A Simple Analytical Expression of a Non-Linear Boundary Value Problem for an Immobilized Oxidase...

Discovery, Volume 21, Number 67, July 3, 2014

Ananthaswamy et al. Approximate analytical solution of thermal criticality for a reactive third-grade liquid in a channel with isothermal walls: homotopy analysis method, Discovery, 2014, 21(67), 53-70, www.discovery.org.in http://www.discovery.org.in/d.htm © 2014 Discovery Publication. All Rights Reserved

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APPROXIMATE ANALYTICAL SOLUTION OF THERMAL CRITICALITY FOR A REACTIVE THIRD-GRADE LIQUID IN A

CHANNEL WITH ISOTHERMAL WALLS: HOMOTOPY ANALYSIS METHOD

V. Ananthaswamy1*, C. Chowmiya1 and M. Subha2

1Assistant Professor, Department of Mathematics, The Madura College, Madurai-625011,

Tamil Nadu, India. E-mail: [email protected] 1M.Phil., Mathematics, The Madura College, Madurai-625011, Tamil Nadu, India.

E-mail:[email protected] 2Assistant Professor, Department of Mathematics, Madurai Sivakasi Nadars Pioneer Meenakshi Women’s

College, Poovanthi, Tamil Nadu, India. Tamil Nadu, India.e-mail: [email protected]

Abstract: In this article we discuss the investigation of thermal criticality for a reactive third-grade liquid flowing steadily between two parallel isothermal plates. The reaction is assumed to be strongly exothermic under Arrhenius kinetics, neglecting the consumption of the material. The non-linear differential equations are solved by using the Homotopy analysis method. The approximate analytical expressions of the dimensionless velocity and temperature fields are derived by using the Homotopy analysis method (HAM). Our analytical results are compared with the previous work and a satisfactory agreement is noted. Keywords: Channel flow; Third grade fluid; Arrhenius kinetics; Thermal criticality; Homotopy analysis method. 1. INTRODUCTION Channel flow (internal flow) refers to the flow of fluids in closed conduits like pipes or air ducts, and the or pipe. Couette flow is the flow of fluid flows are entirely in contact with the rigid boundaries of the conduits. Such flows are of great interest in science, engineering, technology and in everyday life. Most closed conduits in engineering applications are either circular or rectangular in cross section. Poiseuille flow and Couette flow are two examples of channel flow that are frequently encountered. Poiseuille flow is the flow resulting from pressure gradients in a long duct between infinite parallel flat plates driven by the motion of one or more plates. Open Channel Flow is defined as fluid flow with a free surface open to the atmosphere. Examples include streams, rivers and culverts not flowing full. Open channel flow assumes that the pressure at the surface is constant and the hydraulic grade line is at the surface of the fluid. Fluid behavior can be visualized in two ways namely Newtonian or non-Newtonian, depending on their general characteristic response to shear stress. For a Newtonian fluid, shear stress is directly proportional to the rate of strain. If the deformation rate, rate of strain or shearing strain is defined by

dydu , where u is the fluid velocity in the x - direction, then the

shear stress acting on a plane normal to the x -axis in a Newtonian fluid is such that

∝dydu , i.e.

dydu



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where is the absolute or dynamic viscosity. It is the Newton's law of viscosity for one dimensional flow. On the other hand, for a non-Newtonian fluid shear stress is not directly proportional to the rate of strain. Clay suspensions, polymer solutions, grease and toothpaste are examples of non-Newtonian fluids. It is well known that the rheological properties of many fluids in engineering and industrial applications are not well modeled by Navier–Stokes equations due to their non-Newtonian behaviour [25, 28-29]. In the past few decades there has been significant work on flows of non Newtonian fluids, not only because of their non-linearity which occur in the inertial part, but also in the surface forces of the governing equations. The study of thermal criticality and heat transfer plays an important role during the handling and processing of non-Newtonian fluids [6,21]. Rajagopal [26] studied in detail the general thermodynamics, stability and uniqueness of fluids of the differential type with the fluid of third grade being a special case. For problems involving heat transfer for fluid of third grade, a complete thermodynamics analysis of the constitutive function has been performed by Fosdick and Rajagopal [6]. Similar studies with respect to non-Newtonian fluid are also reported in [24, 31]. Mathematically the thermal boundary layer equation for non-Newtonian fluids constitute a nonlinear problem and the long-term behavior of the solutions in space will provide us an insight into inherently complex physical process of thermal criticality in the system. The aim of this article is to find the approximate analytical expressions of velocity and temperature for a reactive third grade liquid flowing steadily between two parallel isothermal plates ([21, 8, 22-23). 2. Mathematical Model We consider the steady flow of an incompressible third grade reactive fluid placed between two parallel isothermal plates (see Fig. 1). It is assumed that the fluid motion is induced by applied axial pressure gradient. We choose x -axis parallel to the plate and y -axis normal to it. For hydrodynamically and thermally developed flow, both velocity and temperature fields depend on y only. Following ([6, 24, 31]) and neglecting the reacting viscous fluid consumption, the one

