New MHD Stagnation Point Flow and Heat Transfer Due to Nano … · 2017. 4. 21. · conditions have been investigated by Carragher and Crane (1982)[1], Grubka and Bobba (1985)[2]

Global Journal of Pure and Applied Mathematics.

ISSN 0973-1768 Volume 13, Number 6 (2017), pp. 1593-1610

© Research India Publications

http://www.ripublication.com

MHD Stagnation Point Flow and Heat Transfer Due

to Nano Fluid over Exponential Radiating Stretching

Sheet

Punnam. Rajendar1*, L. Anand Babu2* and T.Vijaya Laxmi2

1,2Department of Mathematics, Osmania University, Hyderabad, Telangana, India. 2* Department Of Mathematics, School of Engineering, Anurag Group of Institutions

Ghatkeshar (Mdl), Medchal (Dist), 500088, Telangana, India. 2M.V.S. Govt. Arts & Science College, Mahabubnagar-509001, Telangana, India.

Abstract

This paper deals with the MHD stagnation point flow and heat transfer due to

nano fluid over exponential radiating stretching sheet. The basic governing

partial diffential equations are converted into nonlinear ordinary partial

differential equations by employing suitable similarity transformations. The

resulting equations are successfully solved by using implicit finite difference

scheme known as Keller box method. The results are very closely agree with

the existing results in the literature. The major effect of governing parameters

namely, magnetic and radiation parameters on the flow field skin friction

coefficient, nusselt number and Sherwood numbers for several values of

governing parameters examined and are presented in the form of tables and

figure.The results indicate that the local Nusselt number decreases and

Sherwood number increases with an increase in velocity ratio parameter

A.The local Nusselt number and Sherwood number increases with an increase

in Lewis number Le and radiation parameter R. Besides, it is found that the

heat transfer rate at the surface increases with the magnetic parameter when

the free stream velocity exceeds the stretching velocity, i.e. A > 1, and it

decreases when A < 1.

Keywords: MHD, stagnation point, exponential stretching sheet, nano fluid,

radiation parameter, heat transfer, Keller Box Method.

INTRODUCTION

The heat transfer of a viscous fluid over flat surfaces have been investigated in several

technological processes such as hot rolling, metal extrusion, continuous stretching of

1594 Punnam. Rajendar, L. Anand Babu and T.Vijaya Laxmi

plastic films and glass-fibre, polymer extrusion, wires drawing and metal spinning.

Various researchers are engaged in this rich area. Numerous physical phenomena

related to stretched sheet moving with constant velocity under various thermal

conditions have been investigated by Carragher and Crane (1982)[1], Grubka and

Bobba (1985)[2]. The thermal boundary layer of a power law fluid over a stretching

surface was studied by Ali (1995)[3]. A good effort has been made to gain insight

information regarding the stretching flow problem in various situations include

considerations of porous surfaces, MHD fluids, heat and mass transfer, slip effects

etc. Magyari and Keller (1999)[4]investigated the steady boundary layers on an

exponentially stretching continuous surface with an exponential temperature

distribution. A new dimension is added to this investigation by Elbashbeshy

(2001)[5]who examined the flow and heat transfer characteristics over an

exponentially stretching permeable surface.Mukhopadhyay (2013)[6]examined slip

effects on MHD boundary layer flow over an exponentially stretching sheet. Some

recent attempts in this direction are described in wang (2008)[7],Hayat(2008)[8],El-

Aziz (2009)[9], Khan and Pop (2011)[10],Ishak (2011)[11], Cortell (2012)[12].

