A numerical solution of mhd heat transfer in a laminar liquid film on an unstead

International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –
6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 5, September - October (2013) © IAEME
49
A NUMERICAL SOLUTION OF MHD HEAT TRANSFER IN A LAMINAR
LIQUID FILM ON AN UNSTEADY FLAT INCOMPRESSIBLE STRETCHING
SURFACE WITH VISCOUS DISSIPATION AND INTERNAL HEATING
Anand H. Agadi1*
, M. Subhas Abel2
and Jagadish V. Tawade3
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
ABSTRACT
This study deals with the numerical solution of MHD flow and heat transfer to a laminar
liquid film from a horizontal stretching surface. Similarity transformations are used to convert
unsteady boundary layer equations to a system of non-linear ordinary differential equations. The
resulting non-linear differential equations are solved numerically by using efficient numerical
shooting technique with fourth order Runge–Kutta algorithm. Several parameter effects have been
shown with the aid of graphs. The important observation in this study is, for high values of
unsteadiness parameter S reduces the surface temperature and the temperature-dependent heat
absorption is one better suited for effective cooling purpose as temperature-dependent heat
generation enhance the temperature in the boundary layer.
Key words: Internal heat generation, Liquid film, similarity transformation, unsteady stretching
surface, viscous dissipation.
1. INTRODUCTION
Boundary layer flow and heat transfer in a laminar liquid film on an unsteady stretching sheet
has received a considerable attention from researchers because of their numerous practical
applications in many branches of science and technology. The knowledge of flow and heat transfer
within a laminar liquid film is crucial in understanding the coating process and design of various heat
exchangers and chemical processing equipments. Other applications include wire and fiber coating,
food stuff processing reactor fluidization, transpiration cooling and so on. The prime aim in almost
every extrusion applications is to maintain the surface quality of the extrudate. All coating processes
demand a smooth glossy surface to meet the requirements for best appearance and optimum service
INTERNATIONAL JOURNAL OF MECHANICAL ENGINEERING
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ISSN 0976 – 6340 (Print)
ISSN 0976 – 6359 (Online)
Volume 4, Issue 5, September - October (2013), pp. 49-62
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50
properties such as low friction, transparency and strength. The problem of extrusion of thin surface
layers needs special attention to gain some knowledge for controlling the coating product efficiently.
The studies of boundary layer flows of Newtonian and non-Newtonian fluids on stretching
surfaces have become important, not only because of their technological importance but also in view
of the interesting mathematical features presented by the equations governing the flow. Such studies
have considerable practical relevance, for example in the manufacture of plastic film, in the extrusion
of a polymer sheet from a die and in fibre industries, etc. During the manufacture of these films, the
melt issues from a slit and is subsequently stretched to achieve the desired thickness. Such
investigations of magnetohydrodynamic (MHD) flow are very important industrially and have
applications in different areas of research such as petroleum production and metallurgical processes.
Crane [1] 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. The pioneering works of Crane [1] are subsequently
extended by many authors Refs.[2-3] to explore various aspects of the flow and heat transfer
occurring in an infinite domain of the fluid surrounding the stretching sheet. Wang [4, 5], Usha and
Shridharan [6], Chen [7], Andersson et al. [8] and Dandapat et al. [9]. Abel et al [10] have discussed
about the Heat transfer in a liquid film over an unsteady stretching surface with viscous dissipation in
presence of external magnetic field. Aziz et.al [11] have neglected the magnetic field effect and also
used the homotopy analysis method (HAM) for thin film flow and heat transfer on an unsteady
stretching sheet.
If the fluid is very viscous, considerable heat can be produced even though at relatively low
speeds, e.g. in the extrusion of plastic, and hence the heat transfer results may alter appreciably due
to viscous dissipation. To the author’s knowledge, the influence of viscous dissipation on heat
transfer in a finite liquid film over a continuously moving surface has not yet been discussed in the
literature. Aforementioned studies have neglected the viscous dissipation effect on the heat transfer
which is important in view point of desired properties of the outcome. It is the purpose of this
present work to investigate the combined effect of viscous dissipation and internal heat generation
along with an external uniform magnetic field for flow and heat transfer analysis in a thin liquid film
on an unsteady stretching sheet.
2. MATHEMATICAL MODELING
Let us consider a thin elastic sheet which emerges from a narrow slit at the origin of a
Cartesian co-ordinate system for investigations as shown schematically in Fig 1. The continuous
sheet at 0y = is parallel with the x-axis and moves in its own plane with the velocity
( ),
(1 )
bx
U x t
tα
=
−
(1)
where b and α are both positive constants with dimension per time. The surface temperature sT of
the stretching sheet is assumed to vary with the distance x from the slit as
( )
32
2
0, (1 )
2
s ref
bx
T x t T T tα
υ
− 
= − − 
 
