The research paper published by IJSER journal is about Effects of variable viscosity on Three-Dimensional Boundary Layer Flow of Non-Newtonian Fluids over a Stretching Surface with Mass and Heat Transfer 1

ISSN 2229-5518

Effects of variable viscosity on Three- Dimensional Boundary Layer Flow of Non-

Newtonian Fluids over a Stretching Surface with

Mass and Heat Transfer

P. K. Mahanta, G. C. Hazarika

AbstractThree dimensional flow of non-Newtonian viscoelastic fluid with the variation in the viscosity over a stretching surface is investigated. The governing partial differential equations of continuity, momemtum, energy and concentration are transformed into non - linear ordinary differential equations by using similarity transformations. The transformed equations are solved numerically by fourth-order Runge-Kutta shooting method. The effects of viscosity, heat and the stretching ratio parameters with Prandtle and Schmidt numbers on the velocity, temperature and concentration distributions have been discussed and illustrated graphically.

Index TermsBoundary layer, heat transfer, non-newtonian , stretching surface, three-dimensional flow, thermal conductivity , Variable viscosity, Visco-elastic fluid.

—————————— ——————————

1 INTRODUCTION

HE study of flow over a stretching surface has generated much interested in recent years in view of its numerous industrial applications such as extrusion of polymer sheets, continuous casting, glass blowing, rolling and manu- facturing plastic flims and artificial fibers. Sakiadis [1] was probably the first to study the two-dimensional boundary lay- er flow due to a stretching surface in a fluid at rest. An exten- sion of the problem to the case of suction or injection at the surface was investigated by Ericson et al [2]. Crane [3] and Ali [4] carried out a study for a staeching surface subject to suc- tion or injection for uniform and variable surface tempera- tures. Chakrabarti and Gupta [5] studied the temperature dis- tribution in this MHD boundary layer flow over a stretching sheet in the presence of suction. There are several extensions to this problem, which include consideration of more general
stretching velocity and the heat transfer.
The heat transfer in flow over a stretching surface was in-
vestigated by Gupta and Gupta et al. [6], where the surfaces held at constant temperature and is subject to suction and blowing.
Interest of researchers in the flows of non-Newtonian fluids in the presence of heat transfer have relevance in food enggi- neering, petroleum production,
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 P. K. Mahanta is currently working as Associate professor in Math e- matics Department in Namrup College, Dibrugarh -786623, Assam, (INDIA). E-mail: mahantapk@yahoo.co.in

 G. C. Hazarika is currently working as Professor in Mathematics De- partment in Dibrugarh University, Dibrugarh -786004, Assam, INDIA. E-mail: gchazarika@gmail.com

power engineering and in industrial processes including pol- ymer melt and polymer solutions used in the plastic pro- cessing industries. Some recent studies of the topic can be seen in Labropulu et al [7], Sahoo [8], Mustafa et al [9], Hayet et al [10]. It is noted that most studies in the literature discussed the two-dimensional boundary layer flows.
The problem of three-dimensional boundary layer flow of a viscoelastic fluid due to a stretching surface, has been consid- ered before as Hayet et al [11] without mass and heat transfer. Fox et al [12] used both exat and approximate methods to ex- amine the boundary layer flow of a viscoelastic fluid charac- terized by a power law model. Vajravelu and Rollins [13] in- vestigated the heat transfer of the boundary layer flow of se- cond grade fluid. Mahantesh et al [14] discussed the flow and heat transfer charecteristics of a viscoelastic fluid in a porous medium over an impermeable stretching sheet with viscous dissipation.
In most of the studies of this type of problems, the viscosity and thermal conductivity of the ambient fluid were assumed to be constant. However, Hussanien et al [15] revealed that the fluid viscosity and thermal conductivity might function of temperatures. It is known that these physical properties can change significantly with temperature and when the effects of variable viscosity and thermal conductivity are taken in to account, the flow charecteristics are significantly changed compered to the constant property case.
In this paper, we investigate the steady three dimensional
laminar boundary layer flow of viscous incompressible second grade fluid over a stretching surface, when the viscosity and the thermal conductivity are function of temperature. By em-

