ISSN: 0975 -766X CODEN: IJPTFI Available Online through ... · exponentially stretching sheet. Sajid et al [5] examined the influence of thermal radiation on boundary layer flow due

T.Hymavathi*et al. /International Journal of Pharmacy & Technology

IJPT| Dec-2017| Vol. 9 | Issue No.4 | 31088-31102 Page 31088

ISSN: 0975-766X

CODEN: IJPTFI

Available Online through Research Article

www.ijptonline.com NUMERICAL SOLUTION TO FLOW AND HEAT TRANSFER OF A CASSON FLUID

OVER AN EXPONENTIALLY PERMEABLE STRETCHING SURFACE T.Hymavathi

1*and W.Sridhar

2

1,2Department of Mathematics, Adikavi Nannaya University, Rajamundry, A.P., India.

Email: [email protected]

Received on: 23-10-2017 Accepted on: 25-11-2017

Abstract:

The present paper contains the boundary layer flow and heat transfer of a non-Newtonian fluid at an exponentially

stretching permeable surface. Using similarity transformations the governing partial differential equations

corresponding to the equation of continuity, momentum, and energy equations are converted into nonlinear

ordinary differential equations, and numerical solutions to these equations are obtained using an implicit finite

difference scheme known as Keller Box method. It is observed that increasing values of the Casson parameter,

velocity decreases while increasing the temperature field. Also, suction/blowing shows opposite effects with

velocity and temperature. Increase in Prandtl number, temperature decreases. Skin friction is examined and

compared with previous results. The results are found to be good in agreement.

Keywords: Casson fluid, heat transfer, Keller Box method, exponentially stretching sheet, suction/blowing.

1. Introduction

Most of the engineering problems, boundary layer flow over a stretching surface is encountered. Sakiadis[1]

examined the boundary layer flow over a continuous solid surface. L.J.Crane[2]

studied flow past a stretching

surface. Flow and Heat transfer over an exponentially stretching sheet has very important applications in industry.

For example, In case of annealing and thinning of copper wire, the final product depends upon the rate of heat

transfer at the stretching continuous surface with variations in stretching velocity, temperature distribution

exponentially.

The quality of the final product depends on these parameters. Magyari et. al.[3]

studied the heat and mass transfer

on boundary layer flow due to an exponentially stretching sheet. Elbashbeshy[4]

investigated the flow of an

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exponentially stretching sheet. Sajid et al[5]

examined the influence of thermal radiation on boundary layer flow

due to exponentially stretching sheet using HAM.

Bidin et al[6]

analyzed the effect of thermal radiation on the study of laminar two-dimensional boundary layer flow

and heat transfer over an exponentially stretching sheet. Hymavathi et al[7]

examined numerical approach to

magnetohydrodynamic viscoelastic fluid flow and heat transfer over a non isothermal stretching sheet. Swathi

Mukhopadhyay[9]

studied MHD boundary layer flow and heat transfer over an exponentially stretching sheet

embedded in a thermally stratified medium. Swathi Mukhopadhyay[10]

also examined the slip effects on MHD

boundary layer flow over an exponentially stretching sheet with suction/blowing and thermal radiation. The

technique for Keller Box scheme was discussed in detail by Cebeci et al[11]

in Physical and Computational aspects

of convective Heat Transfer.

Mustafa et al[16]

discussed the Stagnation-point flow and heat transfer of a Casson fluid towards a stretching sheet.

Bachok et al[17]

examined the flow and heat transfer over an unsteady stretching sheet in a micropolar fluid with

prescribed heat flux.

Heat transfer analysis is very important in handling and processing of non-Newtonian fluids. In the present study,

we examined Casson fluid model: which is non-Newtonian and exhibits yield stress. If a shear stress is less than

the applied yield stress, it behaves like a solid; whereas if a shear stress is greater than the applied yield stress, it

starts to move. The best examples of Casson fluid are soup, tomato sauce, jelly, honey, human blood etc. It can be

defined as a shear thinning liquid which is assumed to have an infinite viscosity at zero rate of shear; a yield stress

below which no flow occurs and a zero viscosity at an infinite rate of shear [12]

. Eldabe and Salwa [13]

have studied

the Casson fluid for the flow between two rotating cylinders and Boyd et al.[14]

investigated the Casson fluid flow

for the steady and oscillatory blood flow.

