Radiation Effect on the Mixed Convection Flow of a Viscoelastic FluidAlong an Inclined Stretching SheetMuhammad Qasima, Tasawar Hayatb, and Saleem Obaidatc

a Department of Mathematics, COMSATS Institute of Information Technology (CIIT), ParkRoad, Chak Shahzad, Islamabad, Pakistan

b Department of Mathematics, Quaid-I-Azam University 45320, Islamabad 44000, Pakistanc Department of Mathematics, College of Sciences, King Saud University P.O. Box 2455,

Riyadh 11451, Saudi Arabia

Reprint requests to M. Q.; E-mail: mq qau@yahoo.com

Z. Naturforsch. 67a, 195 – 202 (2012) / DOI: 10.5560/ZNA.2012-0006Received June 30, 2011 / revised December 18, 2011

This study concentrates on the heat transfer analysis of the steady flow of viscoelastic fluid alongan inclined stretching surface. Analysis has been carried out in the presence of thermal radiation andthe Rosseland approximation is used to describe the radiative heat flux in the energy equation. Theequations of continuity, momentum and energy are reduced into the system of governing differentialequations and solved by homotopy analysis method (HAM). The velocity and temperature are illus-trated through graphs. Exact and homotopy solutions are compared in a limiting sense. It is noticedthat viscoelastic parameter decreases the velocity and boundary layer thickness. It is also observedthat increasing values of viscoelastic parameter reduces the thickness of momentum boundary layerand increase the heat transfer rate. However, it is found that increasing the radiation parameter hasthe effect of decreasing the local Nusselt number.

Key words: Inclined Stretching; Thermal Radiation; Mixed Convection; Series Solution.

1. Introduction

Recently the heat transfer analysis in boundary layerflow induced by a stretched surface has gained muchinterest because of its engineering and industrial appli-cations. For instance in the extrusion of plastic sheets,spinning of fibers, polymer, cooling of elastic sheetsand many others. The quality of final product stronglydepends upon heat transfer rate and thus cooling pro-cess can be controlled effectively. Further, the con-cept of radiative heat transfer in the flow has an im-portant role in manufacturing industries for the designof reliable equipment, nuclear plants, gas turbines andvarious propulsion devices for aircraft, missiles, satel-lites and space vehicles. The flow of a viscous fluidover a moving surface was initiated by Sakiadis [1, 2].Crane [3] derived similarity solution in closed form forthe flow of viscous fluid induced by a stretching sheet.At the present there is large number of papers deal-ing with the flow of viscous and non-Newtonian fluidsover a stretching sheet. We have just mentioned somerecent representative studies in this direction [4 – 14].

In a very recent study, Haung et al. [15] discussed theheat mass transfer in flow of a viscous fluid along aninclined stretching sheet.

To the best of our knowledge no study has been pre-sented for the heat transfer analysis in the flow of a vis-coelastic fluid along an inclined stretching sheet. Theobject of present work is to address this issue. Hencetwo-dimensional flow of an incompressible fluid inthe presence of thermal radiation has been considered.The flow problem is computed by homotopy analysismethod (HAM) [16 – 25]. The variations of embeddedparameters on the velocity and temperature are dis-played and discussed.

2. Definition of Governing Problem

We consider the steady two-dimensional mixed con-vection flow of an incompressible viscoelastic fluidover an inclined stretching surface. The heat transferis considered in the presence of thermal radiation sub-ject to Rosseland approximation. The x-axis is mea-sured along the stretching surface and y-axis normal

c© 2012 Verlag der Zeitschrift fur Naturforschung, Tubingen · http://znaturforsch.com

196 M. Qasim et al. · Radiation Effect on the Mixed Convection Flow of a Viscoelastic Fluid

Fig. 1. Physical model and coordinate system.

to the x-axis (Fig. 1). The governing equations throughconservation of mass, momentum and energy can bewritten as

∂u∂x

+∂v∂y

= 0, (1)

u∂u∂x

+ v∂u∂y

= ν∂ 2u∂y2 − k0

[u

∂ 3u∂x∂y2 +

∂u∂x

∂ 2u∂y2

+∂u∂y

∂ 2v∂y2 + v

∂ 3u∂y3

]+gβT(T −T∞)cosα,

(2)

ρcp

(u

∂T∂x

+ v∂T∂y

)=

∂

∂y

((16σ∗T 3

∞

3k∗+ k

)∂T∂y

), (3)

in which u and v are the velocity components in the x-and y-directions, σ∗ is the Stefan–Boltzmann constant,T is the fluid temperature, g is gravitational accelera-tion, ν is the kinematic viscosity, σ is the electricalconductivity, ρ is fluid density, βT is thermal expan-sion coefficients of temperature, cp is specific heat andk∗ the mean absorption coefficient.

