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Thermal Boundary Layer Flow on a Stretching Plate with Radiation Effect

Siti Khuzaimah Soida & Anuar Ishakb

aFaculty of Computer and Mathematical Sciences, Universiti Teknologi MARA, 40450 UiTM Shah Alam, Selangor, Malaysia

bSchool of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor, Malaysia

Abstract

A steady two-dimensional laminar forced convection boundary layer flow over a stretching plate immersed in an incompressible viscous fluid is considered. The stretching velocity is assumed to vary linearly with the distance from the leading edge. The governing partial differential equations are first reduced to ordinary differential equations using a similarity transformation, before being solved numerically using MAPLE software, which based on the Runge-Kutta-Fehlberg method. Besides, the solution for the thermal field is also given in terms of Kummer’s function. The effects of the radiation parameter and the Prandtl number on the heat transfer characteristics are obtained and discussed. It is found that the heat transfer rate at the surface increases as the Prandtl number increases but decreases with the radiation parameter.

I. INTRODUCTION

The study of flow and heat transfer over a stretching plate immersed in an incompressible viscous fluid has received a great deal of research interest due to its importance in many manufacturing processes, such as the extrusion of polymers, the aerodynamic extrusion of plastic sheet, paper production, etc. In high temperatures processes, thermal radiation effects play an important role and cannot be neglected [1]. The flow over stretching plates can be observed in the process of blowing and floating or spinning of fiber glass [2]. The effects of radiation and surface stretching on the fluid flow and heat transfer can be used to achieve optimum results and better products of desired characteristics.

Sakiadis [3] was the first investigator who studied the laminar boundary layer flow of a viscous and incompressible fluid over a moving surface with constant velocity. Crane [4] and Grubka and Bobba [5] studied the flow over a linearly stretching plate. Crane [4] found an exact solution for the flow field. Numerous researchers have investigated the fluid flow and heat transfer over a stretching sheet by considering the influence of a magnetic field [6-12], viscoelastic fluid [13] and power-law stretched surface with suction or injection [14].

The effect of radiation on the boundary layer flow was studied by Elbashbeshy and Dimian [15], Hossain et al. [16,17], Bataller [18] and Cortell [19]. Bataller [18] studied

the flow over a static flat plate (Blasius flow) and Cortell [19] studied the boundary layer flow over a moving flat plate (Sakiadis flow) in a quiescent fluid, both with radiation effect is taken into consideration. The problems considered by Bataller [18] and Cortell [19] have been extended by Ishak [20] and he found the existence of dual solutions when the plate and the fluid move in the opposite directions.

In this paper, we extend the work of Grubka and Bobba [5] by including the effect of radiation on the steady two-dimensional laminar forced convection boundary layer flow over a linearly stretching plate immersed in an incompressible viscous fluid.

II. MATHEMATICAL FORMULATION

Consider a steady two-dimensional laminar boundary

layer flow over a linearly stretching plate immersed in an incompressible viscous fluid as shown in Fig. 1. The plate is stretched with velocity wU ax , where a is a positive constant. The ambient temperature T and the surface temperature wT are assumed to be constant where wT T .

Fig. 1. Physical model and coordinate system The boundary layer equations of such type of flow in the

usual notations are

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2

0u vx y

(1)

2

2u u uu vx y y

(2)

2

21 r

p

qT T Tu vx y c yy

(3)

where u and v are the velocity components along the x and y directions, respectively, is the kinematic viscosity, T is the fluid temperature, is the thermal diffusivity, is the fluid density, pc is the specific heat at

constant pressure and rq is the radiative heat flux. By using the Rosseland approximation [1,16,21], the

radiative heat flux rq is written as

44 ,3r

TqK y

(4)

where and K are the Stefan-Bolzman constant and Rosseland mean absorption coefficient, respectively. Assuming that the temperature differences within the flow are sufficiently small such that 4T may be expressed as a linear function of temperature

4 4 3 3 4( )4 4 3 .T T T T T T T T (5)

Using (4) and (5) in (3), we obtain

2 2

2 2 ,T T T Tu v Nx y y y

(6)

where 3(16 / 3 )N T K k is the radiation parameter [19,22]. The boundary conditions for the present problem are , 0,w wu U v T T at 0,y 0,u T T as .y (7) We introduce the similarity transformation as follows:

1/2

,a y

1/2 ( ),a x f

( )w

T TT T

, (8)

where is the similarity variable and is the stream function defined as /u y and /v x which identically satisfies the continuity equation (1). With this transformation, the momentum equation (2) and the energy equation (6) respectively reduce to

2 0, f ff f (9)

1 (1 ) 0 ,Pr

N f (10)

where prime indicates differentiation with respect to and Pr / is the Prandtl number.

The transformed boundary conditions are given by (0) 0, (0) 1, (0) 1f f ,

( ) 0, ( ) 0f as . (11)

The physical quantities of interest are the skin friction coefficient fC and the local Nusselt number xNu which are defined as

2

2 wf

w

CU

,

( )w

xw

x qNu

k T T

, (12)

where the surface shear stress w and the surface heat flux wq are given by

3/2

1/20

(0)w

y

u a x fy

,

1/2

0

(0)w wy

T aq k k T Ty

, (13)

with and k being the dynamic viscosity and the thermal conductivity, respectively. Using the non-dimensional variable in (8) we obtain

1/21 Re (0)2 f xC f , 1/2Re (0),x xNu (14)

where Re /x wU x is the local Reynolds number.

