# EFFECT OF NON-UNIFORM HEAT SOURCE FOR THE UCM FLUID OVER A STRETCHING SHEET WIT-2

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International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN IN –
6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 6, September – October (2013), © IAEME
ENGINEERING AND TECHNOLOGY (IJARET)

ISSN 0976 - 6480 (Print)
IJARET
ISSN 0976 - 6499 (Online)
Volume 4, Issue 6, September – October 2013, pp. 40-49
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EFFECT OF NON-UNIFORM HEAT SOURCE FOR THE UCM FLUID OVER
A STRETCHING SHEET WITH MAGNETIC FIELD

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

stretching sheet with the combined effect of external magnetic field and non-uniform heat
source/sink. By means of similarity transformations, the non-linear equations governing the flow are
reduced to an ordinary differential equation using similarity transformations. These equations are
solved numerically by using standard fourth order Runge–Kutta method with the given set of
parameters. The results are compared with the earlier published results and our results are better in
agreements under some limiting cases. The effect of several parameters controlling the velocity and
temperature profiles are shown graphically and discussed briefly.

Key words: Boundary layer, Elastic Parameter, Eckert number, Maxwell fluid, Magnetic parameter,
Non-uniform heat source.

1. INTRODUCTION

It is generally recognized that rheological properties of material are specified by their
constitutive equations. Recently, non-Newtonian fluids have been receiving a great deal of research
focus and interest due to their engineering applications in a number of processes. The familiar
examples are the extrusion of polymer fluids, solidification of liquid crystals, animal bloods, exotic
lubricants and colloidal and suspension solutions. Because of the complexity of these fluids, there is
not a single constitutive equation which exhibits all properties of non-Newtonian fluids. A steady

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International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –
6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 6, September – October (2013), © IAEME

two-dimensional laminar flow of an incompressible, electrically conducting MHD Visco-elastic
liquid (Walter’s liquid B model) due to a stretching sheet is considered.

y
Boundary Layer
x

Slit                                                                   Force

Stretching sheet
B0    B0      B0      B0

Fig 1. Schematic of the two-dimensional stretching sheet problem

The sheet lies in the plane y = 0 with the flow being confined to y > 0. The coordinate x is
being taken along the stretching sheet and y is normal to the surfaced, two equal and opposite forces
are applied along the x-axis, so that the sheet is stretched, keeping the origin fixed. Under the
boundary layer approximation and the assumption that the contribution due to the normal stress is of
the same order of magnitude as the shear stress.
The pioneering work due to stretching sheets is done by Sakiadis ([1,2]), Sarpakaya ([3])
was the first researcher to study the MHD flow a of non-Newtonian fluid. Prandtl’s boundary layer
theory proved to be of great use in Newtonian fluids as Navier-Stokes equations can be converted
into much simplified boundary layer equation which is easier to handle.
Crane ([4]) was the first among others to consider the steady two-dimensional flow of a
Newtonian fluid driven by a stretching elastic flat sheet which moves in its own plane with a velocity
varying linearly with the distance from a fixed point. Subsequently, various aspects of the flow
and/or heat transfer problems for stretching surfaces moving in the finite fluid medium have been
explored in many investigations, (e.g. Refs. Dutta et al[5], Chakrabarti et al[6], M.S.Abel et al[7]).
Extrusion of molten polymers through a slit die for the production of plastic sheets is an
important process in polymer industry. In a typical sheet production process the extrudate starts to
solidify as soon as it exits from the die. The sheet is then brought into a required shape by a wind-up
roll upon solidification (see Fig. 1). An important aspect of the flow is the extensibility of the sheet
which can be employed effectively to improve its mechanical properties along the sheet. To further
improve sheet mechanical properties, it is necessary to control its cooling rate. Physical properties of
the cooling medium, e.g., its thermal conductivity, can play a decisive role in this regard. The
success of the whole operation can be argued to depend also on the rheological properties of the fluid
above the sheet as it is the fluid viscosity which determines the (drag) force required to pull the
sheet.