Swathy Mukhopadhyay[8]

discussed the Casson fluid flow and heat transfer at an exponentially stretching

permeable surface using Shooting method. Recently, Hymavathi et al.[15]

discussed the numerical solution to mass

transfer on MHD flow of Casson fluid with suction and chemical reaction. In this present study of work, numerical

solution to Casson fluid flow and Heat transfer at an exponentially stretching surface is examined. The Keller Box

method is discussed for numerical computation. The velocity and Temperature profiles were examined.



2. Equations of Motion

Consider the boundary layer flow of a steady incompressible viscous fluid past a flat sheet coinciding with the

plane y = 0. The fluid flow is confined to y > 0. Two equal and opposite forces are applied along the x axis so that

the wall is stretched keeping the origin fixed (see Fig. 1). The rheological equation of state for an isotropic and

incompressible Casson fluid is

Fig 1.

𝜏𝑖𝑗 = {2(𝜇𝐵 + 𝑃𝑦/ 2𝜋)𝑒𝑖𝑗 ,𝜋 > 𝜋𝑐

2(𝜇𝐵 + 𝑃𝑦/ 2𝜋𝑐)𝑒𝑖𝑗 ,𝜋 < 𝜋𝑐

(1)

Here π = eijeij and eij, is the (i,j)th component of the deformation rate, π is the product of the component of the

deformation rate with itself, πc is a critical value of this product based on the non-Newtonian model, µB is plastic

dynamic viscosity of the non-Newtonian fluid, and Py is the yield stress of the fluid. The continuity, momentum

[and energy equations governing such a type of flow problem are

(2)

2

211

y

uv

y

uv

x

uu

(3)

2

2

y

Tk

y

Tv

x

Tu

(4)

Where, u and v are the components of velocity respectively in the x and y directions, v is the kinematic viscosity,

is the fluid density (assumed constant), 𝛽 = 𝜇𝐵 2𝜋𝐶/𝑃𝑦 is the non-Newtonian parameter of the Cass on fluid, k

is the thermal diffusion coefficient of the fluid.

0

y

v

x

u



2.1 Boundary Conditions.

The appropriate boundary conditions for the problem are given by

𝑢 = 𝑈, 𝑣 = −𝑣0(𝑥),𝑇 = 𝑇𝑤 𝑎𝑡 𝑦 = 0 (5)

𝑎𝑠 𝑦 → ∞,𝑢 → 0,𝑇 → 𝑇∞ (6)

Here 𝑈 = 𝑈0𝑒𝑥

𝐿 is the stretching velocity[2], 𝑇𝑤 = 𝑇∞ + 𝑇0𝑒𝑥

2𝐿 is the temperature at the sheet,U0 and T0 are the

reference velocity and temperature respectively. 𝑉(𝑥) = 𝑉0𝑒𝑥

2𝐿 is a special type of velocity at the wall[25] with V0

as constant. The meaning of V0 , V(x) > 0 is the velocity suction and V(x) < 0 is the velocity blowing.

Equations (2)-(6) can be made dimensionless by introducing the following change of variables

𝜂 = 𝑈0

2𝜐𝐿𝑒

𝑥

2𝐿𝑦 (7a)

𝑈 = 𝑈0𝑒𝑥

𝐿𝑓 ′(𝜂) (7b)

𝜐 = − 𝜐𝑈0

2𝐿𝑒

𝑥

2𝐿 [𝑓 𝜂 + 𝜂𝑓 ′ 𝜂 ] (7c)

𝑇 = 𝑇∞ + 𝑇0𝑒𝑥

2𝐿𝜃(𝜂) (7d)

The dimensionless problem satisfies

1 +1

𝛽 𝑓 ′′′ + 𝑓𝑓 ′′ − 2𝑓 ′2 = 0 (8)

𝜃 ′′ + Pr(𝑓𝜙 ′ − 𝑓 ′𝜃) = 0 (9)

and the boundary conditions take the following form

𝑎𝑡 𝜂 = 0,𝑓 = 𝑆,𝜃 = 1,𝑓 ′ = 1 (10)

𝑎𝑠 𝜂 → ∞,𝑓 ′ → 0,𝜃 → 0 (11)

Where 𝑆 = 𝑣0/ 𝑈0𝜐

2𝐿 >0 or (< 0) is the suction or (blowing) parameter and 𝑃𝑟 =

𝜐

𝑘 is the Prandtl number.

