# Similarity Solutions for Flow and Heat Transfer of Non-Newtonian Fluid over a Stretching Surface

Authors: Atta Sojoudi; Ali Mazloomi; Suvash C. Saha; Y. T. Gu
Journal: Journal of Applied Mathematics
 (2014)

---

## Abstract

Similarity solutions are carried out for flow of power law non-Newtonian fluid film on unsteady stretching surface subjected to constant heat flux. Free convection heat transfer induces thermal boundary layer within a semi-infinite layer of Boussinesq fluid. The nonlinear coupled partial differential equations (PDE) governing the flow and the boundary conditions are converted to a system of ordinary differential equations (ODE) using two-parameter groups. This technique reduces the number of independent variables by two, and finally the obtained ordinary differential equations are solved numerically for the temperature and velocity using the shooting method. The thermal and velocity boundary layers are studied by the means of Prandtl number and non-Newtonian power index plotted in curves.


---

## Body

## 1. Introduction

Non-Newtonian fluids such as foodstuffs, pulps, glues, ink, polymers, molten plastics, or slurries are increasingly used in various manufacturing, industrial, and engineering applications, particularly in the chemical engineering processes. Many simultaneous transport processes exist in industrial or engineering applications. For instance, mechanical forming processes include extrusion and melt spinning where the extruded material issues through a die. The ambient fluid status is stagnant but a flow is induced adjacent to the material being extruded, due to the moving surface. Noting that the fluids employed in material processing or protective coatings are in general non-Newtonian, so the study of non-Newtonian liquid films is important.The problem of heat and mass transfer from vertical plate has been the subject of great interest for several researchers in past decades. As an earlier study, Wang [1] considered the flow problem within a finite liquid film of Newtonian fluid over an unsteady stretching sheet. Later, Andersson et al. [2] investigated the heat transfer characteristics of the hydrodynamic problem solved by Wang [1]. Usha and Sridharan [3] regarded a similar problem of axisymmetric flow in a liquid film. The effect of thermocapillarity on the flow and heat transfer in a thin liquid film was studied by Dandapat et al. [4]. The free-surface flow of non-Newtonian liquids in thin films is a widely occurring phenomenon in various industrial applications, for instance, in polymer and plastic fabrication, food processing, and coating equipment. There are limited papers on gravity-driven power-law film flows [5–7] and the studies of non-Newtonian film flows on an unsteady stretching surface remain little. The heat transfer aspect of such problem has also been considered by Chen [8].The most distinctive works of Vajravelu and Hadjinicolaou [9] and Mehmood and Ali [10] are available in the literature describing the heat transfer characteristics in the laminar boundary layer of a viscous fluid over a linearly stretching continuous surface with variable wall temperature and the incompressible generalized three-dimensional viscous flow with heat transfer analysis in the presence of viscous dissipation generated due to uniform stretching of the plane wall, respectively. Unsteadiness of the similar problems is stated by many researchers. For instance, Tsai et al. [11] studied nonuniform heat source/sink effect on the flow and heat transfer from an unsteady stretching sheet through a quiescent fluid medium extending to infinity. The boundary layer equations are converted by using similarity analysis to be a set of ordinary differential equations including unsteadiness parameter.Recently, Sahoo and Do [12] investigated the entrained flow and heat transfer of an electrically conducting non-Newtonian fluid due to a stretching surface subject to partial slip. The constitutive equation of the non-Newtonian fluid is modeled by that for a third grade fluid. They found out that slip decreases the momentum boundary layer thickness and increases the thermal boundary layer thickness, whereas the third grade fluid parameter has an opposite effect on the thermal and velocity boundary layers.Many efforts were made to find analytical and numerical solutions, imposing certain status and using various mathematical concepts to the same problem [13–19]. The mathematical technique employed in the current analysis is the two-parameter group transformation, which leads to a similarity representation of the problem. Morgan [20] presented a theory which led to amendments over earlier similarity methods and Michal [21] extended Morgan’s theory. Group methods, as a class of methods which assuage to a reduction of the number of independent variables, were first presented in [22, 23]. Moran and Gaggioli [24, 25] indicated a general systematic group formalism for similarity study, where a given combination of partial differential equations was reduced to a system of ordinary differential equations [25–34].In the present work, we provide analytical solution for the unsteady free convection non-Newtonian fluids flow over a continuous moving vertical plate subjected to constant heat flux using a two-parameter group. The specified technique is explained at the following parts. Under the employment of a two-parameter group, the governing partial differential equations and boundary conditions are reduced to ordinary differential equations with the appropriate boundary conditions. The obtained differential equations are solved using the shooting method.