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The research paper published by IJSER journal is about Effects of variable viscosity on Three-Dimensional Boundary Layer Flow of Non-Newtonian Fluids over a Stretching Surface with Mass and Heat Transfer 2

ISSN 2229-5518

ploying the similarity transformation, the boundary layer equations governing the flow are reduced to ordinary differ- ential equations and solved numerically using fourth-order Runga-Kutta shooting method.

u C

x

v C

y

w C

z

2C

D

z 2

1 C C

(5)

2 MATHEMATICAL FORMULATION

Where the velocity components in the

x,

y  and z  direc-

We consider the steady three dimensional flow of a viscous incompressible second grade fluid bounded by a stretching
tions are denoted by u,

v and w respectivly, is the fluid
surface. Under the usual boundary layer approximations the
flow is governed by the following equations:

density,  is the Kinemetic viscosity, is the viscosity,

k0 is the material parameter, C p

is the specific heat at constant
The equation of continuity

u  v  w  0

pressure, k is the thermal conductivity, T is temperature of the fluid flow, C is the mass concentration of the species of the flow, Q is the volumetric rate of heat genera-


x y z

(1)
tion/absorption, D is the molecular diffusion coefficient, T
and Care the fluid temperature and concentration far away
The equation of momentum:
and 1 is the reaction rate coefficient.
In most of the studies, of this type of problems, the viscosi- ty and thermal conductivity of the fluid were assumed to be

u u v u w u  1

  u

constant. However, it is known that physical properties can



x y

z z

z

change significantly with temperature and when the effects of

 

  u 2u

u 2 w 

variable viscosity and thermal conductivity are taken in to
account, the flow characteristics are substantially changed

 3u

3u


 x z 2  z

z 2 

compared to the constant property case. Hence in the problem

     

k0 u 2

 xz

w

z3

 u

 2

2u

w 2u 

 2 

under consideration, the viscosity and thermal conductivity have been assumed to be inverse linear functions of tempera-

  z xz

z z 2 

ture. We assume,
1  1 1  T T  or

1  b T T

(6)
(2)

 

1 e

1

u v v v w v  1

  v

Where

b1

and

Te T




x y

z z

z 1




  1  1 1  T T  or

c T T

(7)

  v 2v

 

v 2 w 

 

k k

k 1 r

 3v

v

 y z 2

z z 2  1

k0 v yz 2 w z3  

v 2v

w 2v 

1 k r

Where

c  and

T T

  2

 2 

Where

b , c ,

T and

T are constants and their values de-

 

z yz

z z 2 

1 1 e r

The energy equation:
(3)
pend on the reference state and thermal properties of the fluid
i.e. and . In general b1 > 0, for liquids and b1 < 0 for gases
( the viscosity and thermal conductivity of liquid/gas usually
decrease/increase with increasing temperature).
The appropriate boundary conditions for the present problem

C u T

 x

v T

y

T

w z   k

2T

z 2

Q T T

are given by:

u uw xax,

v vw y by,

w  0

at z  0
And the concentration equation:
(4)
( a , b are positive constants) (8)

T T,

Tw T

C C

Cw C

as z 
and

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The research paper published by IJSER journal is about Effects of variable viscosity on Three-Dimensional Boundary Layer Flow of Non-Newtonian Fluids over a Stretching Surface with Mass and Heat Transfer 3

ISSN 2229-5518

the governing boundary layer three dimensional equation

f  0,

g  0,

f   1,

g  c,

 1,

 1

(16)
with temperature dependent heat generation (absorbtion) as above in equation (4) , the thermal boundary conditions, de-
As 
pend on the type of heating process under considerations, are considered by

f   0,

g  0,

f   0 ,

g  0 ,

 0,

 0

(17)

2 2

T Tw

T

Ax
l
T

B y  ;

l

Where the dimensionless parameters are defined as:
   

2 2

C p

C Cw CAx

CBy
at z  0

Pr 

(Prandtl number)

T T

as z 
(9)

S

c D

Q

(Schmidt number)


Where A , B , A and B are two constants and l is the characteristic length.
In order to reduce the partial differential equations to ordi- nary differential equations, we use the following transfor- mations in this study:
And