(2)
where 0T is the temperature at the slit and refT can be taken as a constant reference temperature such
that 00 refT T≤ ≤ . The term
2
(1 )
bx
tυ α−
can be recognized as the Local Reynolds number based on the

51
surface velocityU . The expression (1) for the velocity of the sheet ( , )U x t reflects that the elastic
sheet which is fixed at the origin is stretched by applying a force in the positive x-direction and the
effective stretching rate
(1 )
b
tα−
increase with time as 0 1α≤ < . With the same analogy the
expression for the surface temperature ( , )sT x t given by equation (2) represents a situation in which
the sheet temperature decreases from 0T at the slit in proportion to 2
x and such that the amount of
temperature reduction along the sheet increases with time. The applied transverse magnetic field is
assumed to be of variable kind and is chosen in its special form as
( ) ( )
1
2
0, 1- .B x t B tα
−
= (3)
The particular form of the expressions for ( , )U x t , ( , )sT x t and ( , )B x t are chosen so as to
facilitate the construction of a new similarity transformation which enables in transforming the
governing partial differential equations of momentum and heat transport into a set of non-linear
ordinary differential equations.
Consider a thin elastic liquid film of uniform thickness ( )h t lying on the horizontal stretching
sheet (Fig.1). The x-axis is chosen in the direction along which the sheet is set to motion and the y-
axis is taken perpendicular to it. The fluid motion within the film is primarily caused solely by
stretching of the sheet. The sheet is stretched by the action of two equal and opposite forces along
the x-axis. The sheet is assumed to have velocity U as defined in equation (1) and the flow field is
exposed to the influence of an external transverse magnetic field of strength B as defined in equation
(3). We have neglected the effect of latent heat due to evaporation by assuming the liquid to be
nonvolatile. Further the buoyancy is neglected due to the relatively thin liquid film, but it is not so
thin that intermolecular forces come into play. The velocity and temperature fields of the liquid film
obey the following boundary layer equations
0,
u v
x y
∂ ∂
+ =
∂ ∂
(4)
2 2
2
,
u u u u B
u v u
t x y y
σ
υ
ρ
∂ ∂ ∂ ∂
+ + = −
∂ ∂ ∂ ∂
(5)
22
02
( ).s
p p
T T T k T u
u v Q T T
t x y C y C y
µ
ρ ρ
 ∂ ∂ ∂ ∂ ∂
+ + = + + − 
∂ ∂ ∂ ∂ ∂ 
(6)
The pressure in the surrounding gas phase is assumed to be uniform and the gravity force
gives rise to a hydrostatic pressure variation in the liquid film. In order to justify the boundary layer
approximation, the length scale in the primary flow direction must be significantly larger than the
length scale in the cross stream direction. We choose the representative measure of the film thickness
to be
1
2
b
υ 
 
 
so that the scale ratio is large enough i.e.,
( )
1
2
1
b
x
>>
υ
. This choice of length scale enables
us to employ the boundary layer approximations. Further it is assumed that the induced magnetic
field is negligibly small. The associated boundary conditions are given by
, 0, at 0,su U v T T y= = = = (7)

