FREE CONVECTION BOUNDARY LAYER FLOW OVER A STRETCHING FLAT PLATE WITH VARIABLE THERMAL CONDUCTIVITY AND RADIATION

Keywords: free convection, stretching plate, radiation, variable thermal conductivity, skin-friction coefficient, and Nusselt number

An analytical approach for the study of effect of thermal radiation on steady boundary layer flow with variable thermal conductivity due stretching sheet has been developed. To describe radiative heat flux, Rosseland diffusion approximation energy equation is used. The governing equation reduced to similarity boundary equation using suitable transformation and then solved by using homotopy analysis method. The velocity profiles and temperature profiles are compared numerically by using Fehlburg method. 

Introduction

The common type of body force which acts on a fluid, is due to gravity. The study of fluid flow gained significant importance in engineering disciplines. For example free convection flow over vertical flat plate has attracted researchers because of its application in cooling by natural convection as in case of electrical heaters and transformer. Also the study of natural convection has supported scientists when stars and planets or cooling reactors is required. 
In a manufacturing process of a flat plate, application of boundary layer flow demands special attention. Example of such technological process of hot rolling, wire drawing, glass fibre and paper production. The moving fiber is governed by rate at which it is cooled. A number of work are presently available that follows the pioneer work of Crane [1], Angew, Chen and char [2], Datta, Roy, Gupta [3]. In 2010 Raftari, Yildrim [5] considered heat transfer problem with variable thermal conductivity on flow towards stretching sheet, Ariel [6] also studied computation of flow past a stretching sheet. In most of cases thermal conductivity is considered to be constant But in this paper we try to solve the boundary layer flow of viscous incompressible flow past a flat plate. 
Many transport processes that occur both in nature and in industries involve fluid flows with combined heat and mass transfer. Such flows are driven by the buoyancy effects arising from the density variations caused by the variations in temperature and/or species concentrations 
In some industrial problems dealing with chemical reactions and dissociating fluids, heat generation/absorption and chemical effects are very important. Vajravelu, Hadjinicolaou [12]. Some of the work related to the paper can be found in the papers by Mukhopadhayay [13], Ali [14], Ishak, Nazar and Pop [15] and chiam [16]. 

Mathematical Formulation

Consider the steady, incompressible and two-dimensional boundary layer free convection flow of a viscous fluid over a flat plate with variable thermal conductivity and radiation effects. Here we assumed that thermal conductivity varies linearly with temperature. The governing boundary equations are 
∂ (u) ∂x + ∂ (v) ∂y = 0, (1) 
u ∂u ∂x + v ∂u ∂y =ν ∂ 2 u ∂y 2 + gβ (T -T ∞ ) , (2) 
ū ∂v ∂ x + v ∂v ∂ ȳ =ν ∂ 2 v ∂ ȳ2 + ∂ 2 u ∂ ȳ2 , (3) 
u ∂T ∂ x + v ∂T ∂ ȳ = 1 ρcp ∂ ∂y (α ∂T ∂y ) - 1 ρc p ∂q r ∂r , (4) 
subjected to the boundary conditions: 
u = 0, v = v w (x), T = T ∞ at x = 0, (5) 
u = 0, v = 0, T = T w (x) as y → ∞. (6) 
In the above expressions, u and v are the velocity components in x and y directions respectively, q r is the radiative heat flux, ν is the kinematic viscosity, ρ is the density, c p is the specific heat, α (T ) is the variable thermal conductivity, T and T ∞ are the temperatures of the fluid and surrounding respectively,. 
There are few forms of thermal conductivity variation available in literature Among them we have considered that one which is appropriate for liquid introduced by Hossain et [16] as follows 
α= α ∞ (1+ ∈ (T -T ∞ )), (7) 
where α ∞ is the thermal conductivity of ambient fluid and 
∈= 1 α f ( ∂α ∂T ) (8) 
Using Roseland approximation the radiative heat flux is simplified as: 
q r = - 4σ * 3k * ∂T 4 ∂y , (9) 
where σ * and k* are the Stefan-Boltzmann constant and the mean absorption coefficient, We assume that the temperature differences within the flow,are assumed to be sufficiently small such as the term T4, may be expressed as a linear function of temperature. Hence, this is accomplished by expanding T4 in a Taylor series about T∞ and neglecting higher-order terms, we get 
T 4 = T 4 ∞ + (T -T ∞ )4T 3 ∞ . ( 10 
)
Using the similarity transformations 
Ψ =x 3 4 (T w(x) ) 1 4 f (x, η), T = T w (x)Θ(x, η), η =T w (x) y x 1 4 (11) 
Equation ( 1) is automatically satisfied while Eqs. ( 2) to (4) takes the form 
f ηηη+ (3 + M ) 4 f f ηη - 1 2 (1 + M )f ′ 2 + θ = 0, ( 12 
) 1 Pr ((1 + ǫ +ǫθ) θ ′′ + 4R 3 )θ ′′ ) + 3 + M 4 f θ η -M f η θ = 0 (13) f (0) =0, f ′ (0) = 1, θ (0) = 1 (14) f ′ (∞) =0, θ (∞) = 0. ( 15 
)
Where M , Pr, R, and ∈. M denotes the Temperature parameter, P r denotes Prandtl number R is representing radiation parameter and ∈ denotes variable thermal conductivity parameter which are given by 
Pr = µc p α ∞ , (16) 
M = x T w (x) dT w dx (17) 

