Copyright © 2013 by Modern Scientific Press Company, Florida, USA

International Journal of Modern Mathematical Sciences, 2013, 8(3): 166-176

International Journal of Modern Mathematical Sciences

Journal homepage:www.ModernScientificPress.com/Journals/ijmms.aspx

ISSN:2166-286X

Florida, USA

Article

Non-Newtonian Fluid Flow with Natural Heat Convection

through Vertical Flat Plates

Naveed Ahmed1, Umar Khan1, Mir Asadullah2, Sanam Ali2, Yang Xiao-Jun3, Syed Tauseef

Mohyud-Din1, *

1Department of Mathematics, Faculty of Sciences, HITEC University, Taxila Cantt, Pakistan

2COMSATS Institute of Information Technology, University Road, Abbottabad, Pakistan

3College of Science, China University of Mining and Technology, Xuzhou, Jiangsu, 221008, China

* Author to whom correspondence should be addressed; Email: syedtauseefs@hotmail.com

Article history: Received 29 September 2013, Received in revised form 8 November, Accepted 12

November 2013, Published 5 December 2013.

Abstract: Natural heat convection for the flow of a third grade fluid through two parallel

vertical infinite flat plates is considered. Similarity transforms combined with conservation

laws are used to obtain the set of coupled nonlinear ordinary differential equations that

govern the flow. Variation of Parameters Method (VPM) is used to determine the semi-exact

solution to the problem. A numerical solution is also sought using Runge-Kutta (RK-4)

method. Both the solutions are compared and an excellent agreement has been found. Effects

of different dimensionless physical parameters on the flow are demonstrated graphically

coupled with comprehensive discussions at the end of the article.

Keywords: Natural heat convection, Variation of Parameters Method (VPM), parallel flat

plates, non-Newtonian fluids.

Mathematics Subject Classification (2010): 76A05; 76A10; 35Q35

Int. J. Modern Math. Sci. 2013, 8(3): 166-176

Copyright © 2013 by Modern Scientific Press Company, Florida, USA

167

1. Introduction

In industries and many practical situations we have to deal with natural convection of heat which

plays a significant role in overall behavior of the flow. In some of the cases it plays such a dominant role

that ignoring it can lead us to very poor and unrealistic flow model. In recent years industrial

development has really enhanced the comfort level of human life and a way better living standard has

been achieved; however further betterment is always welcomed and scope of further improvement is

always there. In similar way despite the great achievement in modeling flow models with heat

convection, still there is a lot of space for new ideas and methodologies.

As in most of the practical cases the fluid under consideration is not simple Newtonian one; there

is a dire need to develop mathematical models for complex fluids also referred as non-Newtonian fluids.

For the flow of a specific class of such fluids between vertically standing flat plates natural heat

convection has been discussed by Bruce and Na [1]. Rajagopal and Na presented a comprehensive

thermodynamic analysis of constitutive relations for some classes of non-Newtonian fluids [2]; one of

them is known as third grade fluid and many researchers have used this constitutive relation to model

the flow of non-Newtonian fluids [3 - 5].

Like most of the other physical and practical situations the equations describing the flow are

highly nonlinear and an exact solution is unlikely, for this purpose many analytical and numerical

schemes have been developed to approximate the solution in better way [6 - 18], one of such techniques

is Variation of Parameters Method (VPM) which is very effective and easy to apply. [19-22] have applied

it to find the solution for many abstract problems; in this paper we have also shown that it can be equally

effective to the problem under consideration and result obtained are enough to back our argument.

Natural heat convection for the flow of a third grade fluid through vertical parallel plates is

considered. VPM solution is compared with the numerical solution obtained by using RK-4 method and

results are shown and compared with the help of tables and graphs. Behavior of flow for ranges of

physical parameters is also discussed in detail.

2. Basic Equations

The basic laws of conservation of mass, momentum and energy for an incompressible third grade

fluid respectively, are given by

,0V (1)

),div( gDt

D

V (2)

Int. J. Modern Math. Sci. 2013, 8(3): 166-176

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168

,2LT

T K

Dt

Dc p (3)

where V , f , , T respectively represent velocity field, body force, stress tensor and temperature.

Further is the constant fluid density, K is the thermal conductivity and pc is the specific heat.