52
0 at ,
u T
y h
y y
∂ ∂
= = =
∂ ∂
(8)
at .
dh
v y h
dt
= = (9)
At this juncture we make a note that the mathematical problem is implicitly formulated only
for 0x ≥ . Further it is assumed that the surface of the planar liquid film is smooth so as to avoid the
complications due to surface waves. The influence of interfacial shear due to the quiescent
atmosphere, in other words the effect of surface tension is assumed to be negligible. The viscous
shear stress
u
y
τ µ
 ∂
=  
∂ 
and the heat flux
T
q k
y
 ∂
= −  
∂ 
vanish at the adiabatic free surface
(at y = h).
Similarity transformations: We now introduce dimensionless variables andf θ and the similarity
variable η as
( ) ( )
1
2
, , ,
1
b
x y t x f
t
υ
ψ η
α
 
=  
− 
(10)
( ) ( ) ( )
2 3
2
0, , 1 ,
2
ref
bx
T x y t T T tα θ η
υ
− 
= − − 
 
(11)
( )
1
2
.
1
b
y
t
η
υ α
 
=   − 
(12)
The physical stream function ( ), ,x y tψ automatically assures mass conversion given in equation (4).
The velocity components are readily obtained as:
( ),
1
bx
u f
y t
ψ
η
α
∂   ′= =  
∂ − 
(13)
( )
1
2
.
1
b
v f
x t
ψ υ
η
α
∂  
= − = −  
∂ − 
(14)
The mathematical problem defined in equations (4) – (8) transforms exactly into a set of ordinary
differential equations and their associated boundary conditions:
( )
2
Mn ,
2
S f f f ff f f
η 
′ ′′ ′ ′′ ′′′ ′+ + − = − 
 
(15)
( ) 2S
Pr 3 (2 ) EcPr ,
2
f f fθ ηθ γ θ θ θ
 ′ ′ ′ ′′ ′′+ + − − = −  
(16)

53
(0) 1, (0) 0, (0) 1,f f θ′ = = = (17)
( ) 0, ( ) 0,f β θ β′′ ′= = (18)
S
( ) .
2
f
β
β = (19)
Where a prime denotes the differentiation with respect toη and S
b
=
α
is the dimensionless
measure of the unsteadiness. Further, the dimensionless film thickness β denotes the value of the
similarity variable η at the free surface so that equation (12) gives
( )
1
2
.
1
b
h
t
β
υ α
 
=   − 
(20)
Yet β is an unknown constant, which should be determined as an integral part of the
boundary value problem. The rate at which film thickness varies can be obtained differentiating
equation (20) with respect to t, in the form
( )
1
2
.
2 1
dh
dt b t
α β υ
α
 
= −   − 
(21)
Thus the kinematic constraint at ( )y h t= given by equation (9) transforms into the free
surface condition (21). It is noteworthy that the momentum boundary layer equation defined by
equation (16) subject to the relevant boundary conditions (17) – (19) is decoupled from the thermal
field; on the other hand the temperature field ( )θ η is coupled with the velocity field ( )f η .
The most important characteristics of flow and heat transfer are the shear stress sτ and the
heat flux sq on the stretching sheet that are defined as
0
0
(22)
(23)
s
y
s
y
u
y
T
q k
y
=
=
 ∂
=  
∂ 
 ∂
= −  
∂ 
τ µ
where µ is the fluid dynamic viscosity. The local skin friction coefficient fC and the local Nusselt
number xNu for fluid flow in a thin film can be expressed as
( )
1
0 2
2
2
2Re 0
y
f x
u
y
C f
U
µ
ρ
−=
 ∂
−  ∂ 
′′≡ = − (24)
( )
1/2 3/2
0
1
1 '(0)Re ,
2
x x
ref y
x T
Nu t
T y
α θ
−
=
 ∂
≡ − = − 
∂ 
(25)