The flow near the stagnation point has attracted the attention of many investigators for

more than a century because of its wide applications. Some of the applications are

cooling of electronic devices by fans, cooling of nuclear reactors during emergency

shutdown, solar central receivers exposed to wind currents, and many hydrodynamic

processes in engineering applications. The study of a stagnation point flow towards a

solid surface in moving fluid traced back to Hiemenz in 1911. He was the pioneer to

analyse two-dimensional stagnation point flow on stationary plate using a similarity

transformation to reduce the Navier–Stokes equations to non-linear ordinary

differential equations. Since then many investigators have extended the idea to

different aspect of the stagnation point flow problems. Stagnation point flow is

continuing to be an interesting area of research among scientists and investigators due

to its importance in a wide variety of applications both in industrial and scientific

applications. Many researchers have been working still on the stagnation- point flow

in various ways. Accordingly, Mahapatra and Gupta [13]numerically analyzed two

dimensional boundary layer flow, stagnation point flow and heat transfer over a

stretching sheet.

The study of magneto-hydrodynamic (MHD) flow of an electrically conducting fluid

is of considerable interest in modern metallurgical and metal-working processes. The

process of fusing of metals in an electrical furnace by applying a magnetic field and

the process of cooling of the wall inside a nuclear reactor containment vessel are good

examples of such fields (Ibrahim et al., 2013)[14]. Some important applications of

radiative heat transfer include MHD accelerators, high temperature plasmas, power

generation systems and cooling of nuclear reactors. Many processes in engineering

areas occur at high temperatures and knowledge of radiation heat transfer becomes

very important for the design of pertinent equipment (Seddeek, 2003)[15]. In

controlling momentum and heat transfers in the boundary layer flow of different

fluids over a stretching sheet, applied magnetic field may play an important

role(Turkyilmazoglu, 2012)[16].Kumaran et al. (2009)[17]investigated that magnetic

MHD Stagnation Point Flow and Heat Transfer Due to Nano Fluid… 1595

field makes the streamlines steeper which results in the velocity boundary layer being

thinner. The heat transfer analysis of boundary layer flow with radiation is also

important in electrical power generation, astrophysical flows, solarpower technology,

space vehicle re-entry and other industrial areas. Raptis et al.(2004)[18] reported the

effect of thermal radiationon the MHD flow of a viscous fluid past a semi-infinite

stationary plate.

Rеnuka Devi et al.[19] presentеd analysis of the radiation and mass transfеr effеcts on

MHD boundary layer flow due to an exponentially stretching sheet with hеat sourcе.

Recently, Poornima and Bhaskar Rеddy [20] presentеd an analysis of the radiation

effеcts on MHD freе convective boundary layеr flow of nanofluids ovеr a nonlinеar

strеtching sheеt. Howevеr, the intеraction of radiation with mass transfеr due to a

strеtching sheеt has receivеd littlе attention. The present paper provides an analytical

solution of MHD boundary layer flow over an exponentially stretching sheet in the

presence of radiation, which has not been considered before. The effects of

controlling parameters on MHD flow and heat transfer characteristics are discussed

and shown graphically.

FORMULATION OF THE PROBLEM:

Consider a two-dimensional stagnation point flow of a nanofluid towards a stretching

sheet kept at a constant temperature Tw and concentration Cw. The ambient

temperature and concentration are T∞ and C∞ respectively. The flow is subjected to a

constant transverse magnetic field of strength B = B0 which is assumed to be applied

in the positive y-direction, normal to the surface. The induced magnetic field is

assumed to be small compared to the applied magnetic field and is neglected. It is

further assumed that the base fluid and the suspended nanoparticles are in thermal

equilibrium and no slip occurs between them. Where Tw, T∞, Cw, C∞ and B0 are

temperature at the surface of the sheet, ambient temperature of the fluid, concentration

at the surface of the sheet, ambient concentration and magnetic field strength

respectively. We choose the coordinate system such that x-axis is along the stretching

sheet and y-axis is normal to the sheet. The flow configuration and coordinate system

is shown in Fig. 1

Figure-1. Flow configuration and coordinate system.


A variable magnetic field B(x) is applied normally to the sheet surface while the

induced magnetic field is negligible, which can be justified for MHD flow at small

magnetic Reynolds number. Under boundary layer approximations, the flow and heat

transfer with radiation effects are governed by the following dimensional form of

equations.