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International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –
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Problems involving fluid flow over a stretching sheet can be found in many manufacturing
processes such as polymer extrusion, wire and fiber coating, foodstuff processing, etc. Essentially,
the quality of the final product depends on the rate of cooling in the process which is significantly
influenced by the fluid flow and heat transfer mechanism. Water is amongst the most-widely used
fluids to be used as the cooling medium. However, the rate of cooling achievable with water is often
realized to be too excessive for certain sheet materials. To have a better control on the rate of
cooling, in recent years it has been proposed that it might be advantageous for water to be made
more or less viscoelastic, say, through the use of polymeric additives ([9]). The idea is to alter flow
kinematics in such a way that it leads to a slower rate of solidification with the price being paid that
fluid’s viscosity is normally increased by such additives. The radiative heat transfer properties of the
cooling medium may also be manipulated to judiciously influence the rate of cooling ([10,11]). In
recent years, MHD flows of viscoelastic fluids above stretching sheets (with and without heat
transfer involved) has also been addressed by various researchers (Pahlavan et al [12], Renardy [13],
Rao and Rajgopal [14], Pahlavan and Sadeghy [15]).
Although there is no doubt about the importance of the theoretical studies cited above, but
they are not above reproach. For example, the viscoelastic fluid models used in these works are
simple models such as second-order model and/or Watler’s B model which are known to be good
only for weakly elastic fluids subject to slow and/or slowly-varying flows (Pahlavan and Sadeghy
[15]). To this should be added the fact that these two fluid models are known to violate certain rules
of thermodynamics (Aliakbar[16]). A non-Newtonian second grade fluid does not give meaning full
results for highly elastic fluids (polymer melts) which occur at high Deborah numbers (Cibeci [17]
and Rajgopal [18]). 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 (Sadegy et. al [11], Pahlavan[12] and Renardy[13]).
Motivated by all the above, in this study, the MHD flow of UCM fluid over a stretching
sheet with the combined effects of Magnetic field and non-uniform heat source is numerically
studied using Runge-Kutta fourth order method with efficient shooting technique. The effects
various parameters of flow and heat transfer coefficients are shown through several plots. It is shown
that the heat fluxes from the liquid to the elastic sheet decreases with S for Pr ≤ 0.1 and increases with
S for Pr ≥ 1 .
The important observation in this study is that, the non-uniform heat sink is one better suited
for effective cooling purpose as heat source enhance the temperature in the boundary layer. On the
other hand it is disclosed that large values of elastic parameter β increase the magnitude of the skin
friction coefficient.

2. MATHEMATICAL FORMULATION

The equations governing the transfer of heat and momentum between a stretching sheet and
the surrounding fluid (see fig.1) can be significantly simplified if it can be assumed that boundary
layer approximations are applicable to both momentum and energy equations. Although this theory is
incomplete for viscoelastic fluids, but has been recently discussed by Renardy [13], it is more
plausible for Maxwell fluids as compared to other viscoelastic fluid models. For MHD flow of an
incompressible Maxwell fluid resting above a stretching sheet, the equations governing transport of
heat and momentum can be written as Pahlavan and Sadeghy [15].

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∂u ∂v
+   = 0,                                                                      (1)
∂x ∂y

∂u    ∂u    ∂ 2u    ∂ 2u      ∂ 2u     ∂ 2u  σ B0 2
u      +v    = υ 2 − λ u 2 2 + v 2 2 + 2uv        −      u,
∂x    ∂y    ∂y      ∂x        ∂y       ∂x∂y 
   ρ                            (2)

∂T    ∂T   k ∂ 2T      q′′′
u      +v    =        2
+       .                                                (3)
∂x    ∂y ρ C p ∂y     ρC p

where B 0 , is the strength of the magnetic field, υ is the kinematic viscosity and λ is the relaxation
time Parameter, k is the thermal conductivity, ρ is the density, T is the temperature, C p is the
specific heat at constant pressure and q′′′ is the space and temperature dependent internal heat
generation/absorption which is modeled as

 ku ( x) 
q′′′ =  w        [ A *(Tw − T∞ ) f (η ) + (T − T∞ ) B*],                     (4)
 xυ 

