)y=0

, qw = −k(∂T∂y

)y=0

, (11)

withµ and k being the dynamic viscosity and thermal conductivity, respectively. Using the dimensionless quantities (6), weobtain

12Cf Re1/2x = f

′′(0), Nux/Re1/2x = −θ′(0). (12)

We note that for A = 0, the problem under consideration reduces to a steady-state flow, where the closed-form solutionfor the flow field and the solution for the thermal field in terms of Kummer’s functions are respectively given by

f (η) = ζ −1ζe−ζη, (13)

θ(η) =M(Pr − 1, Pr + 1,−Pr · e−ζη/ζ 2)M(Pr + 1, Pr − 1,−Pr/ζ 2)

, (14)

where f0 = ζ − 1/ζ (with ζ > 0), and 0 < ζ < 1 and ζ > 1 correspond to injection and suction, respectively. In Eq. (14),M(a, b, z) denotes the confluent hypergeometric function (see Abramowitz and Stegun [21]), with

M(a, b, z) = 1+∞∑n=1

anbn

zn

n!,

an = a(a+ 1)(a+ 2) · · · (a+ n− 1),bn = b(b+ 1)(b+ 2) · · · (b+ n− 1).

By using Eqs. (13) and (14), the skin friction coefficient f ′′(0) and the local Nusselt number −θ ′(0) can be shown to begiven by

f ′′(0) = −ζ ,

θ ′(0) = −ζPr +Pr − 1Pr + 1

Prζ

M(Pr, Pr + 2,−Pr/ζ 2)M(Pr − 1, Pr + 1,−Pr/ζ 2)

. (15)

Moreover, when Pr = 1, the solution θ(η) given in (14) can be expressed as θ(η) = f ′(η) = e−ζη , which implies

− θ ′(0) = ζ . (16)

3. Results and discussion

Theordinary differential equations (7)–(9) have been solvednumerically bymeans of a finite-difference schemeknownasthe Keller-boxmethod, as described in the book by Cebeci and Bradshaw [22]. The results are given to carry out a parametricstudy showing the influence of the non-dimensional parameters, namely the unsteadiness parameter A, suction/injectionparameter ζ and Prandtl number Pr . For validation of the numericalmethod used in this study, the casewhen A = 0 (steady-state flow) has also been considered and compared with the results reported by Grubka and Bobba [3] and Ali [5], as well asthe analytical solution given by Eq. (15). The quantitative comparison is shown in Table 1 and it is found to be in very goodagreement.The temperature profiles presented in Fig. 2 show that the temperature gradient at the surface increases as A increases,

thus the heat transfer rate at the surface increases with A. The same phenomenon is observed for the variation of θ(η)withζ when Pr = 1 and A = 0.5, as can be seen in Fig. 3. This figure shows that the heat transfer rate at the surface is higherfor suction (ζ > 1) compared to injection (ζ < 1). This is due to the fact that the surface shear stress increases whenintroducing suction, which in turn increases the local Nusselt number. From Eqs. (7)–(9), it is clear that the velocity profileis identical to the temperature profile, i.e. f ′(η) = θ(η), when Pr = 1. Thus, Figs. 2 and 3 also represent the velocity profilesfor selected values of A and ζ .In many practical applications, the characteristics involved, such as the heat transfer rate at the surface are vital since

they influence the quality of the final products. Fig. 4 shows the temperature profiles for various values of Pr when the otherparameters are fixed. It is evident from this figure that the temperature gradient at the surface increases as Pr increases,

2912 A. Ishak et al. / Nonlinear Analysis: Real World Applications 10 (2009) 2909–2913

Table 1Values of−θ ′(0) for various values of A, ζ and Pr .

A ζ Pr Grubka and Bobba [3] Ali [5] Exact solutions (Eq. (15)) Present results

0 0.5 0.72 0.4570268328 0.45701 0.5000000000 0.500010 0.645161289 0.6452

1 0.01 0.0197 0.01970635421 0.01970.72 0.8086 0.8058 0.8086313498 0.80861 1.0000 0.9961 1.000000000 1.00003 1.9237 1.9144 1.923682594 1.923710 3.7207 3.7006 3.720673901 3.7207

2 0.72 1.494368413 1.49441 2.000000000 2.000010 16.08421885 16.0842

1 0.5 1 0.80951 1.32052 2.2224

Fig. 2. Temperature profiles θ(η) for different values of Awhen Pr = 1 and ζ = 2 (suction).

Fig. 3. Temperature profiles θ(η) for various values of ζ when Pr = 1 and A = 0.5.

which implies an increase of the heat transfer rate at the surface. This is because a higher Prandtl number fluid has a relativelylow thermal conductivity, which reduces conduction, and thereby reduces the thermal boundary layer thickness, and as aconsequence increases the heat transfer rate at the surface (see Char [6]).Finally, the sample of temperature profiles presented in Figs. 2–4 show that the boundary conditions (9) are satisfied,

thus supporting the numerical results obtained.

4. Conclusions

We have theoretically studied the similarity solutions of the unsteady boundary layer flow and heat transfer due to astretching permeable surface. A new similarity solution has been devised, which transform the time-dependent governingequations to ordinary differential equations. We discussed the effects of the governing parameters A, ζ and Pr on the fluid

A. Ishak et al. / Nonlinear Analysis: Real World Applications 10 (2009) 2909–2913 2913

Fig. 4. Temperature profiles θ(η) for various values of Pr when A = 1 and ζ = 2.

flow and heat transfer characteristics. The numerical results here compared very well with previously reported cases, aswell as the exact solution for the steady-state flow. We found that the heat transfer rate at the surface −θ ′(0) increaseswith A, ζ and Pr .

Acknowledgements

The authors gratefully acknowledge the financial supports received in the form of research grants from the Academyof Sciences Malaysia (SAGA project code: STGL-013-2006) and the Engineering Mathematics Group, Universiti KebangsaanMalaysia (project code: UKM-GUP-BTT-07-25-174).

References

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