0 (1)u v

x y

22

0

2( ) (2)

f

BUu u uu v U U u

x y x y

22

2

1(3)r T

B

p

q DT T T C T Tu v D

x y T c y y y T y

2 2

2 2(4)T

B

DC C C Cu v D

x y y T y

where u and v are the components of the velocity in the x and y directions

respectively, is the kinematic viscosity, is thermal diffusivity, T is the fluid

temperature in the boundary layer, is fluid density, qr is the radiative heat flux, Cp

is the specific heat at constant pressure. By the use of Rosseland approximation for

radiation, we

have4

*

4(5)

3r

Tq

k y

where is Stefan–Boltzman constant, and *k is the absorption coefficient.

We assume the temperature difference within the flow such that 4T may be expanded

in a Taylor series about T, (the free stream temperature) and neglecting terms of

higher order, wehave

4 3 44 3T T T T Then 2

3

* 2

16

3

rq TT

y k y

Eq (3) becomes

22 23

2 * 2

1 16(6)

3

TB

p

DT T T T C T Tu v T D

x y T c k y y y T y


The hydrodynamic boundary conditions are

( ), 0 0 (7)wu U x v at y

0 (8)u as y

Where ( )x

LwU x ae is stretching velocity, ‘a’ is the reference velocity, L is the

characteristic Length. The thermal and solutal boundary conditions are

20 0 (9)

x

LwT T T T e at y and T T as y

20 0 (10)

x

LwC C C C e at y and C C as y

Where wT is the variable temperature at the sheet with T0 being a constant and

wC is

the variable concentration at the sheet with C0 being constant. It is assumed that the

magnetic field B(x) is of the form 20( )

x

LB x B e where B0 is a constant magnetic field.

The continuity Eq (1) is satisfied by introducing a stream function such that

u and vy x

For non dimensionalized form of momentum and energy equations as well as

boundary conditions, the following transformations are introduced.

2 2

2 20 0

, '( ), ( ) '( )2 2

( ) ( ) (11)

x x x

L L L

x x

L L

a ay e u ae f v e f f

L L

T T T e and C C C e

Where is the similarity variable, f( ) is the dimensionless stream function, ( ) is

the dimensionless temperature, ( ) is the dimensionless concentration and prime

denote differentiation with respect to .

Using Eq. (11), the momentum, energy and concentration equations can be reduced

into ordinary differential equations:

2 2

2

''' '' 2( ') ( ') 2 0 (12)

41 '' Pr ' Pr ' Pr ' ' Pr ( ') 0 (13)

3

'' '' ' ' 0 (14)

f ff f M A f A

R f f Nb Nt

NtLef Lef

Nb

The transformed boundary conditions of the problem are:

(0) 0, '(0) 1, '( ) 0, (0) 1, (0) 1, ( ) 0, ( ) 0 (15)f f f


Where 'f is the dimensionless velocity, is the temperature, is the particle

concentration,2

02 B LM

a

is the magnetic parameter ,

bA

a is velocity ratio,

Pr

is the prandtl number,

B

LeD

is the Lewis number,

( )p B w

f

c DNb

c

is

the Brownian motion parameter,( )p T w

f

c D T TNt

c T

is the thermoporesis parameter

and

3

*

4 TR

kk

is the radiation conduction parameter.

The important physical quantities of interest in this problem are the skin friction

coefficient Cf , local Nusselt number Nux and the local Sherwood number Shx are

defined as

2

0 0 0

, ,( ) ( )

, , (16)

w w mf x x

w w B w

w w m B

y y y

xq xC Nu Sh

U k T T D

u T Cwhere q k D

y y y

Here the skin friction w ,wall heat flux

wq and wall mass flux m .

By using the above equations we get

Re ''(0), '(0), '(0)Re Re

x xf x

x x

Nu NuC f

where ’Rex’ local Reynolds number.

NUMERICAL METHOD

The higher order ordinary differential equations with the boundary conditions are

solved numerically by using implicit finite difference scheme known as Keller-Box

method, the following steps are involved to achieve the Numerical solution.