2.2 Numerical Procedure

Equation subject to boundary conditions is solved numerically using an implicit-finite difference scheme

known as Keller box method, as described by cebeci and Bradshaw[11]

. The steps followed are



1. Reduce (8)-(9) to a first order equation

2. Write the difference equations using central differences

3. Linearize the resulting algebraic equation by Newton’s method and write in matrix vector form

4. Use the block tri-diagonal elimination technique to solve the linear system.

Consider the flow equation and concentration equations

1 +1

𝛽 𝑓 ′′′ + 𝑓𝑓 ′′ − 2𝑓 ′2 = 0 (12)

𝑡′ + 𝑃𝑟 𝑓𝑡 − 𝑝𝑔 = 0 (13)

and the boundary conditions

𝑓 𝜂 = 𝑆, 𝑓 ′ 𝜂 = 1, 𝜃 𝜂 = 1 𝑎𝑡 𝜂 = 0 (14)

𝑓 ′ 𝜂 → 0,𝜃 𝜂 → 0 𝑎𝑡 𝜂 → ∞ (15)

Introduce f ′=p, (16)

p′=q, (17)

g′=t (g=ϕ) (18)

equations (12) and (13) reduces to

1 +1

𝛽 𝑞′ + 𝑓𝑞 − 2𝑝2 = 0 (19)

𝑡′ + 𝑃𝑟 𝑓𝑡 − 𝑝𝑔 = 0 (20)

Consider the segment ηj-1,ηj with ηj-1/2 as the midpoint η0=0, ηj = ηj-1+hj, ηj=η∞ (21)

where hj is the ∆η spaces and j=1,2,……..J is a sequence number that indicates the coordinate locations.

2/1

11

2

j

jjjjp

pp

h

ff

j

(22)

2/1

11

2

j

jjjjq

qq

h

pp

j

(23)

2/1

11

2

j

jjjjn

tt

h

gg

j

(24)

02

222

11

2

1111

jjjjjjjj ppqqff

h

qq

j

(25)



022

Pr.22

Pr11111

jjjjjjjjjj ggppttff

h

tt

j (26)

2.3 Newton’s method

Linearizing the non linear system of equations (22) to (26)

Introduce

)()()1( k

j

k

j

k

j fff

)()()1( k

j

k

j

k

j ppp

)()()1( k

j

k

j

k

j qqq

)()()1( k

j

k

j

k

j ggg

)()()1( k

j

k

j

k

j ttt (27)

Substitute in equations (16) to (20)

Write 2

1111 )(2

j

jj

j

jj rpph

ff (28)

2

1)(

2211

j

rqqh

pp jj

j

jj (29)

2

1)(

2311

j

rtth

gg jj

j

jj

(30)

2

1)()()()()()()( 4165143121

j

rpapafafaqaqa jjjjjjjjjjjj (31)

2

1)()()()()()()()()( 5187165143121

j

rgbgbpbpbfbfbtbtb jjjjjjjjjjjjjjjj (32)

Where,

)()1(4

1)( 11

jj

j

j ffh

a

0.2)()( 12 jj aa

)()1(4

)( 13

jj

j

j qqh

a

(33)



jj aa )()( 34

)()1(

)( 15

jj

j

j pph

a

jj aa )()( 56

)(4

Pr1)( 11 jj

j

j ffh

b

0.2)()( 12 jj bb

)(2

Pr)( 13 jj

j

j tth

b

jj bb )()( 34 (34)