## 2. Mathematical Formulation

The unsteady laminar flow of an incompressible fluid induced by a ceaseless moving sheet placed in a fluid at quiescent is considered. The vertical flat sheet illustrated in Figure1 arises from a thin slit at x
=
y
=
0 and is subsequently stretched vertically. The positive x and y  coordinates are measured along the direction of the moving film with the slot as the origin and the normal to the sheet, respectively. Constant heat flux, q
w, is imposed to the flat sheet, giving rise to a buoyancy force, while the ambient fluid is kept at a constant temperature, T
∞. The boundary layer equations governing the free convection flow over the moving sheet are expressed as follows:
(1)
∂
u
*
∂
x
+
∂
v
∂
y
=
0
,
∂
u
*
∂
t
+
u
*
∂
u
*
∂
x
+
v
∂
u
*
∂
y
=
ϑ
n
(
∂
u
*
∂
y
)
n
-
1
∂
2
u
∂
y
2
+
g
β
(
T
-
T
∞
)
,
∂
T
∂
t
+
u
*
∂
T
∂
x
+
v
∂
T
∂
y
=
ϑ
Pr
∂
2
T
∂
y
2
,
where the certain boundary conditions are(2a)
u
*
(
x
,
y
,
0
)
=
u
0
*
(
x
,
y
)
,
v
(
x
,
y
,
0
)
=
v
0
(
x
,
y
)
,
T
(
x
,
y
,
0
)
=
T
0
(
x
,
y
)
,
u
*
(
x
,
0
,
t
)
=
u
w
(
x
,
t
)
,
v
(
x
,
0
,
t
)
=
0
,
(2b)
∂
T
(
x
,
0
,
t
)
∂
y
=
-
q
w
k
,
u
*
(
x
,
y
,
t
)
=
0
,
T
(
x
,
y
,
t
)
=
T
∞
fffffffffffffffffffffffffffffffffffffff
as
y
⟶
+
∞
u
* and v are velocity components related to the x and y coordinates. T, g, and n are fluid temperature, gravity, and power index of non-Newtonian fluid, respectively. Pr is defined as the ratio of kinematic viscosity (υ) to thermal diffusion (α). β is thermal expansion coefficient at constant pressure where u
0
*, v
0, and T
0 are initial velocity components and initial temperature. u
w and T
w are fluid velocity and temperature on vertical moving sheet. Dimensionless x-velocity and temperature are defined using the following relationships: u
=
u
*
/
u
w and θ
(
x
,
y
,
t
)
=
(
(
T
(
x
,
y
,
t
)
-
T
∞
)
/
q
w
x
)
k
Re
x. Now, (1) is rewritten in the form
(3)
u
∂
u
w
∂
x
+
u
w
∂
u
∂
x
+
∂
v
∂
y
=
0
,
(4)
u
∂
u
w
∂
t
+
u
w
∂
u
∂
t
+
u
w
2
u
∂
u
∂
x
+
u
2
u
w
∂
u
w
∂
x
+
v
u
w
∂
u
∂
y
=
ϑ
n
u
w
n
(
∂
u
∂
y
)
n
-
1
(
∂
2
u
∂
y
2
)
+
g
β
q
w
k
Re
x
θ
x
,
(5)
x
∂
θ
∂
t
+
u
w
u
(
x
∂
θ
∂
x
+
θ
)
+
v
∂
θ
∂
y
x
=
ϑ
Pr
x
∂
2
θ
∂
y
2
,
and boundary conditions are (6a)
u
(
x
,
y
,
0
)
=
u
0
(
x
,
y
)
,
v
(
x
,
y
,
0
)
=
v
0
(
x
,
y
)
,
θ
(
x
,
y
,
0
)
=
θ
0
(
x
,
y
)
,
u
(
x
,
0
,
t
)
=
1
,
v
(
x
,
0
,
t
)
=
0
,
(6b)
∂
θ
(
x
,
0
,
t
)
∂
y
=
-
Re
x
x
,
u
(
x
,
y
,
t
)
=
0
,
θ
(
x
,
y
,
t
)
=
0
fffffffffffffffffffffffffffffffffffffff
as
y
⟶
+
∞
.Figure 1
Physical model for laminar flow over stretching sheet.