54
where Rex
Ux
υ
= , the local Reynolds number and refT denotes the same reference temperature
(temperature difference) as in equation (2).
3. NUMERICAL APPROACH
The non-linear differential equations (15) and (16) with appropriate boundary conditions
given in (17) to (19) are solved numerically, by the most efficient numerical shooting technique with
fourth order Runge–Kutta algorithm (see references [12] and [13]). The BVP is equivalent to a
system of five first order differential equations with six boundary conditions. The crucial part of the
numerical solution is to determine the dimensionless film thickness β . Eqs. (15) and (16) are
integrated numerically by fourth order Runge–Kutta scheme from 0 to= =η η β with
(0) 0, (0) 1 and (0) 1f f ′= = =θ and guessed trail values (0), (0) and .f ′′ ′θ β However, the numerical
solution thus obtained will not generally satisfy the right-end boundary conditions
( ) 0, (0) 0 ( ) / 2.f and f S′′ ′= = =β θ β β At this end Newton–Raphson scheme is employed to correct
the three arbitrary guess values such that the numerical solution will eventually satisfy the required
boundary conditions (18) and (19). 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 and the previous iterative values of ( )f β matches with the
value of
2
S
β
up to a tolerance of 6
10−
. For further details on the numerical procedure, the readers are
referred to [12,13,14] .
4. RESULTS AND DISCUSSION
The exact solution do not seem feasible for a complete set of equations (15)-(16) 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. 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 resultant boundary value problem is solved by the efficient
shooting method. It is noteworthy to mention that the solution exists only for small value of
unsteadiness parameter0 2S≤ ≤ . Moreover, when 0S → the solution approaches to the analytical
solution obtained by Crane [1] with infinitely thick layer of fluid ( β → ∞ ). The other limiting
solution corresponding to 2S → represents a liquid film of infinitesimal thickness ( 0β → ). The
numerical results are obtained for 0 2S≤ ≤ . Present results are compared with some of the earlier
published results in some limiting cases are shown in Table 1 and Table 2. The effects of various
parameters influencing the dynamics are shown in Fig.2 – Fig.11.
Fig.2 shows the variation of film thickness β with the unsteadiness parameter S. It is evident
from this plot that the film thickness β decreases monotonically when S is increased from 0 to 2.
This result concurs with that observed by Wang [5]. The variation of film thickness β with respect to
the magnetic parameter Mn is projected in Fig.3 for different values of unsteadiness parameter.
The effect of magnetic parameter Mn, Prandtl number Pr, Eckert number Ec and
temperature-dependent parameter γ on the surface temperature ( )θ β are respectively have been
already discussed by Abel et al [10].
The effect of magnetic parameter Mn on the horizontal velocity profiles are depicted in
Fig.8(a) and 8(b) for two different values of unsteadiness parameter S. From both these plots one
can make out that the increasing values of magnetic parameter decreases the horizontal velocity.