dimensional governing equations for the momentum and heat balance can be written as;

dxdP

dydu

dyud

dyud

2

2

232

26 (1)

0

2

0

2

3

2

2

2

RTE

AeQC

dydu

dydu

dy

Tdk (2)

Subject to the boundary conditions:

,)(,0)(,0)0()0( 0TaTaudydT

dydu

(3)

where the additional Arrhenius kinetics term in energy balance equation is due to [21,7]. Here Tis the absolute temperature, U the fluid characteristic velocity, 0T the plate temperature, k the thermal conductivity of the material, Q the heat of reaction, A the rate constant, E the activation energy, R the universal gas constant, 0C the initial concentration of the reactant species, a the channel half width, 3 the material coefficient, P the modified pressure and is the fluid



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dynamic viscosity coefficient [11, 27, 3]. We introduce the following dimensionless variables into eqns. (1)-(3);

UGuW

RkT

eCQEAaayy

RT

TTE RTE

,,,)(

20

02_

20

0 0

(4)

2

223

20

02

22,,,

0

aGU

dxdP

UaG

ERT

CQAaeUGm

RTE

(5)

And obtain the dimensionless governing equation together with the corresponding boundary conditions as (neglecting the bar symbol for clarity);

162

2

2

2

2

dydW

dyWd

dyWd (6)

0

2122

1

2

2

dydW

dydWm

e

dy

d

(7)

with

,0)1(,0)1(),0()0( W

dyd

dydW (8)

where , , , m represent the Frank–Kamenetskii parameter, activation energy parameter, the dimensionless non-Newtonian parameter and the viscous heating parameter, respectively. In the following sections, Eqns. (6)–(8) are solved using both perturbation and multivariate series summation techniques ([6, 21, 26, 31, 8]). 3. Solution of the non-linear boundary value problem using Homotopy analysis method HAM is a non perturbative analytical method for obtaining series solutions to non-linear equations and has been successfully applied to numerous problems in science and engineering [1-5, 9-20, 27, 30]. In comparison with other perturbative and non- perturbative analytical methods, HAM offers the ability to adjust and control the convergence of a solution via the so-called convergence-control parameter. Because of this, HAM has proved to be the most effective method for obtaining analytical solutions to highly nonlinear differential equations. Previous applications of HAM have mainly focused on non-linear differential equations in which the non-linearity is a polynomial in terms of the unknown function and its derivatives. As seen in (1), the non-linearity present in electro hydrodynamic flow takes the form of a rational function, and thus, poses a greater challenge with respect to finding approximate solutions analytically. Our results show that even in this case, HAM yields excellent results.



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Liao [12-16] proposed a powerful analytical method for non-linear problems, namely the Homotopy analysis method. This method provides an analytical solution in terms of an infinite power series. However, there is a practical need to evaluate this solution and to obtain numerical values from the infinite power series. In order to investigate the accuracy of the Homotopy analysis method (HAM) solution [1-5, 9-11, 17-20, 27, 30] with a finite number of terms, the system of differential equations were solved. The Homotopy analysis method is a good technique comparing to another perturbation method. The approximate analytical expressions of the velocity field and the temperature field using the Homotopy analysis method are as follows:

)1(2

)1(21)( 42 yhyyW

(9)

1512)1(

24)1(

4

1512

)1(24

)1(4

)1(2

)(

22

64

4222

2

mmh

ymmy

yy

h

yy

(10)

4. Previous work The approximate analytical solutions for the velocity and temperature fields using the Hermite-Pade’ approximation are given by;

)()1(

.)1()1(88

)1()1()1(12

)1()1()1(2

)1(21

21

21)(

54234

2342

422

2222

42

Oyyyy

yyyyy

yyy

yyyyy

yyyW

(11)