## 3. Group Formulation of the Problem

### 3.1. Group Formulation

The problem is solved by employing a two-parameter group transformation to the partial differential equations of (3) to (5). This transformation decreases the three independent variables (
x
;
y
;
t
) to one similarity variable; η
(
x
;
y
;
t
) and the governing equations of (3) to (5) are transformed to a system of ordinary differential equations in terms of the similarity variable η. This technique is based on a class of transformation, namely, G, including two parameters (a
1, a
2):
(7)
G
:
S
-
=
C
S
(
a
1
,
a
2
)
S
+
K
S
(
a
1
,
a
2
)
,
where S is representative for x
,
y
,
t
,
u
w
,
v, and θ. C
s and K
s are real valued and at least differentiable in their real arguments (a
1, a
2). Dependent variables and related differentiates are as follows via chain rule operations:
(8)
S
-
i
-
=
(
C
S
C
i
)
S
i
,
S
-
i
j
-
=
(
C
S
C
i
C
j
)
S
i
j
f
f
f
f
f
f
f
i
=
x
,
y
,
t
,
j
=
x
,
y
,
t
.
For instance, (3) is transformed as follows:
(9)
u
-
∂
u
-
w
∂
x
-
+
u
-
w
∂
u
-
∂
x
-
+
∂
v
-
∂
y
-
=
H
1
(
a
1
,
a
2
)
[
u
∂
u
w
∂
x
+
u
w
∂
u
∂
x
+
∂
v
∂
y
]
,
where H
1
(
a
1
,
a
2
) is constant. If (7) and (8) are put in (9),
(10)
[
C
u
C
u
w
C
x
]
u
∂
u
w
∂
x
+
[
C
u
C
u
w
C
x
]
u
w
∂
u
∂
x
+
[
C
v
C
y
]
∂
v
∂
y
+
R
1
=
H
1
(
a
1
,
a
2
)
[
u
∂
u
w
∂
x
+
u
w
∂
u
∂
x
+
∂
v
∂
y
]
and R
1 is [
C
u
w
K
u
/
C
x
]
(
∂
u
w
/
∂
x
)
+
[
C
u
K
u
w
/
C
x
]
(
∂
u
/
∂
x
). For invariant transformation, R
1 is equated to zero. This is satisfied by setting
(11)
K
u
=
K
u
w
=
0
.
Finally, we get
(12)
[
C
u
C
u
w
C
x
]
=
[
C
v
C
y
]
=
H
1
(
a
1
,
a
2
)
.
Simultaneously, after, (4) and (5) transformation:
(13)
K
u
=
K
v
=
K
u
w
=
K
θ
=
K
x
=
K
y
=
0
,
(14a)
[
C
u
C
u
w
C
t
]
=
[
(
C
u
)
2
(
C
u
w
)
2
C
x
]
=
[
C
u
C
v
C
u
w
C
y
]
=
(
C
u
w
)
n
(
C
u
)
n
(
C
y
)
n
+
1
=
C
θ
C
x
=
H
2
(
a
1
,
a
2
)
,
(14b)
[
C
θ
C
x
C
t
]
=
[
C
u
C
u
w
C
θ
]
=
[
C
x
C
v
C
θ
C
y
]
=
[
C
x
C
θ
(
C
y
)
2
]
=
H
3
(
a
1
,
a
2
)and we have
(15)
K
u
=
K
v
=
K
u
w
=
K
θ
=
0
.
Employing invariant transformation for boundary conditions results in
(16)
K
t
=
K
y
=
0
(17)
C
u
=
1
.
Invoking (12) to (15) and (17) reduces to
(18)
C
x
=
C
y
,
C
t
=
(
C
y
)
2
/
(
2
-
n
)
,
C
u
w
=
C
v
=
1
(
C
y
)
n
/
(
2
-
n
)
,
C
θ
=
1
(
C
y
)
4
/
(
2
-
n
)
.
Group G is summarized as follows:
(19)
G
=
{
G
s
{
x
-
=
C
y
x
,
y
-
=
C
y
y
,
t
-
=
(
C
y
)
2
/
(
2
-
n
)
t
,
u
-
=
u
,
u
-
w
=
1
(
C
y
)
n
/
(
2
-
n
)
u
w
,
v
-
=
1
(
C
y
)
n
/
(
2
-
n
)
v
,
θ
-
=
1
(
C
y
)
4
/
(
2
-
n
)
θ
.
This group converts invariantly the differential equations of (3) to (5) and the initial and boundary conditions (6a) and (6b).


