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

1Department of Mathematics, Basaveshwar Engineering College, Bagalkot-587102, INDIA

2Department of Mathematics, Gulbarga University, Gulbarga- 585 106, INDIA

3Department 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

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ISSN 0976 – 6340 (Print)

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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 )

bxU 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

20, (1 )

2s ref

bxT x t T T tα

υ

− = − −

(2)

where 0T is the temperature at the slit and ref

T 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 ( , )s

T x t given by equation (2) represents a situation in which

the sheet temperature decreases from 0T at the slit in proportion to 2x 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

20, 1- .B x t B tα

−= (3)

The particular form of the expressions for ( , )U x t , ( , )s

T 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 Bu v u

t x y y

συ

ρ

∂ ∂ ∂ ∂+ + = −

∂ ∂ ∂ ∂ (5)

22

02( ).

s

p p

T T T k T uu 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., ( )

12

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,s

u U v T T y= = = = (7)



52

0 at ,u T

y hy y

∂ ∂= = =

∂ ∂ (8)

at .dh

v y hdt

= = (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

yτ µ

∂=

∂ and the heat flux

Tq k

y

∂= −

∂ vanish at the adiabatic free surface

(at y = h).

Similarity transformations: We now introduce dimensionless variables and f θ and the similarity

variable η as

( ) ( )

1

2

, , ,1

bx y t x f

t

υψ η

α

=

− (10)

( ) ( ) ( )2 3

20, , 1 ,

2ref

bxT x y t T T tα θ η

υ

− = − −

(11)

( )

1

2

.1

by

tη

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

bxu f

y t

ψη

α

∂ ′= =

∂ − (13)

( )

1

2

.1

bv f

x t

ψ υη

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 fη

′ ′′ ′ ′′ ′′′ ′+ + − = −

(15)

( ) 2SPr 3 (2 ) Ec Pr ,

2f 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( ) .

2f

ββ = (19)

Where a prime denotes the differentiation with respect toη and Sb

=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

bh

tβ

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

d t 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 s

τ and the

heat flux s

q on the stretching sheet that are defined as

0

0

(22)

(23)

s

y

s

y

u

y

Tq k

y

=

=

∂=

∂

∂= −

∂

τ µ

where µ is the fluid dynamic viscosity. The local skin friction coefficient f

C and the local Nusselt

number x

Nu for fluid flow in a thin film can be expressed as

( )1

0 22

2

2Re 0y

f x

u

yC f

U

µ

ρ

−=

∂−

∂ ′′≡ = − (24)

( )1/ 2 3/ 2

0

11 '(0) Re ,

2x x

ref y

x TNu t

T yα θ

−

=

∂≡ − = −

∂ (25)



54

where Rex

Ux

υ= , the local Reynolds number and ref

T 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

Sβ

up to a tolerance of 610− . 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 parameter 0 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 s

T 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 w ith

unstudieness param eter 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)

EcFig.6. Variation of surface tem perature θ(β )

w ith 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 = 1.067175β = 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

d ifferent va lues of m agnetic param eter Mn with S = 1.2

η

S = 1 .2

Mn = 0,1,2 ,3,4

β = 0.627910β = 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 tem perature profiles θ(η )

for different values of Prandtl num ber Pr w ith 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=1.0Pr=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

Fig.10(a). Variation in the temperature profiles θ(η)

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

Fig.10(b). Variation in the temperature profiles θ(η)

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)

Fig.11(a). Variation in the temperature profiles 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)

Fig.11(b). Variation in the temperature profiles 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 1001E-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

REFERENCES

[1]. L.J. Crane, flow past a stretching plate, Z. Angrew. Math. Phys. 21 (1970) 645-647.

[2]. B.K. Dutta, A.S. Gupta, cooling of a stretching sheet in a various flow, Ind. Eng. Chem. Res.

26 (1987) 333-336.

[3]. C.K. Chan, M.I. Char, Heat transfer of a Continuous stretching surface with suction or

Blo3wing, J. math. Anal. Appl. 135 (1988) 568-580.

[4]. C.Y. Wang, Liquid film on an unsteady stretching surface, Quart Appl. Math 48 (1990) 601-

610.

[5]. C. Wang, Analytic solutions for a liquid film on an unsteady stretching surface, Heat Mass

Transfer 42 (2006) 759–766.

[6]. R. Usha, R. Sridharan, on the motion of a liquid film on an unsteady stretching surface,

ASME Fluids Eng. 150 (1993) 43-48.

[7]. Chien-Hsin Chen, Effect of viscous dissipation on heat transfer in a non-Newtonian liquid

film over an unsteady stretching sheet, J. Non-Newtonian Fluid Mech. 135(2006) 128-135

[8]. H.I. Anderson, J.B. Aarseth, B.S. Dandapat, Heat transfer in a liquid film on an unsteady

stretching surface, Int. J. Heat Mass Transfer 43 (2000) 69-74.



62

[9]. B.S. Dandapat, B. Santra, K. Vajravelu, The effects of variable fluid properties and

thermocapillarity on the flow of a thin film on an unsteady stretching sheet, Int. J. Heat Mass

Transfer 50 (2007) 991-996.

[10]. Heat transfer in a liquid film over an unsteady stretching surface with viscous dissipation in

presence of external magnetic field. Published in Applied Mathematical Modeling 33 (2009)

3430-3441.

[11]. R.C. Aziz · I. Hashim · A.K. Alomari, Thin film flow and heat transfer on an unsteady

stretching sheet with internal heating Meccanica (2011) 46: 349–357

[12]. S.M. Roberts, J.S. Shipman, Two point boundary value problems: Shooting Methods,

Elsevier, New York, 1972.

[13]. S. D. Conte, C.de Boor, Elementary Numerical Analysis, McGraw-Hill, NewYork, 1972.

[14]. T. Cebeci, P. Bradshaw, Physical and computational aspects of convective heat transfer,

Springer-Verlag, New York, 1984.

[15]. Dr P.Ravinder Reddy, Dr K.Srihari and Dr S. Raji Reddy, “Combined Heat and Mass

Transfer in MHD Three-Dimensional Porous Flow with Periodic Permeability & Heat

Absorption”, International Journal of Mechanical Engineering & Technology (IJMET),

Volume 3, Issue 2, 2012, pp. 573 - 593, ISSN Print: 0976 – 6340, ISSN Online: 0976 – 6359.

[16]. M N Raja Shekar and Shaik Magbul Hussain, “Effect of Viscous Dissipation on MHD Flow

and Heat Transfer of a Non-Newtonian Power-Law Fluid Past a Stretching Sheet with

Suction/Injection”, International Journal of Advanced Research in Engineering &

Technology (IJARET), Volume 4, Issue 3, 2013, pp. 296 - 301, ISSN Print: 0976-6480,

ISSN Online: 0976-6499.

[17]. M N Raja Shekar and Shaik Magbul Hussain, “Effect of Viscous Dissipation on MHD Flow

of a Free Convection Power-Law Fluid with a Pressure Gradient”, International Journal of

Advanced Research in Engineering & Technology (IJARET), Volume 4, Issue 3, 2013,

pp. 302 - 307, ISSN Print: 0976-6480, ISSN Online: 0976-6499.

[18]. Dr. Sundarammal Kesavan, M. Vidhya and Dr. A. Govindarajan, “Unsteady MHD Free

Convective Flow in a Rotating Porous Medium with Mass Transfer”, International Journal of

Mechanical Engineering & Technology (IJMET), Volume 2, Issue 2, 2011, pp. 99 - 110,

ISSN Print: 0976 – 6340, ISSN Online: 0976 – 6359.

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

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