30120130405018

International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –
6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 5, September - October (2013) © IAEME
156
MHD BOUNDARY LAYER FLOW OF A MAXWELL FLUID PAST
A POROUS STRETCHING SHEET IN PRESENCE OF VISCOUS
DISSIPATION
Anand H. Agadi1*
, M. Subhas Abel2
, Jagadish V. Tawade3
and Ishwar Maharudrappa4
1*
Department of Mathematics, Basaveshwar Engineering College, Bagalkot-587102,INDIA
2
Department of Mathematics, Gulbarga University, Gulbarga- 585 106, INDIA
3
Department of Mathematics, Bheemanna Khandre Institute of Technology,Bhalki-585328
4
Department of Mathematics, Basaveshwar Engineering College, Bagalkot-587102.
ABSTRACT
Present study deals flow of MHD boundary layer of a Maxwell fluid over stretching sheet
with non-uniform heat source in porous medium. The effects of various values of the emerging
dimensionless parameters are discussed in two different cases namely, (i) a surface with prescribed
wall temperature (PST) (ii) a surface with prescribed wall heat flux (PHF). The partial differential
equations governing the momentum and heat transfer are converted in to ordinary differential
equations by suitable similarity transformations. Numerical solutions for boundary value problems
carried out by shooting technique with fourth order Runge-Kutta scheme. The results so obtained are
presented in the form of graphs for different non-dimensional parameters for both PST and PHF
cases and discussed.
Key words: Boundary layer, MHD, Maxwell fluid, porous stretching sheet, viscous dissipation.
1. INTRODUCTION
The study of flow induced by a stretching surface has scientific and engineering applications
such as aerodynamic extrusion of plastic sheets and fibers, drawing-annealing- tinning of copper
wire, paper production, crystal growing and glass blowing. Such applications involve cooling of a
molten liquid by drawing it into a cooling system. In drawing the liquid into the cooling system it is
sometimes stretched as in the case of polymer extrusion process. The fluid mechanical properties
desired for the penultimate outcome of such a process depend mainly on two things one being the
rate of cooling and other being the rate of stretching. The choice of an appropriate cooling liquid is
crucial as it has a direct impact on rate of cooling and care must be taken to exercise optimum
INTERNATIONAL JOURNAL OF MECHANICAL ENGINEERING
AND TECHNOLOGY (IJMET)
ISSN 0976 – 6340 (Print)
ISSN 0976 – 6359 (Online)
Volume 4, Issue 5, September - October (2013), pp. 156-163
© IAEME: www.iaeme.com/ijmet.asp
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157
stretching rate otherwise sudden stretching may spoil the properties desired for the final outcome.
These two aspects demand for a thorough understanding of flow and heat transfer characteristics
which is the main theme of the present investigation.
With the stand point of many applications akin to polymer extrusion process Crane [1]
initiated the analytical study of boundary layer flow due to a stretching sheet. He assumed the
velocity of the sheet to be a linear function of the distance from the slit. The solution so obtained by
Crane for the flow driven by a stretching sheet belongs to an important class of exact solutions of
Navier-Stokes equations and the uniqueness of the solution is well-established. The existence and
uniqueness of solution for the flow caused by a stretching sheet is addressed by many authors (see
McLeod and Rajagopal [2] and Troy et al [3]). The analytical study of McLeod and Rajagopal [2]
throws light on the specification of infinity for solving non-linear differential equations in case of
unbounded domains. Of late the work of Crane was extended by many authors to both Newtonian
and non-Newtonian boundary layer flow subjected to various physical situations. Gupta and
Gupta[4] investigated heat transfer from an isothermal stretching sheet with suction/blowing effects.
Chan and Char [5] extended the works of Gupta and Gupta to that of a non-isothermal stretching
sheet. Grubka and Bobba [6] carried out heat transfer studies by considering the power law variation
of surface temperature. Chiam [7] investigated the MHD heat transfer from a non-isothermal
stretching sheet.
Heat source/sink effect is an important factor that requires attention as it exerts strong
influence on the heat transfer characteristics in such an exothermic process. Many of the authors
have studied heat transfer by considering a uniform heat source/sink or a temperature dependent heat
source/sink (see Vajravelu and Rollins [8], Vajravelu and Hadjinicolaou [9]). Eldahab and El-Aziz
[10] included the effect of non-uniform heat source/sink (space and temperature dependent heat
source/sink) on the heat transfer.
A non-Newtonian second grade fluid does not give meaningful results for highly elastic
fluids (polymer melts) which occur at high Deborah numbers (Refs. Hayat et.al [11], [12]).Therefore,
the significance of the results reported in the above works are limited, at least as far as polymer
industry is concerned. Obviously, for the theoretical results to become of any industrial significance,
more realistic viscoelastic fluid models such as Upper-Convected Maxwell model or Oldroyd-B
model should be invoked in the analysis. Indeed, these two fluid models have recently been used to
study the flow of viscoelastic fluids above stretching and non-stretching sheets but with no heat
transfer effects involved Hayat et al. [11] and Sadeghy et. al [13].
Sadeghy et.al ([13], [14]), Alizadeh-Pahlavan et. al [15], Renardy [16], Rao and
Rajagopal[17], Aliakbar et.al[18] have done the work related to UCM fluid by using HAM- method
and by using numerical methods with no heat transfer. But the effect of thermal conductivity and
non-uniform heat source/sink is very important and cannot be ignored. This fact motivates us to
propose the effect of thermal conductivity and non-uniform heat source/sink on the heat transfer
characteristics of a boundary layer flow of a Maxwell fluid over the stretching sheet in the present
paper. The various effects of different parameters such as Elastic parameter, MHD parameter, porous
parameter, Prandtl number and non-uniform heat source/sink parameter are discussed and shown
with the aid of graphs.
2. MATHEMATICAL FORMULATION
The boundary layer equations can be derived for any viscoelastic fluid starting from Cauchy
equations of motion. For steady two-dimensional flows, these equations governing transport of heat
and momentum can be written as (Refs. Sadeghy et.al[7], Alizadeh-Pahlavan and Sadeghy[11]).