### 3.2. The Complete Set of Absolute Invariants

The complete set of absolute invariants is as follows:(i)
the absolute invariants of the independent variables(
x
,
y
,
t
) are η
=
η
(
x
,
y
,
t
);
(ii)
the absolute invariants of the dependent variables(
u
,
v
,
u
w
,
θ
) are
(20)
g
j
(
x
,
y
,
t
:
u
,
u
w
,
v
,
θ
)
=
F
j
(
η
(
x
,
y
,
t
)
)
,
j
=
1,2
,
3,4
.The basic theorem in group theory asserts that a function g
j
(
x
,
y
,
t
:
u
,
u
w
,
v
,
θ
) is an absolute invariant of a two-parameter group if it satisfies the following two first-order linear differential equations:
(21)
∑
i
=
1
13
(
α
i
S
i
+
α
i
+
1
)
∂
g
∂
S
i
=
0
,
∑
i
=
1
13
(
β
i
S
i
+
β
i
+
1
)
∂
g
∂
S
i
=
0
,
where α
1
=
(
∂
C
x
/
∂
a
1
)
(
a
1
0
,
a
2
0
), α
2
=
(
∂
K
x
/
∂
a
1
)
(
a
1
0
,
a
2
0
), β
1
=
(
∂
C
x
/
∂
a
2
)
(
a
1
0
,
a
2
0
), β
2
=
(
∂
K
x
/
∂
a
2
)
(
a
1
0
,
a
2
0
),  and so forth;  (
a
1
0
,
a
2
0
) are the identity elements of the group.The absolute invariantη
(
x
,
y
,
t
) of the independent variables (
x
,
y
,
t
) is determined using (21):
(22)
(
α
1
x
+
α
2
)
∂
η
∂
x
+
α
3
y
∂
η
∂
y
+
α
5
t
∂
η
∂
t
=
0
,
(
β
1
x
+
β
2
)
∂
η
∂
x
+
β
3
y
∂
η
∂
y
+
β
5
t
∂
η
∂
t
=
0
,
where α
4
=
β
4
=
K
y
=
0 and α
6
=
β
6
=
K
t
=
0.Elimination ofy
(
∂
η
/
∂
y
) and ∂
η
/
∂
x from (22) yields(23a)
(
λ
31
x
+
λ
32
)
∂
η
∂
x
+
λ
35
t
∂
η
∂
t
=
0
,
(23b)
(
λ
31
x
+
λ
32
)
y
∂
η
∂
y
+
(
λ
15
x
+
λ
25
)
t
∂
η
∂
t
=
0
,where λ
i
j
=
α
i
β
j
-
α
j
β
i and i
,
j
=
1,2
,
3,4
,
5. Invoking (21) and definitions of α and β we get α
5
=
2
α
3 and β
5
=
2
β
3; thus λ
35
=
α
3
β
5
-
α
5
β
3 will assuage to zero. Using (23a) we obtain ∂
η
/
∂
x
=
0 indicating that η is dependent on y and t. Solving (23b), we get
(24)
η
=
y
π
(
t
)
,
where π
(
t
)
=
a
t
b and a and b are arbitrary constants.Simultaneously, the absolute invariants of the dependent variablesu
,
v
,
u
w, and θ are obtained from the group transformation (19):
(25)
g
1
(
x
,
y
,
t
;
u
)
=
u
(
η
)
,
g
2
(
x
,
y
,
t
;
u
)
=
θ
(
η
)
.
g
3
(
x
,
t
;
u
w
) is obtained using (21) as(26a)
(
α
1
x
+
α
2
)
∂
g
3
∂
x
+
(
α
5
t
+
α
6
)
∂
g
3
∂
t
+
(
α
11
u
w
+
α
12
)
∂
g
3
∂
u
w
=
0
,
(26b)
(
β
1
x
+
β
2
)
∂
g
3
∂
x
+
(
β
5
t
+
β
6
)
∂
g
3
∂
t
+
(
β
11
u
w
+
β
12
)
∂
g
3
∂
u
w
=
0
.Eliminating terms of ∂
g
3
/
∂
x and ∂
g
3
/
∂
t from ((26a) and (26b)) results in
(27)
g
3
(
x
,
t
;
u
w
)
=
ϕ
1
(
u
w
ω
(
x
,
t
)
)
=
E
(
η
)
.
Similarly, for g
4
(
x
,
t
;
v
),
(28)
g
4
(
x
,
t
;
v
)
=
ϕ
2
(
v
Γ
(
x
,
t
)
)
=
F
(
η
)
,
where ω
(
x
,
t
), Γ
(
x
,
t
), E
(
η
), and F
(
η
) are functions to be determined. Without loss of generality, the ϕ’s in (27) and (28) are selected to be identity functions; hence the functions u
w
(
x
,
t
) and v
(
x
,
y
,
t
) are expressed in terms E
(
η
) and F
(
η
) as
(29)
u
w
(
x
,
t
)
=
ω
(
x
,
t
)
E
(
η
)
,
(30)
v
(
x
,
y
,
t
)
=
Γ
(
x
,
t
)
F
(
η
)
.
Since ω
(
x
,
t
) and u
w
(
x
,
t
) are independent ofy, whereas η is function ofy, we conclude that E
(
η
) is equal to a constant such as E
0. Equation (29) yields
(31)
u
w
(
x
,
t
)
=
E
0
ω
(
x
,
t
)
.
Without loss of generality E
0 is equated to one. The functions of Γ
(
x
,
t
) and ω
(
x
,
t
) will be determined later on, such that the governing equations of (3) to (5) reduce to a set of ordinary differential equations in E
(
η
)
,
F
(
η
), and u
(
η
).