)(2

Pr)( 15 jj

j

j ggh

b

jj bb )()( 56

)(2

Pr)( 17 jj

j

j pph

b

jj bb )()( 78

and

111

2)( jj

j

jjj pph

ffr

112

2)( jj

j

jjj qqh

ppr

1132

)( jj

j

jjj tth

ggr (35)

211114)1(2)1(4

)(

jj

j

jjjj

j

jjj pph

qqffh

qqr

1111154

Pr

4

Pr)( jjjj

j

jjjj

j

jjj ggpph

ttffh

ttr



Taking j=1,2,3…

The system of equations becomes

[A1][δ1]+[C1][δ2]=[r1] (36)

[B2][δ1]+[A2][δ2]+[C2][δ3] = [r2] (37)

…..[BJ-1][δ1]+[AJ-1][δ2]+[CJ-1][δ3] = [rJ-1]

[BJ][δJ-1]+[AJ][δJ]= [rJ]

Where

111312

111312

1

)(0)()(0

0)()(0)(

000

000

00100

bbb

aaa

dd

dd

A

jjjj

jjj

j

bbbb

aaa

d

d

d

A

)(0)()()(

0)()(0)(

0010

0001

0010

1386

136

jj

jj

j

bb

aa

d

d

B

)(0)(00

0)()(00

0000

0000

00100

24

24

000)()(

0000)(

00010

00001

0000

75

5

jj

j

j

bb

a

d

C

(38)

2.4 The Block Elimination Method

The linearized differential equations of the system has a block diagonal structure. This can be written in tri-

diagonal matrix form as

J

J

J

J

JJ

JJJ

r

r

r

r

AB

CAB

CAB

CA

1

2

1

1

2

1

111

222

11

:::

(39)

This is of the form A δ = r (40)

To solve the above system



Write [A] = [L] [U] (41)

Where

JJ

j

L

1

22

1

and

I

I

I

I

U

J 1

22

1

(42)

Where [I] is the identity matrix

ii are determined by the following equations

[α1] = [A1]

[A1][ Γ1] = [C1]

[α j] = [Aj]-[Bj][ Γ j-1] j=2,3,. ........,J

[α j][ Γ j] = [Cj] j=2,3,..........,J-1

Substituting (41) in (40)

LUδ = r

Let U δ = W

then LW = r

where W=

][

][

][

][

1

2

1

J

j

w

w

w

w

Now [α1] [w1] = [r1]

[α j] [wJ] = [rJ]-[BJ][Wj-1] for 2 ≤ j ≤ J

Once the elements of W are found, substitute in Lδ=W and solve for δ



[δJ] = [WJ]

[δJ] = [WJ]-[ Γ J][δJ+1] , 1 ≤ j ≤ J-1

These calculations are repeated until some convergence criterion is satisfied and we stop the calculations when

|δg0(i)

|≤ ε, where ε is very small prescribed value taken to be ε = 0.000001.

3. Results and discussion

Table 1 : Values of wall temperature [–θ’(0)] for various values of Prandtl Number for Newtonian fluid (β→∞)

Pr Margyari and Keller[3]

SwathyMukhopadhyay[4]

Present

Results(Newtonian)

1 0.9548 0.9547 0.954824

3 1.8691 1.8691 1.869089

5 2.5001 2.5001 2.500164

10 3.6604 3.6603 3.660481

The velocity and temperature profiles are plotted graphically using MATLAB for various value of Casson

parameter, suction parameter, Prandtl number It is observed that the velocity is found to be decreasing with

increase in Casson parameter as shown in figure 2a. where the values of S=0, Pr=0.7.Temperature is found to be

increasing in absence of suction/blowing shown in 2(b)

2(a).Velocity profiles in absence of suction

/blowing

2(b) Temperature profiles with Casson

Parameter β in absence of suction/blowing

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

f '(

)

at S=0 and Pr=0.7

=0.5

=1.0

=2.0

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(

)

at S=0 and Pr=0.7

=0.5

=1.0

=2.0



It is observed that velocity is found to decrease with increase in the Casson parameter β in presence of

suction/blowing shown in fig.3(a) and 3(b).where as Temperature is found to be increasing with increase in Casson

parameter in presence of suction/blowing shown in fig 3(c) and 3(d).