Also, L denotes gradient of V and DtD is the material derivative. Constitutive equation depicting a

third grade fluid is

,)()( 1231221231212211 AAAAAAAAAA trpI (4)

where p is the pressure, is coefficient of viscosity and ,1 ,2 ,1 ,2 3 are material constants.

321 ,, AAA are Erickson tensors and are defined by

,1 LLA (5)

,1,)(grad 111 )(grad

ndt

dnn

nn AVA

AA V (6)

where, “

” represents the transpose of a matrix.

3. Formulation of the Problem

Consider incompressible steady flow of a third grade fluid between two parallel flat plates which

are fixed vertically and are distant h2 apart. Left plate is kept at a constant temperature T0 while the

plate on right is kept at )( 100 TTT . Different temperatures at both walls result in an increase in fluid

temperature near the left wall ( hx ) while fluid temperature near the right wall ( hx ) falls.

We seek one-dimensional velocity and temperature fields of the form:

).(],0),(,0[ xTxv TV (7)

Equation of continuity (1) is satisfied identically and, in absence of body forces, momentum and energy

Eqs. (2) and (3) hence become [2]

0)(6 02

22

32

2

gTT

dx

vd

dx

dv

dx

vdm , (8)

,02

4

3

2

2

2

dx

dv

dx

dv

dy

TdK (9)

subject to the boundary conditions

..)(,)(

,0)(,0)(.

10

ThTThT

hvhv (10)

Int. J. Modern Math. Sci. 2013, 8(3): 166-176

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169

We define non-dimensional parameters as [2]

10

**

0

* ,,TT

TTT

h

xx

V

vv m

After using above dimensionless parameters and dropping asterisks, the non-dimensional form of Eqs.

(15-17) become

,062

22

2

2

T

dx

vd

dx

dv

dx

vd (11)

.02Pr

42

2

2

dx

dv

dx

dvEc

dx

Td (12)

Boundary conditions are

,.

2

1)1(,

2

1)1(

,0)1(,0)1(

TT

vv (13)

where,

,

01

20

TTc

VEc

p is Eckert number,

K

cP

pr

, Prandtl number and

2

203

h

V

is

viscoelastic parameter; further pc is the specific heat of fluid under consideration.

4. Solution Procedure

Following the standard procedure proposed for VPM [20-23], Eqs. (11) and (12) give us the

following iterative schemes

For velocity;

,)()()(

6)()(2

22

0211 dssTds

svd

ds

svdsxxCCxv n

nnxk

(14)

where )0(1 vC and )0(2dx

dvC .

For temperature;

,)(

2)(

Pr)()(

42

0431 dsdx

sdv

dx

sdvEcsxxCCxT nnx

k

(15)

where )0(3 TC and )0(4dy

dTC can be determined by using boundary conditions.

First few iterations of the solution are given as under

Int. J. Modern Math. Sci. 2013, 8(3): 166-176

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170

,6

1

2

1 34

23211 xCxCxCCv

,2

1Pr 24

22

2431 xCCEcxCCT

,28

1

4

1

10

3

5

3

2

1Pr

24

1Pr

12

1

2

3

6

123

2

1

734

6243

52

244

23

433

22

42432

344

22

232

23

223212

xCxCCxCCCC

xCCEcCEcCCC

xCCCCCxCCCxCCv

,720

1

72

1

56

1

56

3

7

1

21

2

5

2

10

1

120

1

15

1

5

3

5

2

20

1

3

1

12

1

12

1

3

4

3

1

2

1Pr

1044

9343

82

34

24

23

724324

33

64

232

24

22

24

43

51

22

33243

423

224

32

2342

33

3232

242

22432

xCxCC

xCCCC

xCCCCC

xCCCCCCC

xCDCCCCC

xCCCCCCC

xCCCCxCCEcxCCT

in a similar way we can find further iterations of the solution and Constants (Ci s) can easily be found

using boundary conditions (13).

5. Results and Discussions

Natural heat convection for the flow of a third grade fluid through two vertically positioned

parallel infinite plates is analyzed. Influences of different dimensionless quantities such as Prandtl

number Pr, Eckert number Ec and viscoelastic parameter on the flow are demonstrated graphically in

Figs. 1-5. Fig. 1 shows the effects of on the non-dimensional velocity v of the fluid. Results are as

expected i.e., the velocity decrease with the increase in viscoelastic parameter. Figs. 2 and 3 depict the

behavior of non-dimensional velocity and temperature profiles respectively for increasing Prandtl

Int. J. Modern Math. Sci. 2013, 8(3): 166-176

Copyright © 2013 by Modern Scientific Press Company, Florida, USA

171

number. Both temperature and velocity are increasing functions in this case. Similar effect can be seen

for increasing Eckert number in Figs. 4 and 5.