Reduce the non-linear higher order ordinary differential equations into a system of

first order ordinary differential equations.

Write the finite differences for the first order equations.

Linearize the algebraic equations by Newton’s method, and write them in matrix–

vector form. Solving the linear system by the block tri-diagonal elimination

technique.

In order to solve the above differential equations numerically, we adopt Mat lab

software which is very efficient in using the well known Keller box method.

For getting accuracy of this method to choose appropriate initial guesses.


( ) (1 )(1 ), ( ) , ( )f Ae A e e e

The step size 0.01 is used to obtain numerical solution with four decimal place

accuracy as criterion of convergence.

RESULT AND DISCUSSION

The theme of this section is to discuss the effect of various physical parameters such

as magnetic parameter M, velocity ratio A, prandtl number Pr, the radiation parameter

R, Brownian motion parameter Nb, themoporesis parameter Nt, lewis number Le.

The transformed momentum, energy, concentration Eqs.(12) to (14)and boundary

conditions (15) were numerically solved by using Keller-Box method. The present

results are compared with the existing results in absence of velocity ratio A, Le, Nb,

Nt for local nusselt number '(0) of several values A, M and Pr presented in Table-1

The results so obtained in this vary fairly agree with the previous

results(Mukhopadhyay, Fazle Mabood). Skin friction coefficient ''(0)f for different

values of radiation parameter R and magnetic parameter M calculated and presented

in Table-2 . It is observed that increases R and M, the skin friction coefficient

increased but where as increase in A, the skin friction coefficient slowly decreases.

Calculated the local Nusselt number '(0) and local Sherwood number '(0) for

various parameters presented in in Table-3.

Table-1: Comparison of '(0) for several values of Magnetic, Radiation Parameters

and Pr

R M Pr Mukhopadhyay

(2013)

Fazle Mabood Present Result

0 0 1 0.9547 0.95478 0.954705

2 1.4714 1.47151 1.471551

3 1.8691 1.86909 1.86958

5 2.5001 2.50012 2.50221

10 3.6603 3.66039 3.670012

1 0 1 0.5312 0.53121 0.53108

0 1 1 0.8610 0.86113 0.86096

0.5 0 2 1.0734 1.07352 1.073455

3 1.3807 1.38075 1.38078

1 0 3 1.1213 1.12142 1.12137

1 1 1 0.4505 0.45052 0.45045


We obtained velocity, temperature, concentration profile graph for different values of

governing parameters.The obtained results are displayed through figures.2-4,5-8 and

9-11 for velocity, temperature and concentration profile respectively. Moreover skin

friction coefficient, local nusselt number and local Sherwood number are given in

figures 12-14 respectively. Figures. 2–4show the velocity graphs for different values

of magnetic parameter M and velocity ratio parameter A, when other parameters

remain fixed.

Figure 2 .Effect of several values of A on velocity profile.

Figure. 2 illustrates the influence of velocity ratio parameter A on velocity graph.

When the free stream velocity exceeds the velocity of the stretching sheet, the flow

velocity increases and the boundary layer thickness decreases with increase in A.

Moreover, when the free stream velocity less than stretching velocity, the flow field

velocity decreases and boundary layer thickness also deceases. When A> 1, the flow

has a boundary layer structure and boundary layer thickness decreases as values of A

increases. On the other hand, when A< 1, the flow has an inverted boundary layer

structure, for this case also, as the values A decrease the boundary layer thickness

decreases.

0 1 2 3 4 50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

f'(

)

A = 1.8

A = 1.0

A = 0.2

A = 1.4

A = 0.6


Figure.3 Effect of several values of M on velocity profile

The magnetic parameter M represents the importance of magnetic field on the flow

field. The presence of transverse magnetic field sets in Lorentz force, which results in

retarding force on the velocity field. Therefore, as the values of M increase, so does

the retarding force and hence the velocity decreases. It is shown in Figure. 3.When A

= 3.3 i.e. 1b

a the flow has boundary layer structure and the boundary layer

thickness decreases as the values of M increase.