Homotopy Solutions

Homotopy analysis method depend upon the initial guesses (f 0 , θ 0 ) and linear operators (L f , L θ ) which are given in the forms 
f 0 (η) = 0, θ 0 (η) = exp (-η) , (18) 
L f (η) = d 3 f dη 3 - df dη , L θ (η) = d 2 θ dη 2 -θ, (19) 
with 
L f [C3 * exp (-η) + C1 + C2 * η] = 0, , (20) 
L θ [C1 * exp (-η)] = 0 = 0, (21) 
where A i (i = 1 -3) are the arbitrary constants. The zeroth and mth order deformation problems are 
3.1. Zeroth-Order Problem (1 -p) L f [ f (η; p) -f0 (η)] = p f N f [ f (η; p), θ(η; p)], (22) 
(1 -p) L θ [ θ(η; p) -θ0 (η)] = p θ N θ [ f (η; p), θ(η; p)], (23) 
f (0; p) = 0, f ′ (0; p) = 1, f ′ (∞; p) = 0, (24) 
θ (0; p) = 1, θ (∞; p) = 0, ( 25 
) N f [ f (η; p), θ(η; p)] = ∂ 3 f (η; p) ∂η 3 + ( 3 + M 4 ) f (η; p) ∂ 2 f (η; p) ∂η 2 -( 1 + M 2 ) ∂ f (η; p) ∂η 2 + θ (26) N θ θ (η; p) , f (η; p) = 1 Pr 1 + ǫ θ(η, p) ∂ 2 θ(η, p) ∂η 2 + 1 Pr (1 + 4R 3 ) ∂ 2 θ(η; p) ∂η 2 -( 3 + M 4 ) f (η; p) ∂ θ(η; p) ∂η + (M ) ∂ f (η; p) ∂η θ(η; p) , (27) 
where p ∈ [0, 1] is embedding parameter and f and θ are the non-zero auxiliary parameters. 

m-th-Order Deformation Problems

L f [f m (η) -χ m f m-1 (η)] = f R f m (η) , (28) 
L θ [θ m (η) -χ m θ m-1 (η)] = θ R θ m (η) , (29) 
f m (0) = f ′ m (0) = f ′ m (∞) = 0, ( 30 
)
θ m (0) = 1 θ m (∞) = 0, (31) 
R f m (η) = f ′′′ m-1 + ( 3 + M 4 ) m-1 k=0 f k f ′′ m-1-k + ( 1 + M 2 ) m-1 k=0 f ′ k f ′ m-1-k ′ + θ m-1 (32) R θ m (η) = 1 Pr (1 + 4R 3 )θ ′′ m-1 + 1 Pr (1+ ∈ θ m-1 )θ ′′ m-1 + ( 3 + M 4 ) m-1 k=0 f k θ ′ m-1-k -(M ) m-1 k=0 θ k f ′ m-1-k χ m = 0, m ≤ 1 1, m > 1 . (33) 
For p = 0 and p = 1, we can write 
f (η; 0) = f 0 (η) , f (η; 1) = f (η) , (35) 
θ (η; 0) = θ 0 (η) , θ (η; 1) = θ (η) (36) 
and with the variation of p from 0 to 1, f (η; p) and θ (η; p) vary from the initial solutions f 0 (η) and θ 0 (η) to final solutions f (η) and θ(η) respectively. By Taylor's series we have 
f (η; p) = f 0 (η) + ∞ m=1 f m (η) p m , f m (η) = 1 m! ∂ m f (η; p) ∂p m p=0 (37) θ (η; p) = θ 0 (η) + ∞ m=1 θ m (η) p m , θ m (η) = 1 m! ∂ m θ (η; p) ∂p m p=0 . ( 38 
)
The value of auxiliary parameter is selected so properly that the series (33) and (34) converge at p = 1 i.e. 
f (η) = f 0 (η) + ∞ m=1 f m (η) , (39) 
θ (η) = θ 0 (η) + ∞ m=1 θ m (η) . (40) 
The general solutions (f m , θ m ) of Eqs. ( 23) and (24) in terms of special solutions (f * m , θ * m ) are given by 
f m (η) = f * m (η) + C3 * exp (-η) + C1 + C2 * η, (41) 
θ m (η) = θ * m (η) + c1 * exp (-η) , (42) 
Skin friction and local Nusselt number can be defined as 
C f = τ w ρu 2 w , N u x = xq w k(T w -T ∞ ) , (43) 
τ w = µ ∂u ∂y y=0 , q w = -k ∂T ∂y y=0 . (44) 
While the dimensionless forms of skin friction and local Nusselt number are 
1 √ 2 C f Re 1/2 x = f ′′ (0) , 1 √ 2 N u x Re -1/2 x = -θ ′ (0) , (45) 
where Re x = U w x/ν. 