Where A* and B* are the coefficients of space and temperature dependent internal heat
generation/absorption respectively. Here we make a note that the case A* > 0, B* > 0 corresponds to
internal heat generation and that A* < 0, B* < 0 corresponds to internal heat absorption.
As to the boundary conditions, we are going to assume that the sheet is being stretched linearly.
Therefore the appropriate boundary conditions on the flow are

u = Bx,        v = 0 at y = 0,
u→0         as     y→∞                                                          (5)

where B>0, is the stretching rate. Here x and y are, respectively, the directions along and
perpendicular to the sheet, u and v are the velocity components along x and y directions. The flow is
caused solely by the stretching of the sheet, the free stream velocity being zero. Equations (1) and (2)
admit a selt-similar solution of the form

1
 B 2
u = Bxf ′(η ), v = ν B f (η ),                η =   y,                         (6)
ν 

Where superscript ' denotes the differentiation with respect to η . Clearly u and v satisfy
Equation (1) identically. Substituting these new variables in Eq. (2), we have

2
f ′′′ − M 2 f ′ − ( f ′ ) + f ′′ + β ( 2 ff ′ ′′ − ff ′′′ ) = 0,
f                                   (7)

2          σ B0 2
Here M =                and β = λ B are magnetic and Elastic parameters.
ρB

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International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –
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The boundary conditions (4) become

f ′(0) = 1,  f (0) = 0             at η = 0
f ′(∞) → 0, f ′′(0) → 0            as η → ∞                                                (8)

We define the dimensionless temperature as

2
T − T∞                             x
θ (η ) =         ,        where Tw − T∞ = b   θ (η )               ( PST Case)   (9a)
Tw − T∞                            l

2
T - T∞               D x                        υ
g (η ) =      2
, where Tw − T∞ =                               ( PHF Case)
x 1 ν                  k l                        b                 (9b)
b 
l k b

The thermal boundary conditions depend upon the type of the heating process being
considered. Here, we are considering two general cases of heating namely, (i) Prescribed surface
temperature and (ii) prescribed wall heat flux, varying with the distance.

(i) Governing equation for the prescribed surface temperature case
For this heating process, the prescribed temperature is assumed to be a quadratic function of x
is given by
2
 x
u = Bx, v = 0, T = Tw ( x) = T0 − Ts       at y = 0.
l                                    (10)
u = 0,     T → T∞       as y → ∞

where l is the characteristic length. Using (5), (6) and (10), the dimensionless temperature variable θ
given by (9a), satisfies

Pr [ 2 f ′θ − θ ′ f ] − ( A* f ′ + B*θ ) = θ ′′,                                  (11)

µcp
Where Pr =           is the Prandtl number and corresponding boundary conditions are
k

θ (0) = 1 at η = 0
θ (∞) = 0 as η → ∞                                                                 (12)

(ii) Governing equation for the prescribed heat flux case
The power law heat flux on the wall surface is considered to be a quadratic power of x in the
form
2
 ∂T             x
u = Bx, − k        = qw = D     at y = 0
 ∂y  w          l                                           (13)
u → 0, T → T∞ as                y → ∞.
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International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –
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Here D is constant, k is thermal conductivity. Using (5), (6) and (13), the dimensionless
temperature variable g given by (9b), satisfies

Pr [ 2 f ′ g − g ′ f ] − ( A* f ′ + B* g ) = g ′′,                           (14)

The corresponding boundary conditions are

g ′ (η ) = −1 at η → 0
(15)
g (η ) = 0         as η → ∞ .