 (Heat source/Sink parameter)

aC p

1 (Chemistry reaction parameter)

a

K k0a is the dimensionless viscoelastic parameter,

u axf ,

v ayg

c b

is the the dimensionless stretching ratio. Also
and

w  

af g

(10)

a

Te T

Where

e

Tw

T
is the parameter characterizing the influence of

 a z

(11)
viscosity.
When c  0 , the problem reduces to the two-dimensional case
Here f and g are the dimensionless stream functions, is the similarity variable and prime denotes the differentiation with respect to . Using (10) and (11) , the incompressibility condition (1) is identically satisfied and Equations (2) to (5)
take the dimensionless form as:
( g  0 ), given by

f   f 2 ff   

f   K ff iv f 2  2 f f   0

f   f 2 f g f      f

   

e

(18)

 
e

K f g f iv f   g f   2f   gf   0

(12)
  Pr f   2 Pr f    0
  Sc f   2Sc f    0
(19)
(20)
When

c  1 , the problem reduces to the axi-symmetric flow,

g   g2 f g g      g
where we have ( f g ), the equation becomes

 
e

K f g g iv g   f g   2f   gg   0

f   f 2 ff   

f   2K ff iv  2 f f   0

(21)
  Pr f g   2 Pr f   g 0
  Sc f g   2Sc f   g 0
The boundary conditions (8) and (9) becomes
(13)
(14) (15)

 
e
  2 Pr f   4 Pr f    0
  2Sc f   4Sc f    0
The boundary conditions (16) and (17) becomes
(22) (23)
at  0

f 0 0,

f  0,
g 0,
f 0 0,
g 0,

 0,

0 1,

f  0,

 0

0 1,

(24)
(25)

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The parameters of engineering interest for the present problem
obtained by the study of the temperature and mass concentra-
tion distributions.
are the local skin friction coefficients,

C fx

and

C fy along the

The dimensionless temperature

and dimensionless

x  direction and y  directions respectively, which are de-

mass concentration

have been plotted against the di-
fined as :
mension for several sets of the values of the parameters K ,
C wx ,
C wy ,
(26)
, c , , Pr , Sc
and e .

fx u 2

fy u 2

In Fig 1 we are observing the effect of concentration profile
with the variation of Schmidt number Sc . The values of
Where

wx

is the wall shear stress along the x  direction and

S = 1.00, 2.25, 2.75, 4.50, 5.50 with the values of other pa-

wy

is the wall shear stress along the y  direction. Using

c

rameters Pr = 3.70, K = 2.00, = 0.50, = 2.00, c = 0.25 and
equations (10), we obtain the wall skin friction coefficient in

x  and y  directions respectively as follows:

the thermal conductivity parameter

e =-10. A rise in Sc

C fx

Where

Re 2 f

0
and

C R  2 vw g0

w

(27)
strongly suppreses concentration levels in the boundary layer
regime. All profiles decay monotonically from the surface (wall) to the free stream. Sc embodies the ratio of momentum diffusivity () to molecular diffusivity( D ). It is observed that
the fluids concentration decreases as the mass transfer param-
 u  ,
 v
(28)

wx   

wy   

eters Sc
increases.
and
z z 0
z z 0

R uw x


is the local Reynolds number.

e

To assess the heat transfer ability of the medium the local
Nusselt number and the local heat transfer rate are defined as:

xqw 1

Nu

  R 2 0

k T T e

(29)
Where

q  k  T

(30)

w   

z z 0
Fig 1: Effect of Schmidt number

Sc on concentration.