The boundary conditions relevant to the problem aredefined as follows:

u = cx, v = 0, T = Tw, at y = 0, (4)

u→ 0, T → T∞, as y→ ∞. (5)

with surface temperature Tw equal to

Tw = T∞ +bx, (6)

where b and c are the positive constants. We define thefollowing variables

η =√

cν

y, ψ =√

cνx f (η), θ(η) =T −T∞

Tw−T∞

, (7)

Introducing the stream function ψ by

u =∂ψ

∂yv =−∂ψ

∂x, (8)

the continuity equation is identically satisfied and theresulting problems for f and θ are given by

f ′′′− f ′2 + f f ′′−K[2 f ′ f ′′′− f ′′2− f f ′′′′]+λθ cosα = 0,

(9)

(1+Nr)θ ′′+Pr( f θ′− f ′θ) = 0, (10)

f (0) = 0, f ′(0) = 1, f ′(∞)→ 0, θ(0) = 1,

θ(∞)→ 0.(11)

In the above expressions K =k0cυ

is the dimensionless

fluid parameter, λ =Grx

Re2x

is mixed convection param-

eter, Grx

(=

gβ (Tw−T∞)x3/υ2

u2wx2/υ2

)is the local Grashof

number, Rex = uwx/υ is the local Reynolds number,

Pr =µcp

kis the Prandtl number, Nr

(=

4σ∗T 3∞

k∗k

)is the

radiation parameter and primes indicate the differenti-ation with respect to η .

The skin friction coefficient C f and local Nusseltnumber Nux are presented as follows:

C f =τw

ρu2w, (12)

Nu =xqw

k (Tw−T∞), (13)

where the skin-friction τw and wall heat flux qw aredefined as follows

τw =[µ

∂u∂y

+ k0

(2

∂u∂x

∂u∂y

+u∂ 2u

∂x∂y+ v

∂ 2u∂y2

)]y=0

, (14)

qw =−k

(∂T∂y

)y=0

. (15)

Substituting (7) and (8) into (14) and (15) we obtain

Re1/2x C f = (1+3K) f ′′(0), (16)

Re1/2x Nux =−θ

′(0), (17)

M. Qasim et al. · Radiation Effect on the Mixed Convection Flow of a Viscoelastic Fluid 197

Exact solutions of (9) – (10) for λ = 0 are

f (η) =[

1− exp(−Aη)A

], K =

1√1−K

θ(η) = exp

(− η

A

)1F1[ B

A2 ; BA2 −1;− B

A2 exp(−ηA)]

1F1[ PrA2 ; Pr

A2 −1;− PrA2 ]

,

B =Pr

1+Nr, (18)

where 1F1 are the confluent hypergeometric functions.

3. Solution Expressions

In order to find the homotopy solutions, we expressf and θ in the set of base functions

{ηk exp(−nη)|k ≥ 0,n≥ 0} (19)

by the following definitions

f (η) = a00,0 +

∞

∑n=0

∞

∑k=0

akm,nη

k exp(−nη), (20)

θ(η) =∞

∑n=0

∞

∑k=0

bkm,nη

k exp(−nη), (21)

where am,n and bm,n are the coefficients.Initial guesses f0 and θ0 and auxiliary linear opera-

tors are taken as

f0(η) = 1− exp(−η), θ0(η) = exp(−η), (22)

L f =d3 fdη3 −

d fdη

,Lθ =d2

θ

dη2 −θ , (23)

where

L f [C1 +C2 exp(η)+C3 exp(−η)] = 0,

Lθ [C4 exp(η)+C5 exp(−η)] = 0,(24)

in which Ci (i = 1 – 5) are arbitrary. Introducing p ∈[0,1] as the embedding parameter and h f and hθ thenon-zero auxiliary parameters, the zeroth-order defor-mation problems are presented as follows:

(1− p)L f [ f (η , p)− f0(η)]

= ph fN f [ f (η , p), θ(η , p)],(25)

(1− p)Lθ [θ(η , p)−θ0(η)]

= phθNθ [ f (η , p), θ(η , p)],(26)

f (η ; p)∣∣η=0 = 0,

∂ f (η ; p)∂η

∣∣∣∣η=0

= 1,

∂ f (η ; p)∂η

∣∣∣∣η=∞

= 0,

(27)

θ(η ; p)∣∣η=0 = 1, θ(η ; p)

∣∣η=∞

= 0, (28)

N f [ f (η ; p), θ(η ; p)] =∂ 3 f (η ; p)

∂η3 −(

∂ f (η ; p)∂η

)2

+ f (η ; p)∂ 2 f (η ; p)