## 3.1. Group Formulation

The problem is solved by employing a two-parameter group transformation to the partial differential equations of (3) to (5). This transformation decreases the three independent variables (
x
;
y
;
t
) to one similarity variable; η
(
x
;
y
;
t
) and the governing equations of (3) to (5) are transformed to a system of ordinary differential equations in terms of the similarity variable η. This technique is based on a class of transformation, namely, G, including two parameters (a
1, a
2):
(7)
G
:
S
-
=
C
S
(
a
1
,
a
2
)
S
+
K
S
(
a
1
,
a
2
)
,
where S is representative for x
,
y
,
t
,
u
w
,
v, and θ. C
s and K
s are real valued and at least differentiable in their real arguments (a
1, a
2). Dependent variables and related differentiates are as follows via chain rule operations:
(8)
S
-
i
-
=
(
C
S
C
i
)
S
i
,
S
-
i
j
-
=
(
C
S
C
i
C
j
)
S
i
j
f
f
f
f
f
f
f
i
=
x
,
y
,
t
,
j
=
x
,
y
,
t
.
For instance, (3) is transformed as follows:
(9)
u
-
∂
u
-
w
∂
x
-
+
u
-
w
∂
u
-
∂
x
-
+
∂
v
-
∂
y
-
=
H
1
(
a
1
,
a
2
)
[
u
∂
u
w
∂
x
+
u
w
∂
u
∂
x
+
∂
v
∂
y
]
,
where H
1
(
a
1
,
a
2
) is constant. If (7) and (8) are put in (9),
(10)
[
C
u
C
u
w
C
x
]
u
∂
u
w
∂
x
+
[
C
u
C
u
w
C
x
]
u
w
∂
u
∂
x
+
[
C
v
C
y
]
∂
v
∂
y
+
R
1
=
H
1
(
a
1
,
a
2
)
[
u
∂
u
w
∂
x
+
u
w
∂
u
∂
x
+
∂
v
∂
y
]
and R
1 is [
C
u
w
K
u
/
C
x
]
(
∂
u
w
/
∂
x
)
+
[
C
u
K
u
w
/
C
x
]
(
∂
u
/
∂
x
). For invariant transformation, R
1 is equated to zero. This is satisfied by setting
(11)
K
u
=
K
u
w
=
0
.
Finally, we get
(12)
[
C
u
C
u
w
C
x
]
=
[
C
v
C
y
]
=
H
1
(
a
1
,
a
2
)
.
Simultaneously, after, (4) and (5) transformation:
(13)
K
u
=
K
v
=
K
u
w
=
K
θ
=
K
x
=
K
y
=
0
,
(14a)
[
C
u
C
u
w
C
t
]
=
[
(
C
u
)
2
(
C
u
w
)
2
C
x
]
=
[
C
u
C
v
C
u
w
C
y
]
=
(
C
u
w
)
n
(
C
u
)
n
(
C
y
)
n
+
1
=
C
θ
C
x
=
H
2
(
a
1
,
a
2
)
,
(14b)
[
C
θ
C
x
C
t
]
=
[
C
u
C
u
w
C
θ
]
=
[
C
x
C
v
C
θ
C
y
]
=
[
C
x
C
θ
(
C
y
)
2
]
=
H
3
(
a
1
,
a
2
)and we have
(15)
K
u
=
K
v
=
K
u
w
=
K
θ
=
0
.
Employing invariant transformation for boundary conditions results in
(16)
K
t
=
K
y
=
0
(17)
C
u
=
1
.
Invoking (12) to (15) and (17) reduces to
(18)
C
x
=
C
y
,
C
t
=
(
C
y
)
2
/
(
2
-
n
)
,
C
u
w
=
C
v
=
1
(
C
y
)
n
/
(
2
-
n
)
,
C
θ
=
1
(
C
y
)
4
/
(
2
-
n
)
.
Group G is summarized as follows:
(19)
G
=
{
G
s
{
x
-
=
C
y
x
,
y
-
=
C
y
y
,
t
-
=
(
C
y
)
2
/
(
2
-
n
)
t
,
u
-
=
u
,
u
-
w
=
1
(
C
y
)
n
/
(
2
-
n
)
u
w
,
v
-
=
1
(
C
y
)
n
/
(
2
-
n
)
v
,
θ
-
=
1
(
C
y
)
4
/
(
2
-
n
)
θ
.
This group converts invariantly the differential equations of (3) to (5) and the initial and boundary conditions (6a) and (6b).