The rate of heat transfer between the surface and the fluid conventionally expressed in dimensionless
form as a local Nusselt number and is given by

x        ∂T 
Nu x ≡ −                      = − x Re θ '(0)                            (16)
Tw − T∞     ∂y  y = 0

Similarly, momentum equation is simplified and exact analytic solutions can be derived for
the skin-friction coefficient or frictional drag coefficient as

 ∂u 
µ                                                                   (17)
 dy  y=0                      1
Cf   ≡            = − f ′′ (0 )
ρ ( Bx)2                      Rex
2
ρ Bx
Where Re x =                    is known as local Reynolds number.
µ

3. NUMERICAL SOLUTION

We adopt the most effective shooting method (see Refs. Cebeci [17]) 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 (6) and (11) in the PST case are transformed into a
system of five first order differential equations as follows:

df 0
= f1 ,
dη
df1
= f2 ,
dη
2
df 2 ( f1 ) + M f1 − f 0 f 2 − 2 β f 0 f1 f 2
2

=                                         ,                             (18)
dη               1 − β f02
dθ 0
= θ1 ,
dη
dθ1
= Pr [ 2 f1θ 0 − θ1 f 0 ] − ( A* f ′ + B*θ ).
dη
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International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –
6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 6, September – October (2013), © IAEME

Subsequently the boundary conditions in (7) and (12) take the form,

f 0 (0) = 0,       f1 (0) = 1,         f1 (∞) = 0,
(19)
f 2 (0) = 0,      θ 0 (0) = 0, θ 0 (∞) = 0.

Here f 0 = f (η ) and θ 0 = θ (η ). Aforementioned boundary value problem is first converted
into an initial value problem by appropriately guessing the missing slopes f 2 (0) and θ1 (0) . The
resulting IVP is solved by shooting method for a set of parameters appearing in the governing
equations with a known value of f 2 (0) and θ1 (0) . The convergence criterion largely depends on
fairly good guesses of the initial conditions in the shooting technique. Once the convergence is
achieved we integrate the resultant ordinary differential equations using standard fourth order
Runge–Kutta method to obtain the required solution.

4. RESULTS AND DISCUSSION

The exact solution do not seem feasible for a complete set of equations (6)-(11) because of
the non linear form of the momentum and thermal boundary layer equations. This fact forces one to
obtain the solution of the problem numerically. Present results are compared with Hayat ([10]) some
limiting cases are shown in Table 1. The effect of several parameters controlling the velocity and
temperature profiles are shown graphically and discussed briefly.

1.0                                                                                   1.0
PST-Case                                                                           PHF-Case
β =1.0                                                                            β = 1.0
Pr = 1.0                                                                           Pr =1.0
0.8                                                    *                              0.8                                                 *
A=0.5                                                                              A=0.5
*                                                                                  *
B=0.5                                                                              B=0.5

0.6                                                                                   0.6
g(η)
θ(η)

0.4                                                                                   0.4               M= 0, 1, 2
M= 0, 1, 2

0.2                                                                                   0.2

0.0                                                                                   0.0
0           2              4              6             8                             0          2             4             6             8
η                                                                                   η
Fig.2(a). The effect of Magnetic parameter Mon temperature destribution θ(η)         Fig.2(b). The effect of M             eter
agnetic param Mon temperature destribution g(η)

Figs. 2(a) and 2(b) show the effect of magnetic parameter on the temperature profiles above
the sheet for both PST and PHF cases. An increase in the magnetic parameter is seen to increase the
fluid temperature above the sheet. That is, the thermal boundary layer becomes thicker for larger the
magnetic parameter.