∂η2 +λ cosαθ(η , p)

−K

[+2

∂ f (η ; p)∂η

∂ 3 f (η ; p)∂η3

− ∂ f (η ; p)∂η

∂ 4 f (η ; p)∂η4 −

(∂ 2 f (η ; p)

∂η2

)2],

(29)

Nθ [ f (η ; p), θ(η ; p)] =(

1+43

Nr

)∂ 2θ(η , p)

∂η2

+Pr

(f (η ; p)

∂ θ(η ; p)∂η

− θ(η ; p)∂ f (η ; p)

∂η

) (30)

Substituting p = 0 and p = 1, we have

f (η ;0) = f0(η), f (η ;1) = f (η), (31)

θ(η ;0) = θ0(η), θ(η ;1) = θ(η) (32)

and when p increases from 0 to 1, f (η ; p) and θ(η ; p)deforms from f0(η) and θ0(η) to f (η) and θ(η) re-spectively. Further Taylors’ series expansion results

f (η ; p) = f0(η)+∞

∑m=1

fm(η)pm, (33)

θ(η ; p) = θ0(η)+∞

∑m=1

θm(η)pm, (34)

fm(η) =1

m!∂ m f (η ; p)

∂ pm

∣∣∣∣∣p=0

,

θm(η) =1

m!∂ mθ(η ; p)

∂ pm

∣∣∣∣∣p=0

,

(35)

and the auxiliary parameters h f and hθ are selected insuch a manner that the series (20) and (21) converge atp = 1. Hence

f (η) = f0(η)+∞

∑m=1

fm(η),

θ(η) = θ0(η)+∞

∑m=1

θm(η),(36)

and the associated problem at mth order are

L f [ fm(η)−χm fm−1(η)] = h fR fm(η),

Lθ [θm(η)−χmθm−1(η)] = hθRθm(η),

(37)

198 M. Qasim et al. · Radiation Effect on the Mixed Convection Flow of a Viscoelastic Fluid

fm(0) = 0, f ′m(0) = 0, f ′m(∞) = 0,

θm(0) = 0,θm(∞) = 0,φm(0) = 0,φm(∞) = 0,(38)

R fm(η) = f ′′′m−1

+m−1

∑k=0

(fm−1−k f ′′k − f ′m−1−k f ′k

−K(2 fm−1−k f ′′′k − f ′′m−1−k f ′′k − fm−1−k f ′′′′k

))+λ cosaθm−1−k, (39)

Rθm(η) =

(1+

43

Nr

)θ′′m−1

+Prm−1

∑k=0

(fm−1−kθ

′k− f ′m−1−kθk

) (40)

χm =

{0, m≤ 1

1, m > 1. (41)

The general solutions of (39) – (41) are obtained as fol-lows:

fm(η) = f ∗m(η)+C1 +C2 exp(η)+C3 exp(−η), (42)

θm(η) = θ∗m(η)+C4 exp(η)+C5 exp(−η), (43)

where f ∗m(η) and θ ∗m(η) are the special solutions and

C2 = C4 = 0,

C1 =−C3− f ∗m(0), C3 =∂ f ∗m(η)

∂η

∣∣∣∣η=0

,

C5 =−θ∗m(0).

(44)

4. Convergence of the Series Solutions

Clearly the non-zero auxiliary parameters h f and hθ

are present in the solutions (33) and (34). These param-

Fig. 2. h-curves of the functions f ′′(0) and θ ′(0) at 20th orderof approximation.

Table 1. Convergence of homotopy solutions for different or-der of approximations when α = π/4, K = 0.2, λ = 1.0,Pr = 1.0 and Nr = 0.3.

Order of approximation − f ′′(0) −θ ′(0)0 0.82251 0.85177

10 0.67463 0.8759520 0.67339 0.8760530 0.67334 0.8760335 0.67334 0.8760240 0.67334 0.8760

eters have indispensable role in controlling and adjust-ing the permissible range of convergence, the h f andhθ curves are presented for 20th-order of approxima-tion. The admissible values of h f and hθ are−1.1≤ h f ,hθ ≤ −0.3 (Fig. 2). The series given by (36) con-verge in the whole region of η when h f = hθ =−0.6. Table 1 indicates the convergence of the homo-topy analysis solutions for different order of approxi-mations.