158
0,
u v
x y
∂ ∂
+ =
∂ ∂
(1)
2 2 2 2
2 2
2 2 2
2 ,
'
u u u u u u
u v u v uv u
x y x y x y y k
ν
λ υ
 ∂ ∂ ∂ ∂ ∂ ∂
+ + + + = − ∂ ∂ ∂ ∂ ∂ ∂ ∂ 
(2)
22
2
.
p p
T T k T u
u v
x y C y C y
µ
ρ ρ
 ∂ ∂ ∂ ∂
+ = +  
∂ ∂ ∂ ∂  (3)
where u and v are the velocity components along x and y directions respectively, t is the
temperature of the fluid, σ is the density, υ is the kinematic viscosity, k′ is the porosity parameter,
Cp is the specific heat at constant pressure, k is the thermal conductivity of the liquid far away from
the sheet, 0B , is the strength of the magnetic field, υ is the kinematic viscosity of the fluid andλ is
the relaxation time Parameter of the fluid.
The boundary conditions applicable to the flow problem are
2
2
, 0
0
0, 0, ,
w
w
y
x
u bx v T T T A PST Case
l
T x
K Q D PHF Case at y
y l
u u T T as y
∞
∞
 
= = = = +  
 
∂  
− = = = 
∂  
→ → → → ∞
(4)
Where A and D are constants, b is the constant known as the stretching rate, l the
characteristic length, Tw is the wall temperature and T∞ constant temperature far away from the
sheet.
In order to obtain dimensionless form of the solution we define following variables
y
b
Wherefbvfxbu
γ
ηηγηη =−== ),(),(
2
2
( ) , w
w
T T x
where T T A PST Case
T T l
x
D PHF Case
l
θ η ∞
∞
∞
−  
= − =  
−  
 
=  
 
(5)
where subscript η denotes the derivative with respect to η. Clearly u & v satisfy the equation (1)
identically. Substituting these new variables in equation (2) and (3), we have,
( ) ( )
2
22 0,f f f ff f ff k fβ′′′ ′ ′′ ′ ′′ ′′′ ′− + + − + = (6)
[ ] 2
Pr 2 Pr ,f f Ec fθ θ θ′ ′ ′′ ′′− = + (7)
[ ] 2
Pr 2 Pr ,f g g f g Ec f′ ′ ′′ ′′− = + (8)