55
This is due to the fact that applied transverse magnetic field produces a drag in the form of Lorentz
force thereby decreasing the magnitude of velocity. The drop in horizontal velocity as a consequence
of increase in the strength of magnetic field is observed for S = 0.8 as well as S = 1.2.
Fig.9(a) and 9(b) demonstrate the effect of Prandtl number Pr on the temperature profiles for
two different values of unsteadiness parameter S. 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 as observed by Anderson et al [8]. The thermal boundary layer thickness decreases
drastically for high values of Pr i.e., low thermal diffusivity. From these figure we observe that
Prandtl number Pr will speed up the cooling of the thin film flow.
Fig.10(a) and 10(b) project the effect of Eckert number Ec on the temperature profiles for
two different values of unsteadiness parameter S. The effect of viscous dissipation is to enhance the
temperature in the fluid film. 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.
Fig.11(a) and Fin.11(b) presents the effect of temperature-dependent heat
generation/absorption γ on the temperature profile for different values of unsteadiness parameter S.
For 0<γ reduces the temperature and for 0>γ enhances the temperature in the fluid. The
dimensionless wall temperature gradient '(0)θ− takes a higher value at a large Prandtl number Pr.
The effect '(0)θ− for 1.2S = only marginally exceeds that for 0.8S = for Pr 1> (see fig.12). The
dimensionless wall temperature gradient '(0)θ− takes a uniform value at certain moderate values of
Eckert number Ec, while the effect of '(0)θ− decreases with their increasing Ec (see fig. 13).
Table 1 and Table 2 give the comparison of present results with that of Wang [5] and Aziz
et.al [11]. Without any doubt, from these tables, we can claim that our results are in excellent
agreement with that of references [5 & 11] under some limiting cases. Table.3 tabulates the values
of surface temperature ( )1θ for various values of Mn, Pr, Ec andγ . This table also reveals that Mn
and γ proportionately increase the surface temperature whereas Pr and Ec decreases the surface
temperature.
5. CONCLUSIONS
The present method gives solutions for steady incompressible boundary layer flow of a
laminar liquid film over a heated stretching surface in the presence of a transverse magnetic field
including the viscous dissipation and internal heating effect. Present results reveal that Magnetic
field and viscous dissipative effects play significant role on controlling the heat transfer from
stretching sheet to the liquid film.
The important findings pertaining to the present analysis are,
i) The effect of transverse magnetic field on a viscous incompressible fluid is to suppress the
velocity field which in turn causes the enhancement of the temperature field.
ii) The viscous dissipation effect is characterized by Eckert number (Ec) in the present analysis. It is
observed that the dimensionless temperature will increases when the fluid is being heated
( 0)Ec > but decreases when the fluid is being cooled ( 0)Ec < . This is the effect of viscous
dissipation is to enhance the temperature in the boundary layer.
iii) For a wide range of Pr, the effect of viscous dissipation is found to increase the dimensionless
free surface temperature (1)θ for the fluid cooling case. The impact of viscous dissipation on
(1)θ diminishes in the two limiting cases: Pr 0 and Pr→ → ∞ , in which situations (1)θ
approaches unity and zero respectively.

56
iv) The effect of internal heat generation/absorption is to generate temperature for increasing
positive values and absorb temperature for decreasing negative values. However negative value
of temperature dependent parameter is better suited for cooling purpose.
Fig.1. Schematic representation of a liquid film on an elastic sheet
TABLE 1: Comparison of values of skin friction coefficient ( )0f ′′ with Mn = 0.0
S
Wang [5] Aziz et.al [11] Present work
Β ( )0f ′′−
( )0f
β
′′−
β ( )0f ′′−
( )0f
β
′′−
β ( )0f ′′−
0.4 5.122490 6.699120 1.307785 - - - 4.981455 1.134098
0.6 3.131250 3.742330 1.195155 - - - 3.131710 1.195128
0.8 2.151990 2.680940 1.245795 2.151994 2.680943 1.245794 2.151990 1.245805
1.0 1.543620 1.972380 1.277762 1.543616 1.972384 1.277768 1.543617 1.277769
1.2 1.127780 1.442631 1.279177 1.127780 1.442625 1.174986 1.127780 1.279171
1.4 0.821032 1.012784 1.233549 0.821032 1.012784 1.233549 0.821033 1.233545
1.6 0.576173 0.642397 1.114937 0.576173 0.642397 1.114937 0.576176 1.114941
1.8 0.356389 0.309137 0.867414 0.356389 0.309137 0.867414 0.356390 0.867416
Note: Wang [5] and Aziz [11] have used different similarity transformation due to which the value of
( )0f
β
′′
in his paper is the same as ( )0f ′′ of our results.
For
Sli
t
u
T
h(t
y =
0
y =
y
x