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)(

245

1807

2809

10511

9926

1001240

19517

41

241

301

283

154

338

918

241

3601

8401

4201

4952

30038

13651

21

121

151

143

158

3316

9116

21

121

151

143

158

3316

9116

)(

3

2

23

452

22

2232

4252

468

102123

144165

2

345

246

82103

124145

O

m

mmm

mmy

mymy

mymy

mymy

ymyym

ymym

ymym

mmm

mmm

ymyym

ymym

ymym

y

(12)

5. Results and Discussions

Figure.1 is the Geometry of the problem of Third grade reactive fluid. Figure. 2 shows that the dimensionless velocity )(yW versus the dimensionless distance y . From Fig.3, it is evident that when the dimensionless non- Newtonian parameter increases the dimensionless velocity

)(yW decreases. Figure.3 shows that the dimensionless temperature )(y versus the dimensionless distancey . From Fig.3, it is evident that when the Frank- Kamenetskii parameter increases the

dimensionless temperature )(y also increases for some fixed values of the activation energy parameter , dimensionless non- Newtonian parameter and the viscous heating parameter m . Figure.4 shows that the dimensionless temperature )(y versus the dimensionless distancey . From Fig.4 (a),(b) and (c) it is evident that when the viscous heating parameter m increases

the dimensionless temperature )(y also increases for some fixed values of the activation energy parameter , dimensionless non- Newtonian parameter and when the Frank- Kamenetskii parameter (i) 1.0 (ii) 15.0 and (iii) 3.0 .



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Figure.5 shows that the dimensionless temperature )( y versus the dimensionless distance y . From Fig.5 (a), (b) and (c) it is evident that when the dimensionless non-Newtonian parameter increases the dimensionless temperature )(y decreases for some fixed values of the activation energy parameter , the viscous heating parameter m and when the Frank- Kamenetskii parameter (i) 1.0 (ii) 15.0 and (iii) 3.0 . Figure.6 shows that the dimensionless temperature )(y versus the dimensionless distancey . From Fig.6 (a), (b) and (c) it is evident that when the activation energy parameter increases

the dimensionless temperature )(y will coincide for some fixed values of the viscous heating parameter m dimensionless non-Newtonian parameter and when the Frank- Kamenetskii parameter (i) 1.0 (ii) 15.0 and (iii) 3.0 . 6. Conclusion

The non-linear boundary value problem for the thermal criticality of a reactive third-grade liquid flowing steadily between parallel isothermal plates has been solved analytically and compared with the Hermite-Pade approximation. Analytical expressions of the velocity and temperature fields can be derived by using the Homotopy analysis method (HAM). The primary result of this work is simple and approximate expressions of the dimensionless velocity and temperature for all values of the dimensionless parameters and m , respectively. These analytical results reveal that the non-Newtonian third-grade reactive fluid is more thermally stable than the Newtonian reactive fluid in industrial and thermal engineering processes. This method is an extremely simple and is also a promising method to solve other strongly non-linear boundary value problem in science and engineering. 7. References

1. Abbasbandy. S., “Soliton solutions for the FitzhughNagumo equation with the homotopy analysis method”, Appl Math Model 32, (2008) 2706–2714.

2. Ananthaswamy. V., Eswari. A. and Rajendran. L., Non‐linear reaction ‐diffusion process in a thin membrane and Homotopy analysis method, International Journal of Automation and Control Engineering, 2(1), p. 10 (2013).

3. Ananthaswamy. V., Ganesan. S.P. and Rajendran. L., Approximate analytical solution of non-linear reaction-diffusion equation in microwave heating model in a slab: Homotopy analysis method, International Journal of Mathematical Archive, 4(7), p.178 (2013).

4. Cheng. S., Liao. J., Mohapatra. R.N. and Vajravelu. K., “Series solutions of nano boundary layer flow by means of the Homotopy analysis method. J Math Anal Appl, 343, (2008) 233–245.

5. Domairry. G., Bararnia. H., “An approximation of the analytical solution of some nonlinear heat transfer equations: a survey by using Homotopy analysis method”,Adv. Studies TheorPhys. 2, (2008) 507-518.

6. Fosdick. R.L., Rajagopal. K.R., Thermodynamics and stability of fluids of third grade, Proc.R. Soc. Lond., A 339 (1980) 351.

7. Frank Kamenetskii. D.A., Diffusion and heat transfer in chemical kinetics, Plenum Press, New York, 1969.



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8. Guttamann. A.J., in: Domb. C., Lebowitz. J.K., (Eds.), Asymptotic analysis of power series expansions-Phase transitions and critical phenomena, Academic Press, New York, 1989, pp.1–234.