## 3.2. The Complete Set of Absolute Invariants

The complete set of absolute invariants is as follows:(i)
the absolute invariants of the independent variables(
x
,
y
,
t
) are η
=
η
(
x
,
y
,
t
);
(ii)
the absolute invariants of the dependent variables(
u
,
v
,
u
w
,
θ
) are
(20)
g
j
(
x
,
y
,
t
:
u
,
u
w
,
v
,
θ
)
=
F
j
(
η
(
x
,
y
,
t
)
)
,
j
=
1,2
,
3,4
.The basic theorem in group theory asserts that a function g
j
(
x
,
y
,
t
:
u
,
u
w
,
v
,
θ
) is an absolute invariant of a two-parameter group if it satisfies the following two first-order linear differential equations:
(21)
∑
i
=
1
13
(
α
i
S
i
+
α
i
+
1
)
∂
g
∂
S
i
=
0
,
∑
i
=
1
13
(
β
i
S
i
+
β
i
+
1
)
∂
g
∂
S
i
=
0
,
where α
1
=
(
∂
C
x
/
∂
a
1
)
(
a
1
0
,
a
2
0
), α
2
=
(
∂
K
x
/
∂
a
1
)
(
a
1
0
,
a
2
0
), β
1
=
(
∂
C
x
/
∂
a
2
)
(
a
1
0
,
a
2
0
), β
2
=
(
∂
K
x
/
∂
a
2
)
(
a
1
0
,
a
2
0
),  and so forth;  (
a
1
0
,
a
2
0
) are the identity elements of the group.The absolute invariantη
(
x
,
y
,
t
) of the independent variables (
x
,
y
,
t
) is determined using (21):
(22)
(
α
1
x
+
α
2
)
∂
η
∂
x
+
α
3
y
∂
η
∂
y
+
α
5
t
∂
η
∂
t
=
0
,
(
β
1
x
+
β
2
)
∂
η
∂
x
+
β
3
y
∂
η
∂
y
+
β
5
t
∂
η
∂
t
=
0
,
where α
4
=
β
4
=
K
y
=
0 and α
6
=
β
6
=
K
t
=
0.Elimination ofy
(
∂
η
/
∂
y
) and ∂
η
/
∂
x from (22) yields(23a)
(
λ
31
x
+
λ
32
)
∂
η
∂
x
+
λ
35
t
∂
η
∂
t
=
0
,
(23b)
(
λ
31
x
+
λ
32
)
y
∂
η
∂
y
+
(
λ
15
x
+
λ
25
)
t
∂
η
∂
t
=
0
,where λ
i
j
=
α
i
β
j
-
α
j
β
i and i
,
j
=
1,2
,
3,4
,
5. Invoking (21) and definitions of α and β we get α
5
=
2
α
3 and β
5
=
2
β
3; thus λ
35
=
α
3
β
5
-
α
5
β
3 will assuage to zero. Using (23a) we obtain ∂
η
/
∂
x
=
0 indicating that η is dependent on y and t. Solving (23b), we get
(24)
η
=
y
π
(
t
)
,
where π
(
t
)
=
a
t
b and a and b are arbitrary constants.Simultaneously, the absolute invariants of the dependent variablesu
,
v
,
u
w, and θ are obtained from the group transformation (19):
(25)
g
1
(
x
,
y
,
t
;
u
)
=
u
(
η
)
,
g
2
(
x
,
y
,
t
;
u
)
=
θ
(
η
)
.
g
3
(
x
,
t
;
u
w
) is obtained using (21) as(26a)
(
α
1
x
+
α
2
)
∂
g
3
∂
x
+
(
α
5
t
+
α
6
)
∂
g
3
∂
t
+
(
α
11
u
w
+
α
12
)
∂
g
3
∂
u
w
=
0
,
(26b)
(
β
1
x
+
β
2
)
∂
g
3
∂
x
+
(
β
5
t
+
β
6
)
∂
g
3
∂
t
+
(
β
11
u
w
+
β
12
)
∂
g
3
∂
u
w
=
0
.Eliminating terms of ∂
g
3
/
∂
x and ∂
g
3
/
∂
t from ((26a) and (26b)) results in
(27)
g
3
(
x
,
t
;
u
w
)
=
ϕ
1
(
u
w
ω
(
x
,
t
)
)
=
E
(
η
)
.
Similarly, for g
4
(
x
,
t
;
v
),
(28)
g
4
(
x
,
t
;
v
)
=
ϕ
2
(
v
Γ
(
x
,
t
)
)
=
F
(
η
)
,
where ω
(
x
,
t
), Γ
(
x
,
t
), E
(
η
), and F
(
η
) are functions to be determined. Without loss of generality, the ϕ’s in (27) and (28) are selected to be identity functions; hence the functions u
w
(
x
,
t
) and v
(
x
,
y
,
t
) are expressed in terms E
(
η
) and F
(
η
) as
(29)
u
w
(
x
,
t
)
=
ω
(
x
,
t
)
E
(
η
)
,
(30)
v
(
x
,
y
,
t
)
=
Γ
(
x
,
t
)
F
(
η
)
.
Since ω
(
x
,
t
) and u
w
(
x
,
t
) are independent ofy, whereas η is function ofy, we conclude that E
(
η
) is equal to a constant such as E
0. Equation (29) yields
(31)
u
w
(
x
,
t
)
=
E
0
ω
(
x
,
t
)
.
Without loss of generality E
0 is equated to one. The functions of Γ
(
x
,
t
) and ω
(
x
,
t
) will be determined later on, such that the governing equations of (3) to (5) reduce to a set of ordinary differential equations in E
(
η
)
,
F
(
η
), and u
(
η
).