5. Results and Discussion

The object of this section is to examine the velocityand concentration for various parameters of interest.A comparative study between exact and the homotopysolution for the velocity f ′(η) and the concentrationfield field θ(η) is presented in the Figures 3 and 4when λ = 0. These figures show an excellent agree-ment between the exact solution and homotopy solu-tion at 15th-order of approximations. The variationsof K, λ , α, Pr and Nr have been sketched in Fig-ures 5 – 9. Figure 5 depicts the velocity profile f ′(η)and temperature θ for various values of viscoelasticparameter K. It is found that velocity f ′(η) decreasesas the viscoelastic parameter increases (Fig. 5a) buttemperature increases in this case (Fig. 5b). Figure 6plots the effects of mixed convection parameters λ onthe velocity and temperature profiles. The influence ofbuoyancy parameter λ decreases the thermal bound-ary layer. Increasing buoyancy parameter correspondsto the stronger buoyancy force and thus lead to thelarger velocity. The larger velocity accompanies withdecreasing boundary layer thickness for temperature. Itis interesting to observe that for large inclination α theboundary layer thickness increases (Fig. 7a) whereasopposite effects are found in case of temperature dis-tribution Figure 8. Infact increasing values of λ corre-sponds to the stronger buoyancy force which causes an

M. Qasim et al. · Radiation Effect on the Mixed Convection Flow of a Viscoelastic Fluid 199

Fig. 3. Comparison of f ′(η) for the analytical approximation with an exact solution when λ = 0. Filled circle: exact solution;solid line: 15th-order HAM solution.

Fig. 4. Comparison of θ(η) for the analytical approximation with the numerical solutions when λ = 0. Filled circle: numericalsolution; solid line: 15th-order HAM solution.

Fig. 5. Influence of viscoelastic parameter K on f ′ and θ .

200 M. Qasim et al. · Radiation Effect on the Mixed Convection Flow of a Viscoelastic Fluid

Fig. 6. Influence of mixed convection parameter λ on f ′ and θ .

Fig. 7. Influence of sheet inclination α on f ′ and θ .

Fig. 8. Influence of Prandtl number Pr on f ′ and θ .

M. Qasim et al. · Radiation Effect on the Mixed Convection Flow of a Viscoelastic Fluid 201

Fig. 9. Influence of radiation parameter Nr on f ′ and θ .

Table 2. Local Nusselt number Re−1/2x Nux for some values

of λ , Pr, K, and Nr when α = π/4.

λ K Pr Nr −Re−1/2x Nux

0.0 1.0 1.0 0.3 0.77372

1.0 0.87565

2.0 0.93747

1.5 0.0 1.0 0.3 0.90985

0.2 0.90926

0.4 0.90912

1.2 0.1 0.5 0.3 0.61227

0.7 0.73526

1.2 0.98682

1.5 0.1 0.7 0.0 0.88224

0.5 0.66862

1.0 0.55747

increase in flow velocity. Figure 8 illustrates the effectsof Prandtl number on the velocity and temperatureprofiles. It is observed that the effects of the Prandtlnumber decreases both velocity and thermal bound-ary layer thickness. Infact an increase in the Prandtlnumber leads to an increase in fluid viscosity whichcauses a decrease in the flow velocity. Note that Pr < 1corresponds to the flows for which momentum diffu-sivity is less than the thermal diffusivity. An increasein the weaker thermal diffusivity therefore results inthinning the thermal boundary layer. Note that Fig-ure 9 represent the variations of velocity and temper-ature profiles for various values of the radiation pa-

rameter Nr. Figure 9a depicts that velocity increaseswith the increase of radiation parameter Nr. It is foundthat an increase in Nr significantly increases the tem-perature θ (Fig. 9b). Thus radiation should be min-imized to have the cooling process at a faster rate.Numerical values of local Nusselt number for vari-ous values of embedding parameters are computed inTable 2. It is noticed that local Nusselt number is an in-creasing function of λ and Pr. However an increase inNr causes a reduction in the magnitude of local Nusseltnumber.

6. Concluding Remarks

In this work, the effect of thermal radiation on themixed convection boundary layer flow and heat trans-fer in a viscoelastic fluid over an inclined stretchingsheet is studied. The main observations are presentedas follows.

• The HAM solutions for velocity and temperaturefields are in an excellent agreement with the exactsolution.

• The effect of viscoelastic parameter K on velocityand thermal boundary layer thickness are quite op-posite.

• An increase in the viscoelastic parameter K resultsa decrease in the velocity and the associated bound-ary layer thickness. However the temperature andthe thermal boundary layer thickness increase whenK increases.

202 M. Qasim et al. · Radiation Effect on the Mixed Convection Flow of a Viscoelastic Fluid

• An increase in Prandtl number Pr reduces the tem-perature and the thermal boundary layer thickness.

• The values for Nusselt number for viscoelastic fluidare more than the viscous fluid.

Acknowledgement

Second author as a visiting Professor thanks theKing Saud University for the support (KSU-VPP-117).

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