57
TABLE 2: Comparison of values of surface temperature ( )1θ and wall temperature gradient
( )0θ′− with Mn = Ec =γ = 0.0
Pr
Wang [5] Aziz et. al[11] Present work
( )1θ ( )0θ′−
( )0θ
β
′−
( )1θ ( )0θ′−
( )0θ
β
′−
( )1θ ( )0θ′−
S = 0.8 and β = 2.15199
0.01 0.960480 0.090474 0.042042 - - - 0.960438 0.042120
0.1 0.692533 0.756162 0.351378 - - - 0.692296 0.351920
1 0.097884 3.595790 1.670913 0.097956 3.591125 1.668746 0.097825 1.671919
2 0.024941 5.244150 2.436884 0.025083 5.074186 2.357904 0.024869 2.443914
3 0.008785 6.514440 3.027170 0.008545 5.926547 2.753984 0.008324 3.034915
S = 1.2 and β = 1.127780
0.01 0.982331 0.037734 0.033458 - - - 0.982312 0.033515
0.1 0.843622 0.343931 0.304962 - - - 0.843485 0.305409
1 0.286717 1.999590 1.773032 - - - 0.286634 1.773772
2 0.128124 2.975450 2.638324 - - - 0.128174 2.638431
3 0.067658 3.698830 3.279744 - - - 0.067737 3.280329
Note: Wang [5] has used different similarity transformation due to which the value of
( )0θ
β
′−
in his
paper is the same as ( )0θ′− of our results.

58
TABLE 3: Values of surface temperature ( )1θ for various values of Mn, Pr, Ec, γ and S.
Mn Pr Ec γ ( )1θ
S = 0.8 S = 1.2
0.0 1.0 0.02 0.1 0.118639 0.296847
1.0 1.0 0.02 0.1 0.250815 0.413568
2.0 1.0 0.02 0.1 0.358547 0.495749
3.0 1.0 0.02 0.1 0.439666 0.557227
4.0 1.0 0.02 0.1 0.506920 0.604382
5.0 1.0 0.02 0.1 0.564159 0.642261
6.0 1.0 0.02 0.1 0.605107 0.673786
7.0 1.0 0.02 0.1 0.644046 0.699351
8.0 1.0 0.02 0.1 0.676447 0.721720
1.0 0.001 0.02 0.1 0.997829 0.998886
1.0 0.01 0.02 0.1 0.978616 0.988952
1.0 0.1 0.02 0.1 0.814440 0.897785
1.0 1.0 0.02 0.1 0.225360 0.421320
1.0 2.0 0.02 0.1 0.085194 0.228930
1.0 5.0 0.02 0.1 0.009701 0.061819
1.0 10.0 0.02 0.1 -0.000264 0.012560
1.0 100.0 0.02 0.1 -0.001574 -0.000572
1.0 1.0 0.01 0.1 0.226444 0.422094
1.0 1.0 0.1 0.1 0.216691 0.415427
1.0 1.0 0.2 0.1 0.205854 0.407387
1.0 1.0 0.5 0.1 0.173345 0.384166
1.0 1.0 1.0 0.1 0.119162 0.345464
1.0 1.0 2.0 0.1 0.010796 0.268060
1.0 1.0 3.0 0.1 -0.097570 0.190646
1.0 1.0 4.0 0.1 -0.205937 0.113252
1.0 1.0 5.0 0.1 -0.314303 0.035849
1.0 1.0 0.02 -0.5 0.190930 0.366775
1.0 1.0 0.02 -0.2 0.223926 0.393744
1.0 1.0 0.02 -0.1 0.236696 0.403400
1.0 1.0 0.02 0.0 0.250515 0.413420
1.0 1.0 0.02 0.1 0.265505 0.423823
1.0 1.0 0.02 0.2 0.281804 0.434630
1.0 1.0 0.02 0.5 0.340312 0.469708

59
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8
0
1
2
3
4
5
S
β
Fig.2 Variation of film thickness β with
unstudieness parameter S with Mn = 0.0
0 2 4 6 8
0.0
0.4
0.8
1.2
1.6
2.0
Fig.3. Variation of film thickeness β
with magnetic parameter Mn
β
S = 1.2
S = 0.8
Mn
0 2 4 6 8
0.0
0.2
0.4
0.6
0.8
S=1.2
S=0.8θ(β)
Mn
Fig.4. Variation of surface temperature θ(β)
with the Magnetic parameter Mn
1E-3 0.01 0.1 1 10 100 1000
0.0
0.2
0.4
0.6
0.8
1.0
Fig.5. Variation of surface temperature θ(β) for S=0.8(solid line)
and S=1.2(broken line) with the Prandtl number Pr
Pr
θ(β)
S = 0.8
S = 1.2
0.01 0.1 1
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
S = 1.2
S = 0.8
θ(β)
Ec
Fig.6. Variation of surface tem perature θ(β)
with the Eckert num ber Ec
-0.4 -0.2 0.0 0.2 0.4
0.0
0.1
0.2
0.3
0.4
0.5
S=1.2
S=0.8
Q
θ(β)
Fig.7. Variation of surface temperature θ(β)
with the Heat source/sink parameter Q