## 4. Reduction to a Set of Ordinary Problem

Now we defineη in general form of η
=
y
π
(
x
,
t
). Invoking (30) and (31) and (3) to (5),
(32)
d
F
d
η
+
C
1
η
d
u
d
η
+
C
2
u
=
0
,
n
(
d
u
d
η
)
n
-
1
d
2
u
d
η
2
-
[
η
(
C
3
+
C
4
u
)
+
C
5
F
]
d
u
d
η
-
C
6
u
-
C
7
u
2
+
C
8
θ
=
0
,
d
2
θ
d
η
2
-
Pr
[
d
θ
d
η
{
η
(
C
10
+
C
11
u
)
+
C
12
F
}
-
C
9
u
θ
]
=
0
.
Constants of C
1 to C
12 are defined as follows:
(33)
C
1
=
ω
Γ
π
2
∂
π
∂
x
C
2
=
1
Γ
π
∂
ω
∂
x
C
3
=
1
ϑ
ω
n
-
1
π
n
+
2
∂
π
∂
t
C
4
=
1
ϑ
ω
n
-
2
π
n
+
2
∂
π
∂
x
C
5
=
Γ
ϑ
ω
n
-
1
π
n
C
6
=
1
ϑ
ω
n
π
n
+
1
∂
ω
∂
t
C
7
=
1
ϑ
ω
n
-
1
π
n
+
1
∂
ω
∂
x
C
8
=
x
ϑ
ω
n
π
n
+
1
g
β
q
w
k
Re
x
C
9
=
ω
ϑ
x
π
2
C
10
=
1
ϑ
π
3
∂
π
∂
t
C
11
=
ω
ϑ
π
3
∂
π
∂
x
C
12
=
Γ
ϑ
π
.
Now we obtain the exact value of constants as
(34)
C
7
C
8
=
1
⟶
ω
(
x
)
=
(
g
β
q
w
k
Re
x
)
1
/
2
x
.
From (31),
(35)
u
w
(
x
)
=
(
g
β
q
w
k
Re
x
)
1
/
2
x
.
For C
9
=
1, we have
(36)
π
=
1
ϑ
1
/
2
(
g
β
q
w
k
Re
x
)
1
/
4
.
According to the definition of similarity variable,
(37)
η
=
y
(
g
β
q
w
ϑ
2
k
Re
x
)
1
/
4
.
For C
2
=
1, we get
(38)
Γ
=
ϑ
1
/
2
(
g
β
q
w
k
Re
x
)
1
/
4
.
Using (30), the horizontal component of velocity is
(39)
v
(
η
)
=
ϑ
1
/
2
(
g
β
q
w
k
Re
x
)
1
/
4
F
(
η
)
.
As π is invariant and ω is independent of time,
(40)
C
1
=
C
3
=
C
4
=
C
6
=
C
10
=
C
11
=
0
.
On the other hand, due to definitions of π
,
ω
,
and
Γ,
(41)
C
12
=
1
,
C
5
=
C
7
=
C
8
=
(
ω
π
)
1
-
n
.
Substituting the above constants in (32), they finally reduce to
(42)
d
F
d
η
+
u
=
0
,
(43)
n
(
ω
π
)
n
-
1
(
d
u
d
η
)
n
-
1
d
2
u
d
η
2
-
F
d
u
d
η
-
u
2
+
θ
=
0
,
(44)
d
2
θ
d
η
2
-
Pr
(
d
θ
d
η
F
+
u
θ
)
=
0
.
The new forms of boundary conditions are
(45)
F
(
0
)
=
0
,
u
(
0
)
=
1
,
u
(
∞
)
=
0
,
∂
θ
(
0
)
∂
η
=
-
1
,
θ
(
∞
)
=
0
.
It can be proved that ∂
θ
(
0
)
/
∂
η
=
-
1. According to (2b) and definition of θ, (2b) will change into
(46)
∂
θ
(
x
,
0
,
t
)
∂
y
=
-
Re
x
x
,
where Re
x
=
u
s
x
/
ϑ and u
s is average velocity iny direction. Assuming u
s
=
u
w,
(47)
Re
x
=
x
2
π
2
.
Using chain rule operation,
(48)
∂
θ
(
0
)
∂
η
∂
η
∂
y
=
-
Re
x
x
.
So, according to Re
x and η, ∂
θ
(
0
)
/
∂
η will be −1.For instance, assumingn
=
1 for a case of Newtonian fluid flow, (43) will reduce to
(49)
d
2
u
d
η
2
+
η
2
d
u
d
η
-
u
=
-
erfc
⁡
(
Pr
2
η
)
,
where
(50)
u
(
0
)
=
1
,
u
(
∞
)
=
0
.


