Approximate analytical solution of non-linear boundary value problem in steady state flow of a...

Int. Journal of Applied Sciences and Engineering Research, Vol. 2, No. 5, 2013 www.ijaser.com

© Copyright 2013 - Integrated Publishing Association [email protected]

Research article ISSN 2277 – 8442

—————————————

*Corresponding author (e-mail:[email protected])

Received on Sep. 15, 2013; Accepted on Sep. 20, 2013; Published on Oct. 29, 2013

Approximate analytical solution of non-linear boundary value

problem in steady state flow of a liquid film: Homotopy

perturbation method

( 1V. Ananthaswamy, 2SP. Ganesan, 3L. Rajendran*)

1,3 The Madura College, Madurai, Tamil Nadu, India, 2Syed Ammal Engineering College, Ramnad,

Tamil Nadu, India

DOI: 10.6088/ijaser.020500010

Abstract: In this paper the non-linear boundary value problem in thermal stability of boundary layer flows

of a temperature-dependent viscosity liquid film with adiabatic free surface along an inclined heated plate

is discussed. The analytical expression of the temperature and velocity can be obtained using Homotopy

perturbation method (HPM) for various values of the relevant parameters. We also compared our analytical

result with perturbation method and show that the present approach is less computational and are

applicable for solving other non-linear boundary value problem.

Keywords: Inclined plate; Liquid film; Variable viscosity; Adiabatic free surface; Thermal criticality;

Homotopy perturbation method.

1. Introduction

The study of thermal boundary layer flows of variable viscosity fluids on a heated inclined plate not only

possesses a theoretical appeal but also models some fluid transport mechanisms encountered in industries

and engineering systems (Schlichting, 1968). Amongst others, we can name hot rolling, wire drawing,

fiberglass and paper production, gluing of labels on hot bodies, drawing of plastic films, etc. When a cooler

fluid flows around a hot body, the temperature of the fluid will rise in a thin layer around the body and in a

wake behind it. This thin layer is known as the thermal boundary layer. In this layer, flow and thermal

phenomena interact nonlinearly and are governed by the so-called thermal boundary layer equations

(Makinde et al, 2006 and 2007, Lawrence et al, 1992). In classical treatment of thermal boundary layers,

the kinematic viscosity is assumed to be constant; however, experiments indicate that this assumption only

makes sense if temperature does not change rapidly for the application of interest. Indeed, for liquids,

experimental data (Bender et al, 1978, Cebeci et al, 1984, Guttamann et al, 1989, Hunter et al, 1979 and

Makinde et al, 2001) shows that viscosity decreases with temperature.

Recently, the second law analysis of heat transfer of a laminar falling liquid film of constant viscosity along

an inclined heated plate was investigated by (Makinde et al, 2005 and Saouli et al, 2004). Meanwhile,

several authors have investigated the effects of temperature-dependent viscosity on the flow of

non-Newtonian fluids in a channel under various conditions, e.g. (Makinde et al, 2004, Massoudi et al,

1995, Szeri et al,1985 and Yurusoy et al, 2002), etc. The objective of this paper is to examine the effects of

temperature-dependent viscosity and viscous dissipation on the overall flow structure including the

bifurcation study in hydrodynamically and thermally developed flow on an inclined heated plate. The

mathematical formulation of the problem is established and solved in the following section. Both present

mailto:[email protected]

Approximate analytical solution of non-linear boundary value problem in steady state

flow of a liquid film: Homotopy perturbation method

Ananthaswamy et al.,

Int. Journal of Applied Sciences and Engineering Research, Vol. 5, Issue 6, 2013

570

and previous methods are presented and discussed with respect to the relevant parameters.

2. Mathematical formulation of the problem

Consider an inclined heated plate placed in a parallel stream of a hydrodynamically and thermally

developed variable viscosity liquid film. It is assumed that the characteristic length in flow direction is

typically large as compared with that across the film. This suggests that lubrication theory can be employed

and the inertia terms in the governing momentum and energy balance equations can be easily neglected

since we are dealing with a very small aspect ratio problem (see Fig. 1). Under these conditions, the

governing momentum and energy balance equations are of the form (Makinde et al, 2006, Massoudi et al

1995, Saouli et al, 2004 and Schlichting, 1968)

0)sin(,0

2

2

2

gyd

ud

yd

d

yd

ud

kyd

Td (1)

,0yd

ud,0

yd

Td on y (2)

and

,0,0 TTu on 0y (3)

where u is the axial velocity, T is the absolute temperature, 0T the incline plane wall temperature,

k the thermal conductivity, liquid film thickness, inclination angle, fluid density, g

gravitational acceleration, y vertical distance. The temperature dependency of dynamic viscosity

can be expressed as

)(0

0TTe (4)

where 0 is the fluid viscosity at reference temperature 0T and the coefficient determines the

strength of dependency between and T . We introduce the dimensionless variables in Eqs. (1) – (3)

as follows:

02

0

00

22

0

0 ,)sin(

,))sin((

, Tg

uu

Tk

gyy

T

TT

(5)

Using these dimensionless variables we can obtain the dimensionless governing equation together with the

corresponding boundary condition as follows:

eydy

duey

dy

d)1(,0)1( 2

2

2

(6)

with

0)0(,0)1(

,0)0( udy

d (7)

where , represents Brinkmann number and variable viscosity parameter respectively.