We remark that the exact solution of (9) subject to the associated boundary conditions (11) was given by Crane [4] as ( ) 1f e , (15)

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3 while the solution for the energy equation is given by

Pr

1

Pr Pr Pr, 1,1 1 1( )

Pr Pr Pr, 1,1 1 1

NM e

N N NeM

N N N

, (16)

where ( , , )M a b z denotes the Kummer’s function (see Abramowitz and Stegun [23]), with

1

( , , ) 1 ,!

nn

nn

a zM a b zb n

( 1) ( 2) ( 1),na a a a a n ( 1) ( 2) ( 1).nb b b b b n (17)

Equations (12) and (13) yield the skin friction coefficient

(0)f and the heat transfer rate at the surface (0) given by

(0) 1f , Pr Pr PrPr 1, 2,

Pr Pr 1 1 11(0) .Pr Pr Pr Pr1 1 1 , 1,1 1 1 1

MN N NN

N N MN N N N

(18)

III. RESULTS AND DISCUSSION

The nonlinear ordinary differential equations (9) and (10)

along with the boundary conditions (11) were solved numerically using “dsolve” in MAPLE software, which based on the Runge-Kutta-Fehlberg method. We also have solved the system of equations (9)-(11) analytically, and obtained a series solution in terms of Kummer’s function. Comparative study with the numerical results of Grubka and Bobba [5] for the case N = 0 (without radiation) was also carried out to validate the numerical and the series solutions obtained. The comparison shows an excellent agreement, as can be seen from Table 1.

Table 1 presents the values of the heat transfer rate at the surface (0) for various values of Pr and N. It is seen that the heat transfer rate at the surface increases as the Prandtl number Pr increases. This is because increasing Pr is to increase the fluid viscosity but reduces the thermal conductivity, and in consequent increases the heat transfer rate at the surface (0) . Opposite behaviors are observed for the effect of the radiation parameter N. As the radiation parameter increases, the fluid temperature inside the boundary layer increases, and this reduces the temperature gradient at the surface.

Fig. 2 shows the temperature profiles for different values of Pr when 1N . It is seen that as Pr increases, the fluid temperature inside the boundary layer as well as the thermal boundary layer thickness decreases. This results in increasing

manner of the magnitude of the temperature gradient at the surface | (0) | . Thus, the heat transfer rate at the surface increases as Pr increases.

Figs. 3 and 4 depict the temperature profiles for different values of N when Pr = 0.7 and 10, respectively. It is seen that when the radiation parameter N increases, the fluid temperature inside the boundary layer ( ) increases. The thermal boundary layer thickness also increases and cause the surface temperature gradient (0) decreases. Thus, the rate of heat transfer at the surface decreases in the presence of radiation parameter N.

Table 1: Values of (0) for various values of Pr and N . Values in parentheses are from Grubka and Bobba [5].

Heat Transfer Coefficient,

(0)

Pr N Numerical solution

Exact solution

(18) 0.7 0 0.4539 0.4539

0.5 0.3364 0.3364

1 0.2688 0.2688

5 0.1051 0.1051

10 0.0599 0.0599

1 0 0.5820(0.5820) 0.5820

0.5 0.4383 0.4383

1 0.3544 0.3544

5 0.1444 0.1444

10 0.0836 0.0836

3 0 1.1652(1.1652) 1.1652

0.5 0.9114 0.9114

1 0.7603 0.7603

5 0.3544 0.3544

10 0.2195 0.2195

7 0 1.8954 1.8954

0.5 1.5075 1.5075

1 1.2762 1.2762

5 0.6454 0.6454

10 0.4238 0.4238

10 0 2.3080(2.3080) 2.3080

0.5 1.8445 1.8445

1 1.5681 1.5681

5 0.8131 0.8131

10 0.5453 0.5453

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Fig. 2. Temperature profiles ( ) for different values of Pr when N=1

Fig. 3. Temperature profile ( ) for different values of N when Pr = 0.7

Fig. 4. Temperature profile ( ) for different values of N when Pr = 10

Fig. 5 presents the velocity profiles ( )f . We note that the parameters N and Pr have no effects on the flow field due to their inexistence in Eq. (9). The negative velocity gradient at the surface, 0 0,f means the stretching sheet exerts a

drag force on the fluid that cause the movement of the fluid on the surface.

IV. CONCLUSIONS The steady laminar two-dimensional boundary layer flow

and heat transfer over a linearly stretching sheet with radiation effects was investigated. The effects of the radiation parameter N and the Prandtl number Pr on the heat transfer characteristics were discussed. It was found that the heat transfer rate at the surface decreases as the radiation parameter N increases. Different behaviors are observed for the effect of the Prandtl number, i.e. increasing the Prandtl number is to increase the heat transfer rate at the surface.

Fig. 5. Velocity profile f

IV. ACKNOWLEDGMENT The authors would like to express their very sincere

thanks to the editor and the anonymous referees for their valuable comments and suggestions. The financial support received from the Centre of Excellence, Universiti Teknologi MARA is gratefully acknowledged.

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