60
0.0
0.5
1.0
1.5
2.0
0.0 0.2 0.4 0.6 0.8 1.0
η
Fig. 8(a). Variation in the velocity profiles f '(η) for
different values of m agnetic parameter Mn with S = 0.8
S = 0.8
Mn = 0,1,2,3,4
β = 1.067175
β = 1.184197
β = 1.350880
β = 1.616880
β = 2.151992
f '(η)
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0.0 0.2 0.4 0.6 0.8 1.0
Fig. 8(b). Variation in the velocity profiles f '(η) for
different values of m agnetic param eter Mn with S = 1.2
η
S = 1.2
Mn = 0,1,2,3,4
β = 0.627910
β = 0.690238
β = 0.775795
β = 0.903878
β = 1.127780
f '(η)
0.0
0.4
0.8
1.2
1.6
0.0 0.2 0.4 0.6 0.8 1.0
S = 0.8
Pr=0.001
Pr=0.01
Pr=0.1
Pr=1.0
Pr=2.0
Pr=5.0
Pr=10.0
Pr=100
β = 1.616880
Fig.9(a). Variation in the temperature profiles θ(η)
for different values of Prandtl number Pr with S = 0.8
θ(η)
η
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.2 0.4 0.6 0.8 1.0
S = 1.2
θ(η)
Fig.9(b). Variation in the temperature profiles θ(η)
for different values of Prandtl number Pr with S = 1.2
η
Pr=100.0
Pr=10.0
Pr=5.0
Pr=2.0
Pr=1.0
Pr=0.1
Pr=0.001
Pr=0.01
β = 0.903878
0.0
0.4
0.8
1.2
1.6
0.0 0.2 0.4 0.6 0.8 1.0
for different values of Eckert number Ec with S = 0.8
θ(η)
η
S = 0.8
Ec=0.01,0.1,0.5,1,2
β = 1.616880
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.2 0.4 0.6 0.8 1.0
S = 1.2
β = 0.903878
for different values of Eckert number Ec with S = 1.2
θ(η)
η
Ec = 0.01,0.1,0.5,1,2

61
0.0
0.4
0.8
1.2
1.6
0.0 0.2 0.4 0.6 0.8 1.0
S = 0.8
γ = - 0.5
γ = -0.2
γ = -0.1
γ = 0.0
γ = 0.1
γ = 0.2
γ = 0.5
η
θ(η)
for different values of Heat source/sink γ with S = 0.8
β = 1.616880
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.2 0.4 0.6 0.8 1.0
S = 1.2
η
θ(η)
for different values of Heat source/sink γ with S = 1.2
β = 0.903878
γ = 0.0
γ = -0.1
γ= -0.2
γ = -0.5
γ = 0.1
γ = 0.2
γ = 0.5
1E-3 0.01 0.1 1 10 100
1E-3
0.01
0.1
1
10
100
Pr
−θ'(0)
Fig.12. Dimensionless temperature gradient −θ'(0) at the sheet vs
Prandtl number for S=0.8 (solid lines) and S=1.2 (broken lines)
S=0.8
S=1.2
0.01 0.1 1
0.01
0.1
1
Ec
−θ'(0)
Fig.13. Dimensionless temperature gradient −θ'(0) at the sheet vs
Eckert number for S=0.8 (solid lines) and S=1.2 (broken lines)
S = 1.2
S = 0.8
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