9. Hayat. T., Naz. R., Sajid. M., “On the omotopy solution for poiseuille flow of a fourth grade Fluid”, Commun Nonlinear Sci Numer Simul. 15, (2010) 581–589.

10. Hayat. T., Abbas. Z., “Heat transfer analysis on MHD flow of a second grade fluid in a channel with porous medium” Chaos Solitons Fractals, 38, (2008) 556–567.

11. Jafari. H., Chun. C., Saeidy S.M., “Analytical solution for non-linear gas dynamic Homotopy analysis method”, Appl. Math. 4, (2009) 149-154.

12. Liao. S.J., “The proposed Homotopy analysis technique for the solution of non linear Problems”, Ph.D. Thesis, Shanghai Jiao Tong University, (1992).

13. Liao. S.J., “An approximate solution technique which does not depend upon small parameters: a special example”, Int. J. Non-Linear Mech. 30, (1995), 371-380.

14. Liao. S.J., “Beyond perturbation introduction to the Homotopy analysis method”, 1st edn., Chapman and Hall, CRC press, Boca Raton (2003) 336.

15. Liao. S.J., “On the Homotopy analysis method for non-linear problems”, Appl. Math. Comput.147, (2004) 499-513.

16. Liao. S.J., An optimal Homotopy-analysis approach for strongly non-linear differential equations, Commun.Nonlinear Sci. Numer. Simulat. 15, (2010) 2003-2016.

17. Liao. S.J., “The Homotopy analysis method in non-linear differential equations”, Springer and Higher education press, (2012).

18. Liao. S.J., “An explicit totally analytic approximation of blasius viscous flow problems”, Int J Nonlinear Mech, 34, (1999)759–778.

19. Liao. S.J., “On the analytic solution of magnetohydrodynamic flows non-Newtonian fluids over a stretching sheet”, J Fluid Mech 488, (2003) 189–212.

20. Liao. S.J., “A new branch of boundary layer flows over a permeable stretching plate”, Int J Nonlinear Mech; 42 (2007), 819–830.

21. Makinde. O.D., On steady flow of a reactive variable viscosity fluid in a cylindrical pipe with an isothermal wall, Int. J. Numer. Meth. Heat Fluid Flow 17 (2) (2007) 187–194.

22. Makinde. O.D., Exothermic explosions in a slab: a case study of series summation technique, Int. Commun. Heat Mass Transf. 31 (8) (2004) 1227–1231.

23. Makinde. O.D., Thermal ignition in a reactive viscous flow through a channel filled with a porous medium, ASME, J. Heat Transfer, 128 (2006) 601–604.

24. Massoudi. M., Christe. I., Effects of variable viscosity and viscous dissipation on the flow of a third grade fluid in a pipe, Int. J. Non-Linear Mech. 30 (1995) 687.

25. Oluwole Daniel Makinde, Hermite- Pade approximation approach to thermal criticality for a reactive third- grade liquid in a channel with isothermal walls, Int. Commu. Heat and Mass Transfer 34 (7), 870-877, 2007.

26. Rajagopal. K.R., On boundary conditions for fluids of the differential type: Navier-Stokes equations and related non-linear problems, Plenum press, New York, 1995, p. 273.

27. Saravanakumar. K., Ananthaswamy. V., Subha. M., and Rajendran. L., Analytical Solution of nonlinear boundary value problem for fin efficiency of convective straight Fins with temperature-dependent thermal conductivity, ISRN Thermodynamics, Article ID 282481, p.1 (2013).

28. Siddiqui. A.M., Ahmed. M., Ghori. Q.K., Couette and Poiseuille flows for non Newtonian fluids, Int. J. Nonlinear Sci. Numer. Simul. 7 (1) (2006) 15–26.



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29. Siddiqui.A.M., Mahmood. R., Ghori. Q.K., Thin film flow of a third grade fluid on a moving belt by He’s homotopy perturbation method, Int. J. Nonlinear Sci. Numer. Simul. 7(1) (2006) 1–8.

30. Tan Y. Y., H. Xu, S. J. Liao, “Explicit series solution of travelling waves with a front of fisher equation”, Chaos Solitons Fractals, 31, (2007) 462–472.

31. Yurusoy. M., Pakdemirli. M., Approximate analytical solutions for the flow of a third grade fluid in a pipe, Int. J. Non-Linear Mech. 37 (2002) 187.