159
Boundary conditions of equation (4) transform to
PST CASE
( ) 1, ( ) 1 ( ) 0 0
( ) 0, ( ) 0 ( ) 0
f f at
f f as
η
η ηη
η θ η η η
η θ η η η
= = = =
→ → → →∞
(9)
PHF CASE
( ) 1, ( ) 1 ( ) 0 0
( ) 0, ( ) 0 ( ) 0
f f at
f f as
η η
η ηη
η θ η η η
η θ η η η
= = − = =
→ → → →∞
(10)
Where subscript η denotes the differentiation with respect to η . β denotes elastic parameter, 2k is
the porosity parameter, Pr and Ec denotes the Prandtl number and Eckert number respectively, The
physical quantities are defined as,
kb
k
′
=
γ
2 ,
∞
=
K
Cpµ
Pr ,
2 2
p
b l
Ec
Ac
=
PHYSICAL QUANTITIES
Our interest lies in investigation of the flow behavior and heat transfer characteristics by
analyzing the non-dimensional local shear stress )( wτ and Nusselt number (Nu). These non-
dimensional parameters are defined as :
0
),0(
=
∗
∗






∂
∂
−===
y
w
y
u
Wheref
bxb
µτ
γµ
τ
τ ηη (11)






=
−
−
=
∞ CasePHF
CasePST
T
TT
h
Nu y
w )0(1
)0(
θ
θη
(12)
3. NUMERICAL SOLUTION OF THE PROBLEM
We adopt the most effective shooting method (see Refs. Conte and De Boor[20], Cebeci and
Bradshaw[21]) with fourth order Runge-Kutta integration scheme to solve boundary value problems
in PST and PHF cases mentioned in the previous section. The non-linear equations (1) and (3) in the
PST case are transformed into a system of five first order differential equations as follows:
( )
[ ]
0
1
1
2
2
1 0 2 0 1 2 22
2
0
0
1
21
1 0 1 0 2
,
,
2
,
1
,
Pr 2 EcPr .
df
f
d
df
f
d
f f f f f f k fdf
d f
d
d
d
f f f
d
η
η
β
η β
θ
θ
η
θ
θ θ
η
=
=
′− − −
=
−
=
= − −
(13)

160
Subsequently the boundary conditions (8) take the form,
0 1 1
2 0 0
(0) 0, (0) 1, ( ) 0,
(0) 0, (0) 0, ( ) 0.
f f f
f θ θ
= = ∞ =
= = ∞ =
(14)
Here 0 0( ) and ( ).f f η θ θ η= = Aforementioned boundary value problem is first converted
into an initial value problem by appropriately guessing the missing slopes 2 1(0) and (0)f θ . The
resulting IVP is solved by shooting method for a set of parameters appearing in the governing
equations with a known value of 2 1(0) and (0)f θ . The convergence criterion largely depends on
fairly good guesses of the initial conditions in the shooting technique. The iterative process is
terminated until the relative difference between the current iterative values of 2 (0)f matches with
the previous iterative value of 2 (0)f up to a tolerance of 6
10−
. Once the convergence is achieved we
integrate the resultant ordinary differential equations using standard fourth order Runge–Kutta
method with the given set of parameters to obtain the required solution.
4. RESULTS AND DISCUSSION
Numerical computation has been carried out for different physical parameters like Elastic
parameter ( )β , Porosity parameter (k2), Prandtl number (Pr), Eckert number (Ec), which are
presented graphically in figures (1 – 7). The non-linear ordinary differential equations (6), (7) and (8)
subject to the boundary conditions (4), (8) and (9) were solved numerically using the most effective
numerical fourth-order Runge-Kutta method with efficient shooting technique. Appropriate
similarity transformation is adopted to transform the governing partial differential equations of flow
and heat transfer into a system of non-linear ordinary differential equations. The effect of several
parameters controlling the velocity and temperature profiles are shown graphically and discussed
briefly.
Figs.1 and 2 show the effect of Elastic parameter β , on the velocity profile above the sheet.
An increase in the Elastic parameter is noticed to decrease both u- and v- velocity components at any
given point above the sheet.
Figs.3 and 4 revels that, the effect of porosity γ in presence of magnetic number and Elastics
parameter (at 1)β = on the velocity profile above the sheet. An increase in the porous parameter
leads to increase both u- and v- velocity components above the sheet.
Fig.5(a) and 5(b) demonstrate the effect of Prandtl number Pr on the temperature profiles for
two different PST and PHF. These plots reveals the fact that for a particular value of Pr the
temperature increases monotonically from the free surface temperature sT to wall velocity the 0T .
The thermal boundary layer thickness decreases drastically for high values of Pr i.e., low thermal
diffusivity.
Fig.6(a) and 6(b) project the effect of Eckert number Ec on the temperature profiles for
both PST and PHF cases. The effect of viscous dissipation is to enhance the temperature of the fluid.
i.e., increasing values of Ec contributes in thickening of thermal boundary layer. For effective
cooling of the sheet a fluid of low viscosity is preferable.
Figs.7(a) and 7(b) revels that, the effect of porosity γ in presence of magnetic number and
Elastics parameter (at 1)β = on the temperature profile above the sheet. An increase in the porous
parameter leads to decrease the temperature in PST case whereas opposite effect is seen in PHF case.