3(a)

Velocity profiles with Casson Parameter β in

presence of suction

3(b)

Velocity profiles with Casson Parameter β in

presence of blowing

3(c) Temperature profiles with Casson

Parameter β in presence of suction

3(d) Temperature profiles with Casson

Parameter β in presence of blowing

Fig.4(a) and 4(b) denotes the effect of suction/blowing parameter on velocity and temperature profiles. It is

observed that the velocity decreases with increasing in suction where as fluid velocity increases when blowing.

S=0 represents the case of non porous stretching sheet. Opposite behavior is noted. This happens because when

stronger blowing is provided, the heated fluid is pushed farther from the wall due to less influence of viscosity and

hence the flow is increasing. The same principle applies but in opposite direction in the case of suction.

Fig4(c) and 4(d) represents the temperature profiles for various suction/blowing parameters S. It is observed that

temperature decreases with increase in suction and increases with increase in blowing.

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

f '(

)

S=0.5,Pr=0.7

=0.5

=1.0

=2.0

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

f '(

)

S=-0.5,Pr=0.7

=0.5

=1.0

=2.0

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(

)

S=0.5,Pr=0.7

=0.5

=1.0

=2.0

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(

)

S=-0.5,Pr=0.7

=0.5

=1.0

=2.0



4(a) Velocity profiles with blowing

parameter S

4(b) Velocity Profiles with suction parameter S

4(c) Temperature profiles with blowing

parameter S at β=0.5, Pr=0.7

4(d) Temperature profiles with suction parameter

S at β=0.5, Pr=0.7

Fig.5(a). Represents the effect of Prandtl number Pr on temperature profiles in absence of suction and observed

that as Prandtl number increases, Temperature decreases i.e., increase in Prandtl number reduces the thermal

boundary layer thickness.

5(a) Temperature profiles with Prandtl number(Newtonian case).

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

f'(

)

=5,Pr=0.7

S=-0.5

S=-1

S=0

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

f'(

)

=5,Pr=0.7

S=0.5

S=1.0

S=0

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(

)

S=-1.0

S=-0.5

S=0

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(

)

S=0.5

S=1.0

S=0

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(

)

at S=0,=2

Pr=0.5

Pr=0.7

Pr=1.0

Pr=2.0



Fig 5(b) and 5(c) represents the effect of Temperature with variations in Prandtl number in presence of

suction/blowing. In both the cases as Prandtl number increases , Temperature decreases.

5(b) Temperature profiles with Prandtl

number in presence of suction(non-

Newtonian case)

5(c) Temperature profiles with Prandtl

number in presence of blowing (non-

Newtonian case)

4. Conclusions

The present work concludes the numerical solution for steady boundary layer flow and heat transfer analysis for a

Casson fluid over an exponentially stretching surface. It is observed that

1. Velocity is found to be decreasing with increase in Casson parameter where as Temperature is found to be

increasing in absence of suction.

2. Velocity is found to be decreasing with increase in Casson parameter where as Temperature is found to be

increasing in presence of suction/blowing.

3. Momentum boundary layer thickness decreases with increasing in Casson parameter, but thermal boundary

layer thickness increases.

4. The surface shear stress increases with increase in Casson parameter.

5. Increase in Pr, Temperature decreases and in turn reduces the thermal boundary layer thickness.

References:

1. Sakiadis B.C. ,“Boundary layer bahaviour on continous solid surface: I boundary layer equation for two

dimensional and axisymmetric flow.” – AIChE J., vol.7,1961, pp.26-28.

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(

)S=0.5,=2

Pr=0.5

Pr=0.7

Pr=1.0

Pr=2.0

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(

)

S=-0.5,=2

Pr=0.5

Pr=0.7

Pr=1.0

Pr=2.0



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prescribed heat flux”,Meccanica,46(5),2011,pp935-942.

Corresponding Author:

T.Hymavathi *

Email: [email protected]

mailto:[email protected]

ISSN: 0975 -766X CODEN: IJPTFI Available Online through ... · exponentially stretching sheet. Sajid et al [5] examined the influence of thermal radiation on boundary layer flow due

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