Figure: 1 Geometry of the problem

Figure: 2 Dimensionless velocity )(yW versus the dimensionless distance y . The velocity profile were computed using the eqn. (9) for various values of the dimensionless non -Newtonian parameter and 525.0h



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Figure: 3 Dimensionless temperature )(y versus the dimensionless distance y . The temperature profile were computed using the eqn. (10) for various values of the Frank- Kamenetskii parameter and 183.0h



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Figure: 4 Dimensionless Temperature )( y profile versus the dimensionless distance y . The temperature profile were computed using the eqn. (10) for various values of the viscous heating parameter m and 183.0h when the Frank- Kamenetskii parameter (i) 1.0 (ii) 15.0 and (iii) 3.0



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Figure: 5 Dimensionless Temperature )( y profile versus the dimensionless distance y . The temperature profile were computed using the eqn. (10) for various values of the dimensionless non- Newtonian paramameter and

183.0h when the Frank- Kamenetskii parameter (i) 1.0 (ii) 15.0 and (iii) 3.0



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Figure: 6 Dimensionless Temperature )( y profile versus the dimensionless distance y . The temperature profile were computed using eqn. (10) for various values of the activation energy parameter and 183.0h when the Frank- Kamenetskii parameter (i) 1.0 (ii) 15.0 and (iii) 3.0

Figure: 7 The h curve to indicate the convergence region for )9.0(W when 15.0



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Figure: 8 The h curve to indicate the convergence region for )9.0('W when 15.0

Figure: 9 The h curve to indicate the convergence region for )9.0( when 3 , 1.0 , 2m , 3.0 .

Figure: 10 The h curve to indicate the convergence region for )9.0(' when 3 , 1.0 , 2m , 3.0 Appendix A: Basic concept of Homotopy analysis method Consider the following differential equation [12-14]:

0)]([ tuN (A.1) Where N is a nonlinear operator, t denotes an independent variable, )(tu is an unknown function. For simplicity, we ignore all boundary or initial conditions, which can be treated in the similar way. By means of generalizing the conventional Homotopy method, Liao [15] constructed the so-called zero-order deformation equation as:

)];([)()]();([)1( 0 ptNtphHtuptLp (A.2)



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where 1,0p is the embedding parameter, 0h is a nonzero auxiliary parameter, 0)( tH is an auxiliary function, L an auxiliary linear operator, )(0 tu is an initial guess of )(tu , ):( pt is an unknown function. It is important, that one has great freedom to choose auxiliary unknowns in HAM. Obviously, when 0p and 1p , it holds:

)()0;( 0 tut and )()1;( tut (A.3) respectively. Thus, as p increases from 0 to 1, the solution );( pt varies from the initial guess

)(0 tu to the solution )(tu . Expanding );( pt in Taylor series with respect to p , we have:

1

0 )()();(m

mm ptutupt (A.4)

where

0);(

!1)(

pm

mm

p

ptm

tu (A.5)

If the auxiliary linear operator, the initial guess, the auxiliary parameter ,h and the auxiliary function are so properly chosen, the series (A.4) converges at 1p then we have:

1

0 )()()(m

m tututu . (A.6)

Differentiating the eqn. (A.2) for m times with respect to the embedding parameter p , and then setting 0p and finally dividing them by m !, we will have the so-called thm order deformation equation as:

)()(][ 11

mmmmm uthHuuL (A.7) where

1

11

)];([)!1(

1)(

m

mmm

p

ptNm

u (A.8)

and

.1 1,,1 ,0

mm

m (A.9)

Applying 1L on both side of the eqn. (A.7), we get

)]()([)()( 11

1

mmmmm utHhLtutu (A.10)

In this way, it is easily to obtain mu for ,1m at thM order, we have

M

mm tutu

0)()( (A.11)

When M , we get an accurate approximation of the original eqn. (A.1). For the convergence of the above method we refer the reader to Liao [12]. If the eqn. (A.1) admits unique solution, then this method will produce the unique solution. Appendix B: Solution of the Boundary value problem using HAM



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162

2

2

2

2

dydW

dy

Wd

dy

Wd (B.1)

0

2122

1

2

2

dydW

dydWm

e

dy

d

(B.2)

When is small, then the eqn. (B.2) can be written as

0

2

)1(

4

2

2

2

2

dydWm

dydWm

dy

d

(B.3)

We construct the Homotopy for the eqns. (B.1) and (B.3) are as follows:

161

)1( 2

2

2

2

2

2

2

dydW

dy

Wd

dy

Wd

hpdy

Wdp

(B.4)

4

2

2

2

2

2

2

2

)1()1(

dydWm

dydWm

dy

d

hpdy

dp

(B.5)

The approximate solution of the eqns. (B.3) and (B.4) are as follows: .....2

210 WppWWW (B.6)

.....22

10 pp (B.7) The initial approximations are as follows:

0)1(;0)0(,0)1(;0)0( 0'00

'0 WW (B.8)

,...3,2,1,0)1(;0)0(

,0)1(;0)0('

'

i

WW

ii

ii

(B.9)

Substituting the eqns. (B.6) and (B.7) into the eqns. (B.4) and (B.5) we get



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22

210

22

210

2

22

210

2

22

210

2

...)(

....)(

6

1...)(

1...)(

)1(

dyWppWWd

dy

WppWWd

dy

WppWWd

hp

dy

WppWWdp

(B.10)

42

210

22

210

22

210

22

10

22

210

2

22

210

2

....)(

.2

....)(

.....)(

....)(1

....)(

....)()1(

dyWppWWd

m

dyWppWWd

mpp

pp

dy

ppd

ph

dy

ppdp

(B.11)

Comparing the coefficients of like powers p in the eqns. (B.10) and (B.11) we get

01:2

20

dy

Wdp (B.12)

0: 2

20

dydp (B.13)

0

1

6

:

2

20

2

20

2

21

21

dydW

dy

Wd

dy

Wdh

dy

Wdp

(B.14)



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0

2

)(

:

40

20

200

21

21

dydW

m

dydW

mhdy

dp

(B.15)

Solving the eqns. (B.12)-(B.14), (B.15) and using the boundary conditions eqns. (B.8) and (B.9) we can obtain the following results:

2)1()(

20

yyW (B.16)

)1(2

)( 41 yhyW

(B.17)

)1(2

)( 20 yy

(B.18)

)1(15

)1(12

)1()1(24

)1()1(4

)(

6

442

22

1

ymh

ymhyh

yhy

(B.19)

According to the HAM, we can conclude that ...lim 10

1

WWWW

p (B.20)

...lim 101

p

(B.21)

After putting the eqns. (B.16) and (B.17) into an eqn. (B.20) and (B.18) and (B.19) into an eqn. (B.21) we get the solutions in the text eqns. (9) and (10).

Appendix C: Determining the region of h for validity The analytical solution should converge. It should be noted that the auxiliary parameter h controls the convergence and accuracy of the solution series. The analytical solution represented by (9) and (10) contains the auxiliary parameter h which gives the convergence region and rate of approximation for the Homotopy analysis method. In order to define region such that the solution series is independent of h a multiple of h curves are plotted. The region where the velocity profiles )(yW and )(' yW , )9.0( and )9.0(' versus h is a horizontal line known as the convergence region for the corresponding function. The common region )(yW among and its derivatives are known as the overall convergence region. To study the influence of h on the convergence of solution, h- curves of )9.0(W and )9.0('W are plotted in Figs.(7) and (8) respectively for 15.0 these figures clearly indicate that the valid region of h is about (-0.55 to -0.46). Similarly we can find the value of the convergence control parameter for different values of the constant parameters. The common region )(y among and its derivatives are known as the overall convergence region. To study the influence of h on the convergence of solution, h- curves of



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)9.0( and )9.0(' are plotted in Figs. (9) and (10) respectively for 3 , 1.0 , 2m and 3.0 these figures clearly indicate that the valid region of h is about (-0.22 to -0.18). Similarly we can find the value of the convergence control parameter for different values of the constant parameters. Appendix D: Nomenclature

Symbol Meaning W Dimensionless velocity

Dimensionless temperature

T Absolute Temperature

U Fluid characteristic velocity

0T Plate temperature

k Thermal conductivity of the material

Q Heat of reaction

A Rate constant

E Activation energy

R Universal gas constant

0C The initial concentration of the reactant species

a Channel half width

3 Material coefficient

P Modified pressure Fluid dynamic viscosity coefficient

Frank- Kamenetskii parameter Action energy parameter Dimensionless non Newtonian parameter

m Viscous heating parameter

fC Skin- friction parameter

Nu Nusselt number

wl Shear stress

wq Heat flux evaluated at the wall

)(U

Local representation of an algebraic function of

H Constant

c Critical point

Exponent Algebraic function

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