Fig. 1: Effects of on non-dimensional velocity

Fig. 2: Effects of Pr on non-dimensional velocity

Fig. 3: Effects of Pr on non-dimensional temperature

Int. J. Modern Math. Sci. 2013, 8(3): 166-176

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172

Fig. 5: Effects of Ec on non-dimensional temperature

To support our analytical solution we also solve this problem by using numerical method known

as RK-4 method. Comparison is shown with the help of tables and an excellent agreement can be easily

seem between both the solutions which approve the effectiveness of VPM. Table. 1 displays the

numerical and analytical solutions to the velocity profile while Table.2 shows the results for temperature

distribution.

Fig. 4: Effects of Ec on non-dimensional velocity

Int. J. Modern Math. Sci. 2013, 8(3): 166-176

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173

Table 1: Comparison of VPM and numerical solution for 1,1Pr,5.0 Ec

x )(xv (VPM) )(xv (Numerical) Error

-1 0 0 0

-0.9 0.01412053 0.0141168 3.7301E-06

-0.8 0.023923905 0.02391934 4.56506E-06

-0.7 0.029795346 0.029790688 4.65734E-06

-0.6 0.032177244 0.032172681 4.56328E-06

-0.5 0.031549454 0.031545028 4.42565E-06

-0.4 0.028411138 0.028406861 4.27696E-06

-0.3 0.023267439 0.023263313 4.12667E-06

-0.2 0.016621612 0.016617631 3.98062E-06

-0.1 0.008971818 0.008967981 3.83722E-06

0 0.000811305 0.000807609 3.70E-06

0.1 -0.007369355 -0.007372911 3.55607E-06

0.2 -0.015079063 -0.015082486 3.42234E-06

0.3 -0.021823032 -0.02182632 3.28808E-06

0.4 -0.027100605 -0.027103751 3.1462E-06

0.5 -0.030405165 -0.030408152 2.98731E-06

0.6 -0.031227397 -0.031230199 2.80181E-06

0.7 -0.029063185 -0.029065745 2.56033E-06

0.8 -0.023426954 -0.023429111 2.15639E-06

0.9 -0.013869914 -0.013871225 1.31107E-06

1 0 0 0.00E+00

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174

Table 2: Comparison of VPM and numerical solution for 1,1Pr,5.0 Ec

x )(xT (VPM) )(xT (Numerical) Error

-1 0.5 0.5 0

-0.9 0.450440489 0.450441811 1.32208E-06

-0.8 0.400734304 0.400735882 1.57897E-06

-0.7 0.350964537 0.350966031 1.49412E-06

-0.6 0.301176070 0.301177386 1.31626E-06

-0.5 0.251385403 0.251386529 1.12636E-06

-0.4 0.201589966 0.20159091 9.44183E-07

-0.3 0.151776311 0.151777083 7.71513E-07

-0.2 0.101926898 0.101927505 6.07713E-07

-0.1 0.052025353 0.052025804 4.50801E-07

0 0.002060220 0.002060518 2.98E-07

0.1 -0.047972798 -0.04797265 1.47787E-07

0.2 -0.098070044 -0.098070043 1.26481E-09

0.3 -0.148220564 -0.148220705 1.40255E-07

0.4 -0.198408225 -0.198408503 2.78122E-07

0.5 -0.248615355 -0.248615773 4.1801E-07

0.6 -0.298827947 -0.29882851 5.63319E-07

0.7 -0.349042435 -0.349043134 6.99057E-07

0.8 -0.399273961 -0.399274722 7.60951E-07

0.9 -0.449565886 -0.449566475 5.88539E-07

1 -0.5 -0.5 0

6. Conclusions

Flow of a non-Newtonian fluid between two vertically placed infinite parallel plates is

considered. Natural heat convection for the flow is examined and Variation of Parameters Method is

used to solve equations governing the flow. Numerical solution is also sought and both solutions are

compared which shows that there is an excellent agreement between both numerical and analytical

solutions. Effects of involved physical parameters are also discussed graphically coupled with

comprehensive discussions and explanations.

Int. J. Modern Math. Sci. 2013, 8(3): 166-176

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