Figure.4. Effect of several values of M and A on velocity profile.

Figure. 4 shows the variation of flow field velocity with magnetic field and velocity

ratio A. Here consider three different values of A i.e. A = 3.3, A = 1.0 , A = 0.2 and A

= 0.3. When A = 3.3 i.e. 1b

a , the flow has boundary layer structure and the

boundary layer thickness is decreasing with M.When A = 1, the velocity graph for

different values of M are coincide, this indicate that the velocity graph is not

0 1 2 3 4 51

1.5

2

2.5

3

f'(

)

M = 0.0, 1.0, 5.0, 10.0

0 1 2 3 4

0.5

1

1.5

2

2.5

3

f'(

)

A =3.3

A = 0.2

A = 0.3

M = 0, 1, 5, 10A = 1.0

M = 0.0

M = 1.0

M = 5.0

M = 10. 0


influenced by the magnetic field. On the other hand, when A < 1, the flow has an

inverted boundary layer structure, which results from the fact that when A < 1, the

stretching velocity “ax’ of the surface exceeds the velocity “bx” of the free stream

velocity. For this case also the thickness of the boundary layer decrease with M,

which implies increasing manner of the magnitude of the velocity gradient at the

surface.

Figure.5 Effect of several values of A on temperature profile

Figure. 5 shows the variation of temperature profile in response to a change in the

values of velocity ratio parameter A. It shows that as velocity ratio parameter

increases the thermal boundary layer thickness decreases. Moreover, the temperature

gradient at the surface increase (in absolute value) as A increases. As a result,

temperature profile decreases.

Figure.6 Effect of several values of Pr on temperature profile

0 5 10 150

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(

)

A = 0.0

A = 0.3

A = 0.6

A = 0.9

0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(

)

Pr = 0.5

Pr = 1.0

Pr = 1.5

Pr = 2.0


Figure.6. The graph depicts that the temperature decreases when the values of Prandtl

number Pr increase at a fixed value of . This is due to the fact that a higher Prandtl

number fluid has relatively low thermal conductivity, which reduces conduction and

thereby the thermal boundary layer thickness and as a result, temperature decreases.

Figure.7 Effect of several values of Nb on temperature profile

Figure.7.shows the influence of the change of Brownian motion parameter Nb on

temperature profile graph. It is noticed that as Brownian motion parameter increases

the thermal boundary layer thickness increases and the temperature gradient at the

surface decrease (in absolute value) as Nb. Consequently, temperature on the surface

of a plate increases.

Figure.8 Effect of several values of R on temperature profile when Pr = 2.2

0 2 4 6 80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(

)

Nb = 0.1

Nb = 0.3

Nb = 0.6

Nb = 1.0

0 1 2 3 4 5 60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(

)

Pr = 2.2

R = 0.0, 0.5, 1.5, 2.0


It is also noticed from Figure. 8 that the dimensionless temperature increases with an

increase in radiation parameter R.The Lorentz force has the tendency to increase the

temperature and consequently, the thermal boundary layer thickness becomes thicker

for stronger magnetic field.

Figure.9 Effect of several values of A on concentration profile

Figure.9 depicts the influence of velocity ratio parameter A on concentration graph.

As the values of A increase, the concentration boundary layer thickness decreases.

Moreover, it is recognize from the graph that the magnitude of temperature gradient

on the surface of a plate increases as A increases.

Figure.10 Effect of several values of Nb on concentration profile

Figure.10 reveals variation of concentration graph in response to a change in

Brownian motion parameter Nb. The influence of Brownian motion on concentration

0 2 4 6 8 10 12 14 160

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(

)

A = 0.0

A = 0.2

A = 0.4

A = 0.8

0 1 2 3 4 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(

)

Nb = 0.5

Nb = 1.0

Nb = 1.5

Nb = 2.0


profile graph is as the values of Brownian motion parameter increase, the

concentration boundary layer thickness is decreasing. The graph also reveals that the

thermal boundary layer thickness does not change much when the values of Nb

increases.