161
5. CONCLUSIONS
The viscous dissipation effect is characterized by Eckert number (Ec) in the present analysis.
Comparing to the results without viscous dissipation, one can see that the dimensionless temperature
will increase when the fluid is being heated (Ec > 0) but decreases when the fluid is being cooled (Ec
< 0). This reveals that effect of viscous dissipation is to enhance the temperature in the thermal
boundary layer.
0 1 2 3 4 5
0.0
0.2
0.4
0.6
0.8
1.0
Fig.1. The effect of elastic parameter β on u-velocity component f' at β =1
f'(η)
η
β = 1
β = 2
β = 3
0 1 2 3 4 5 6 7
0.0
0.2
0.4
0.6
0.8
1.0
Fig.2. The effect of elastic parameter β on v-velocity component f at β =1f(η)
η
β = 1
β = 2
β = 3
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
0.0
0.2
0.4
0.6
0.8
1.0
Fig.3. The effect of Porous parameter γ on u-velocity component f' at β=1
f'(η)
η
γ = 0.1
γ = 0.2
γ = 0.3
0 1 2 3 4
0.0
0.2
0.4
0.6
0.8
1.0
Fig.4. The effect of Porous parameter γ on v-velocity component f at M=β=1
f(η)
η
γ = 0.1
γ = 0.2
γ = 0.3
0 1 2 3 4 5
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
Fig. 5b. The effect of Prandtl number Vs temperature profile
g(η)
η
PHF-Case
Pr=1
Pr=5
Pr=10
0 1 2 3 4 5 6
0.0
0.2
0.4
0.6
0.8
1.0
Fig. 5a. The effect of Prandtl number Vs Temperature profile
PST Case
Pr = 5
Pr = 10
θ(η)
η

162
0 1 2 3 4 5 6 7
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
Fig. 6b. The effect of Eckert number Vs Temperature profile
g(η)
η
PHF-Case
Ec=0.2
Ec=1.0
Ec=2.0
0 1 2 3 4 5 6
0.0
0.2
0.4
0.6
0.8
1.0
Fig.7a. Effect of porous parameter γ on temperature profile in PST Case
θ(η)
η
PST Case
γ = 0.1
γ = 0.2
γ = 0.3
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
Fig.7b. Effect of Porous parameter γ on temperature profile in PHF Case
g(η)
η
PHF Case
γ=0.1
γ=0.2
γ=0.3
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1.2
Fig. 6a. The effect of Eckert number Vs temperature profile
θ(η)
η
PST-Case
Ec=1
Ec=2
Ec=5

163
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Surface with Viscous Dissipation and Internal Heating”, International Journal of Mechanical
Engineering & Technology (IJMET), Volume 4, Issue 5, 2013, pp. 49 - 62, ISSN Print:
0976 – 6340, ISSN Online: 0976 – 6359.
[23] Anand H. Agadi, M. Subhas Abel, Jagadish V. Tawade and Ishwar Maharudrappa, “Effect of
Non-Uniform Heat Source for the UCM Fluid Over a Stretching Sheet With Magnetic Field”,
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Volume 4, Issue 6, 2013, pp. 40 - 49, ISSN Print: 0976-6480, ISSN Online: 0976-6499.
[24] Anand H. Agadi, M. Subhas Abel and Jagadish V. Tawade, “MHD and Heat Transfer in a
Thin Film Over an Unsteady Stretching Surface with Combined Effect of Viscous
Dissipation and Non-Uniform Heat Source”, International Journal of Mechanical Engineering
& Technology (IJMET), Volume 4, Issue 4, 2013, pp. 387 - 400, ISSN Print: 0976 – 6340,
ISSN Online: 0976 – 6359.
[25] Anand H. Agadi, M. Subhas Abel, Jagadish V. Tawade and Ishwar Maharudrappa, “MHD
Flow and Heat Transfer for the Upper Convected Maxwell Fluid Over a Stretching Sheet
with Viscous Dissipation”, International Journal of Advanced Research in Engineering &
Technology (IJARET), Volume 4, Issue 5, 2013, pp. 231 - 242, ISSN Print: 0976-6480,
ISSN Online: 0976-6499.

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