3 RESULTS AND DISCUSSION

In Fig 2 we study the effect of

e the variable viscosity prame-

The system of differential equations (12) to (15) governed by boundary conditions (16) and (17) are solved numerically by applying an efficient numerical technique based on the fourth order Runge-Kutta shooting method and an iterative method. It is experienced that the convergence of the iteration process is quite rapid.
The purpose of this study is to bring out the effects of the
variable viscosity on the governing flow with the combina- tions of the other flow parameters. The three dimensional flow of the present problem is governed by seven parameters, namely K , the viscoelastic parameter, the heat parameter,
the chemical reaction parameter, c the dimensionless
ter on concentration profile. The values of e = -15, -12, -10 has been considered and the other parameters are taken as Pr = 3.70, K = 2.00, = 0.50, = 2.00, c = 0.25 and Sc = 2.50. It is observed that the concentration profile decreases with the increase of the variable viscosity parameter e .
stretching ratio, Pr the Prandtl number, Sc
the Schmidt num-
ber and

e the dimensionless viscosity parameter. An insight

into the effects of these parameters of the flow field can be

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Fig 2: Effect of variable viscosity e on concentration.
Fig 4: Variation of concentration profile with Pr.

In Fig 5, the temperature profile for various values of Pr has been studied. It is observed that the temperature profile de- creases with the increase of the Prandtl number.
Fig 3: Effect of viscoelastic parameter K on concentration.
Fig 3 illustrates the effects of the viscoelastic parameter K on the concentration profile. Substituting various values of

K =2.00, 3.00, 3.50 at Pr = 3.70, Sc

= 2.50, = 0.50, = 2.00,
c = 0.50 and e =-10, it is observed that the concentration pro-
file increases with the increase of viscoelastic parameter K .
In Fig 4, it has been investigated that the concentration profile increases with the change of Prandtl number Pr. The study revels that concentration increases with the increase of Pr.
Fig 5: Variation of temperature profile with Pr.
Figures 6 and 7 exhibits the effects of heat parameter on temperature distribution. It is evident from the Fig 6, the tem-
perature distribution

increases with an increase in the

heat parameter +ve and the inverse is true seing in Fig 7.

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Fig 8: Concertration distribution for varies values of the dimensionless stretching ratio c .
In the Fig 9 we observe the effect of chemical reaction parame- ter on concentration profile . Substituting various values
Fig 6: Temperature distribution for varies values of the
for Pr = 3.70, K = 2.00, = 0.50,

Sc = 2.50, c = 0.25 and the

heat parameter (positive).
thermal conductivity parameter

e =-10, it is observed that the


Fig 7: Temperature distribution for varies values of the heat parameter (negative).
concentration profile increases with the increase of reaction parameter of the fluid flow.

Fig 9: Concentration distribution for varies values of the Chemical reaction parameter .
In the Fig 8, the effect of dimensionless stretching ratio c on concentration profile has been studied. It is observed that the

Missing values of f  0, 0

and 0for various val-

concentration profile decreases with the increase of starching
parameter.
ues of e , K , , and c have been derived. In Table I, miss-
ing values of

f  0, 0and 0were found for e = -

15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4. It is observed that
the missing values of

f  0decreases while that of 0

and 0increases. In Table II we observed the missing val-

ues of

f  0, 0and 0for K = 2, 2.5, 3, 3.5, 4, 4.5, 5,

5.5, 6. The study revels that the missing values of

f  0

in-

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ISSN 2229-5518

creases while that of 0and 0decreases.

In Table III it is observed that for increasing values of c = 0,

0.25, 0.50, 0.75, 1.00, 1.25 the missing values of creases while that of 0and 0decrease.

f  0

in-
Table I: Missing values of ious values of e

f  0, 0and 0for var-

4 CONCLUSION

A numerical study of the effect of variable viscosity on bound- ary layer flow of second order fluids over a stretching surface with mass and heat transfer has been offered.
The subsequent outcome may be drawn as:
1. The temperature profile within the boundary flim rises for the decreasing values of Prandtl number Pr .
2. The concentration profile within the boundary flim raises considerable for the increasing values of Pr , K , and decreases for e .
3. The temperature profile within the boundary flim raises considerable for the increasing values of heat (Source/sink) parameter .
Table II: Missing values of various values of K .

f  0,

0

and

0

for

5 Acknowledgments

Dr Mahanta wants to thank the financial support re- ceived in the form of MRP grant from the UGC (NERO), Gu- wahati, India (Ref No: F-5-106/2010-2011/MRP(NERO)/5731
Dated 16 MAR 2011).

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ISSN 2229-5518

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(1)

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