Figure.11 Effect of several values of Le on concentration profile

As it is noticed from Figure.11 as Lewis number increases the concentration graph

decreases. Moreover, the concentration boundary layer thickness decreases as Lewis

number increases. This is probably due to the fact that mass transfer rate increases as

Lewis number increases. It also reveals that the concentration gradient at surface of

the plate increases.

Table-2: Calculation of skin friction coefficient - ''(0)f for various

parameters M, R, A.

M R A - ''(0)f

1.0 0.5 0.2 1.4381

1.5 1.5454

2.0 1.6456

1.0 1.4381

1.5 1.4880

2.0 1.8298

0.4 1.1707

0.6 0.8373

0.8 0.4452

0 1 2 3 4 5 6 7 8 90

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(

)

Le = 0.5

Le = 1.0

Le = 2.0

Le = 5.0


Table-3: Calculation of Nusselt number & Sherwood number for various values of

M, R, A, Le, Pr, Nb, Nt.

M R A Le Pr Nb Nt '(0) '(0)

0 0.5 0.2 2.0 1.0 0.5 0.5 0.5789 1.2952

1.0 0.5560 1.2508

5.0 0.5069 1.1492

1.0 0.4354 1.1892

1.5 0.3862 1.2155

2.0 0.3498 1.2343

0.4 0.6126 1.3471

0.6 0.6613 1.4337

0.8 0.7055 1.5142

3.0 0.5441 1.6803

4.0 0.5364 2.0383

5.0 0.5309 2.6513

2.0 0.6534 2.3236

3.0 0.6985 2.3252

4.0 0.7134 2.3368

0.2 0.5973 2.0444

0.3 0.5743 2.2161

0.4 0.5521 2.3010

0.2 0.5831 1.3731

0.3 0.5738 1.3303

0.4 0.5647 1.2896


0.6 0.8 1.0 1.2 1.4

0.30

0.35

0.40

0.45

0.50

0.55

0.60

0.65

R

Pr = 0.5

Pr = 1.0

Pr = 1.5

'(0)

Figure. 12 represents a variation of local Nusselt number with respect to Radiation

Parameter R and Prandtl number Pr.

Figure.13 Demonstrate the variation of local Sherwood number with respect to

Brownian motion parameter Nb and Lewis number Le.

0.20 0.25 0.30 0.35 0.40

0.8

0.9

1.0

1.1

1.2

1.3

1.4

1.5

1.6

Nb

Le = 2.0

Le = 2.5

Le = 3.0


1.0 1.2 1.4 1.6 1.8 2.00.8

0.9

1.0

1.1

1.2

1.3

1.4

1.5

-f''(0

)

M

A = 0.6

A = 1.0

A = 1.5

Figure.14. Illustrate the influence of Magnetic parameter M and velocity

ratio A on skin friction co efficient.

CONCLUSION

In this work, the MHD stagnation point flow of a nano fluid and heat transfer over a

exponential radiating stretching sheet was analysed. The effects of magnetic

parameter M, Radiation parameter R, velocity ratio A, Prandtl number Pr, Lewis

number Le, Brownian motion parameter Nb, thermophoresis parameter Nt on the

fluid flow and heat transfer characteristics of the MHD stagnation point flow of a

Nano fluid over exponential radiating stretching sheet were investigated. The

numerical results obtained are agreed very well with the previously published data in

limiting condition and for some particular cases of the present study.

An increase in velocity ratio A is to stagnate in the velocity decreases in concentration

and temperature profiles. As Lewis number Le increases, concentration profile

decreases. As magnetic parameter M increases, velocity profile increases.As

Radiation parameter R increases, the temperature of nano fluid increases. As Nb

increases, the temperature of nano fluid increases, and the concentration decreases. As

Pr increases, the temperature of nano fluid decreases.

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Documents

New MHD Stagnation Point Flow and Heat Transfer Due to Nano … · 2017. 4. 21. · conditions have been investigated by Carragher and Crane (1982)[1], Grubka and Bobba (1985)[2]