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1.25                                                                                       7
PST-Case                                                                                     PHF-Case
M = 1.0                                                                                      β = 1.0
6
A* = 0.5                                                                                     M=1.0
1.00                                                           *                                                                                            *
B = 0.5                                                                                      A = 0.5
5                                                               B* = 0.5

0.75
Pr = 0.01, 0.1, 1, 5                                  4
θ(η)                                                                                       g(η)
Pr = 0.01, 0.1, 1.0, 5
3
0.50

2

0.25
1

0.00                                                                                       0
0                1                2 η        3                4                        0       1         2      3       4         5         6         7       8
η
Fig. 3(a). Effect of Prandtl number on temperature destribution θ(η)
Fig. 3(b). Effect of Prandtl number on temperature destribution g(η)

Figs.3(a) and 3(b) show the temperature profile θ (η ) and g (η ) versus η from the sheet, for
different values of Pr. We infer from these figures that temperature decreases with increase in Pr
which implies viscous boundary layer is thicker than the thermal boundary layer. Temperature in
both PST and PHF cases asymptotically approaches to zero in free stream region.

1.2                                                                                        2.0
PST-Case                                                                                    PHF-Case
β 1.0                                                                                      β = 1.0
M = 1.0                                                                                     M = 1.0
*                            1.6                                                            *
B = 0.5                                                                                     B = 0.5
Pr = 1.0                                                   *
A = -0.5, 0, 0.5                   Pr = 1.0
0.8
*
A = -0.5, 0, 0.5                                      1.2

θ ( η)                                                                                    g(η)

0.8
0.4

0.4

0.0                                                                                        0.0
0.0         0.4         0.8
η 1.2           1.6         2.0                                   0.0       0.4        0.8
η
1.2          1.6           2.0
*
Fig.4(a). The effect of Space dependent A on temperature destribution θ(η)                                                         *
Fig.4(b). The effect of space dependent A on temperature destribution g(η)

Figs. 4(a) and 4(b), are graphs of temperature profiles θ (η ) and g (η ) versus distance η for
different values of A*. For A* > 0, it can be seen that the thermal boundary layer generates the
energy, and this causes the temperature θ (η ) and g (η ) of the fluid to increase with increase in the
value of A* > 0 (heat source), where as for A*<0 (absorption) the temperature θ (η ) decreases with
increase in the value of A*.
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1.0
1.6                                                PHF-Case
PST-Case
β = 1.0
M = 1.0                                                                                   M= 1.0
*
0.8                                            A = 0.5                                                                                   A* = 0.5
Pr = 1.0                                                                                  Pr = 1.0
1.2

0.6

θ(η )                                                                                  g(η ) 0.8

0.4
B* = -0.05, -0.09, 0
B* = -0.5, 0.0, 0.5
0.4
0.2

0.0                                                                                  0.0
0      1             2              3               4                              0                2              4               6
η                                                                                  η
*
*
Fig.5(a). The effect of temperature dependent B on temperature dstribution θ(η)        Fig.5(b). The effect of Temperature dependent B on temperature destribution g(η)

Figs. 5(a) and 5(b), depicts the temperature profiles θ (η ) and g (η ) versus distance η , for
different values of B*. The explanation is similar to that given for A*.
The present work analyses, the MHD flow and heat transfer within a boundary layer of UCM
fluid above a stretching sheet in presence of non-uniform heat source. Numerical results are
presented to illustrate the details of the flow and heat transfer characteristics and their dependence on
the various parameters.
The results of PST and PHF cases infer that the boundary layer temperature is quantitatively
higher in PST case as compared to PHF case and the results are in tune with what happens in regions
away from the sheet.

Table 1: Comparison of values of skin friction coefficient f ′′ ( 0 ) with M= 0.0 and M= 0.2

Hayat et. al [10]                                                     Present Results
S
M=0.0                        M=0.2                   M=0.0                            M=0.2

0.0                 -1.90250                       -1.94211            -0.999962                         -1.095445
0.4                 -2.19206                       -2.23023            -1.101850                         -1.188270
0.8                 -2.50598                       -2.55180            -1.196692                         -1.275878

1.2                 -2.89841                       -2.96086            -1.285257                         -1.358733

1.6                 -3.42262                       -3.51050            -1.368641                         -1.437369
2.0                 -4.13099                       -4.25324            -1.447617                         -1.512280

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International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –
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