Convergence of the Homotopy Solutions

Homotopy analysis method is applied to find the convergence of desired equations which is proposed by Liao [17] . Obviously the series solutions obtained by homotopy analysis method contain the convergence control parameter . This parameter controls the convergence region and the rate of approximation of the HAM solution. To ensure the convergence of the solutions in the admissible range of the values of the auxiliary parameters f and θ ,curve for 12-thorder approximations have been made. It is evident from Figs. 
(2) to (3), that the admissible range of values of f ,and θ are -1.4 ≤ f ≤ --0. x Re x so we conclude that as Reynolds number increases, Re x = U x v increases when viscosity decreases, so decrease in viscosity enhances the magnitude of rate of convectional heat transfer Also, the coefficient of convectional heat depends on Prandtl number P r, radiation parameter R, thermal variable parameter ǫ and temperature parameter M. The behavior of the coefficient of convectional heat transfer is studied in • The behavior of radiation parameter R on velocity and temperature fields are same i.e. both velocity and temperature fields increase with velocity and temperature. 
• Temperature field and velocity field decreases for temperature parameter. 
• Velocity field increases for values of variable thermal conductivity parameter 

Figure 1 :

 1 Figure 1: Effect of R on velocity profile 


Figure 2 :Figure 3 :Figure 4 :Figure 5 :

 2345 Figure 2: Effect of ∈ on velocity profile 


Figure 6 :

 6 Figure 6: Effect of ∈ on temperature profile M Pr ∈ R f ′′ (0) 0.1 2 1 1 0.781798 0.3 2 1 1 0.774325 0.5 2 1 1 0.766929 0.7 2 1 1 0.759609 


Table 1 :

 1 Convergence table for velocity and temperature profile 


	to

8, -1.2 ≤, θ ≤ -0.3The convergence table 1. is prepared for each of the function up 
Table 2 :

 2 The variation of 1 2 C f √ Re x with respect to M Reading the Table 2, we see that C f Re | decreases. It is well known that as Re x increases the viscous forces start reducing, in turn |C f | reduces. In the present case, as we increase temperature parameter M the 1 2 |C f Re x | decreases. As we increase the Pr 1 2 |C f Re x | decreases. But in the case of variable thermal conductivity parameter the 1 2 |C f Re x | increases. Reading the Table 3, we see that N u 2 


	-1 2 x . Thus, as Reynold number

Table 3 .

 3 By increasing value of prandtl number value of viscosity increases because it 


	M Pr ∈ 2 1 1	R 1	-θ ′ (0) 0.600000
	2	2	1	1	0.630000
	2	3	1	1	0.677000
	2	1	0.1 1	0.404400
	2	1	0.2 1	0.408019
	2	1	0.3 1	0.393589
	2	1	0.4 1	0.391581
	1	1	1	1	0.362597
	2	1	1	1	0.377245
	3	1	1	1	0.391758
	2	1	1	0.1 0.444500
	2	1	1	0.2 0.437632

Table 3 :

 3 Temprature gradient -θ ′ (0) at the outer surface of plate for different values of ǫ, P r and R 


		M =1			M = 2
	ǫ HAM Result F ehlburgResult HAM Result F ehlburgResult
	2	0.8730	0.87301	0.78432	0.78471
	4	0.8834	0.88368	0.79522	0.79511
	6	0.8938	0.89287	0.8043	0.80413
	8	0.9025	0.90093	0.81209	0.81209

Table 4 :

 4 The comparison of HAM and Fehlburg method for f " (η) when η → 0 for variable thermal conductivity ǫ and temperature parameter M is ratio b/w momentum diffusitivity and thermal diffusivity there fore Nusselt number increases because Reynolds number decreases.Table4and Table5are the caparison of numerical and HAM solution. 


	5. Conclusions
	• The main findings of present analysis are listed below

Table 5 :

 5 The comparison of HAM and Fehlburg method for θ(η) when η → 0 for Prandtl number Pr and Radiation parameter R
M Y Malik, S Bilal, Farzana Khan
FREE CONVECTION BOUNDARY LAYER FLOW OVER A STRETCHING FLAT PLATE WITH VARIABLE THERMAL CONDUCTIVITY AND RADIATION