## 5. Results and Discussion

In this section, initially, results of Newtonian fluid flow over stretching sheet are compared with Kassem [34]. Table 1 shows the results obtained by numerical solution of ODE equation, (49), and [34] for power index, n equal to unity and Pr number of 1. There is a satisfying agreement among the present evaluations. Complementary results for different values of Pr are shown in Figures 2 and 3. It can be understood that, as the fluid momentum diffusivity increases with respect to the thermal diffusivity, large Pr numbers, vertical velocity decreases. In other words, the velocity u
(
η
) inside the boundary layer decreases with the increase of fluid viscosity. Also, the temperature plot of boundary layer reveals a decrease of temperature at the wall temperature for larger Pr. So heat loss increases for larger Pr as the boundary layer gets thinner.Table 1
Comparison of present problem solution and case of Kassem [34] for Pr = 1, 2, 5, and 10 of Newtonian fluid.
Pr
Kassem [34]
Present work
η
u (η)
Θ (η)
u (η)
Θ (η)
1
0.5
0.669
0.507
0.667
0.510
1
0.391
0.256
0.381
0.261
1.5
0.165
0.103
0.161
0.105
2
0.000
0.003
0.003
0.005
2
0.3
0.736
0.398
0.742
0.401
0.6
0.515
0.222
0.520
0.225
0.9
0.315
0.092
0.317
0.095
1.2
0.143
0.009
0.150
0.011
5
0.25
0.745
0.206
0.750
0.208
0.5
0.525
0.095
0.531
0.011
0.75
0.333
0.040
0.335
0.042
1
0.171
0.017
0.171
0.017
10
0.2
0.768
0.124
0.771
0.125
0.4
0.565
0.046
0.571
0.048
0.6
0.381
0.015
0.391
0.017
0.8
0.216
0.007
0.218
0.008Figure 2
Vertical velocity of Newtonian fluid at the vicinity of stretching sheet for different values of Pr.Figure 3
Temperature of Newtonian fluid at the vicinity of stretching sheet for different values of Pr.For non-Newtonian case of the fluid, the vertical velocity and temperature of fluid adjacent to the stretching sheet are evaluated by solving (42) to (44) with the certain boundary conditions of (45). Results of the shooting method solution are illustrated in Figures 4 and 5 for Pr
=
5. As the power index increases, vertical velocity component encounters with resistance to rise due to increased value of shear stress. But, for lower power index, vertical component of velocity moves faster due to reduced apparent viscosity. Similarly, thinner thermal boundary layer for higher power index will increase the amount of heat loss.Figure 4
Vertical velocity of non-Newtonian fluid at the vicinity of stretching sheet for constant values ofPr
=
5.Figure 5
Temperature of non-Newtonian fluid at the vicinity of stretching sheet for constant values ofPr
=
5.


## 6. Conclusion

Similarity solutions are performed for flow of power law non-Newtonian fluid film on unsteady stretching surface subjected to constant heat flux. The nonlinear coupled partial differential equations (PDE) governing the flow and the boundary conditions are transformed to a set of ordinary differential equations (ODE) using two-parameter groups. This technique reduces the number of independent variables by two, and conclusively the obtained ordinary differential equations are solved numerically for the temperature and velocity using the shooting method. Results show that higher Pr number and higher power index of non-Newtonian fluid encounter with difficulty to move as faster as lower Pr and power index. Enhanced amount of shear stress explains the reason of the predicted flow.



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*Source: 718319-2014-05-11.xml*