571

2.1 Solution of the boundary value problem using HPM

Linear and non-linear phenomena are of fundamental importance in various fields of science and

engineering. Most models of real – life problems are still very difficult to solve. Therefore, approximate

analytical solutions such as Homotopy perturbation method (HPM) (Ghori et al, 2007, Ozis et al, 2007, Li

et al, 2006, Mousa et al, 2008, He, 1999, 2003, Ariel, 2010, Loghambal et al, 2010, Meena et al, 2010,

Anitha et al, 2011 and Ananthaswamy et al, 2012, 2013) were introduced. This method is the most

effective and convenient ones for both linear and non-linear equations. Perturbation method is based on

assuming a small parameter. The majority of non-linear problems, especially those having strong

non-linearity, have no small parameters at all and the approximate solutions obtained by the perturbation

methods, in most cases, are valid only for small values of the small parameter. Generally, the perturbation

solutions are uniformly valid as long as a scientific system parameter is small. However, we cannot rely

fully on the approximations, because there is no criterion on which the small parameter should exists. Thus,

it is essential to check the validity of the approximations numerically and/or experimentally. To overcome

these difficulties, HPM have been proposed recently.Recently, many authors have applied the Homotopy

perturbation method (HPM) to solve the non-linear boundary value problem in physics and engineering

sciences (Ghori et al, 2007, Ozis et al, 2007, Li et al, 2006, Mousa et al, 2008). Recently this method is

also used to solve some of the non-linear problem in physical sciences (He, 1999, 2003). This method is a

combination of Homotopy in topology and classic perturbation techniques. Ji-Huan He used to solve the

Lighthill equation(He, 1999), the Diffusion equation (He, 2003) and the Blasius equation (He, 2003 and

Ariel, 2010). The HPM is unique in its applicability, accuracy and efficiency. The HPM uses the imbedding

parameter p as a small parameter, and only a few iterations are needed to search for an asymptotic solution.

The approximate analytical solution of Eqns. (6) and (7) using Homotopy perturbation method is given by

28

)1(

132

)1(

20

)1(

288

56

)1(

12

)1(

721330560

101

2016

11)1(1

12)(

812523

8422324

yyy

yyyy

(8)

610222

622

)1(10)1(315308640

120

71)1(36

722)(

yyyy

yyyy

yyu

(9)

3. Previous works

The solution for the temperature and velocity fields using Hermite-Pad’e approximation is as follows:

)()81218123()22()2(2016

)22()2(12

)(

323422

2

Oyyyyyyyy

yyyyy

(10)





572

)()1232768451183()2(6048

)32()2(72

)2(2

1)(

3234562222

222

Oyyyyyyy

yyyyyyyu

(11)

4. Results and Discussion

The main interest in this section is to investigate the effects of Brinkmann number and the variable

viscosity parameter using Homotopy perturbation method. Figures (2) and (3) show the dimensionless

radial distance y versus the dimensionless temperature )(y . From Fig.(2), it is evident that, when the

variable viscosity parameter increases, the dimensionless temperature )(y also increases for the

fixed value of . From Fig. (3), it is clear that when the Brinkmann number increases, the

dimensionless temperature )(y also increases for all values . In Figures (4) and (5) the

dimensionless velocity )(yu versus the radial distance y is plotted. From Fig. (4), we notice the

dimensionless velocity )(yu increases, when increases for the fixed values of the Brinkmann

number . From Fig. (5), we conclude that when the Brinkmann number increases, the

dimensionless velocity )(yu also increases for the various values of .

Figure 1: Geometry of Problem

Figure 2: Dimensionless temperatures )(y versus the dimensionless vertical distance y . The

velocity were computed using eqn. (8) for various values of the dimensionless parameters and





573

Figure 3: Dimensionless temperature )(y versus the dimensionless vertical distance y . The velocity

were computed using eqn. (8) for various values of the dimensionless parameters and

Figure 4: Dimensionless velocity )(yu versus the dimensionless vertical distance y . The velocity

were computed using eqn. (9) for various values of the dimensionless parameters and

Figure 5: Dimensionless velocity )(yu versus the dimensionless vertical distance y . The velocity were

computed using eqn. (9) for various values of the dimensionless parameters and





574

5. Conclusions

The analytical expressions of the temperature )(y and the velocity fields )(yu in the

hydrodynamically and thermally developed variable viscosity liquid film along an inclined heated plate are

derived by using the HPM for all values of dimensionless parameters and . We compared our

analytical results to the Perturbation technique. The HPM is an extremely simple compared to other

method and it is also a promising method to solve other non-linear equations. This method can be easily

extended to find the solution of all other non-linear equations.

Acknowledgement

This work is supported by the University Grant Commission (UGC) Minor project No: F. MRP-4122/12

(MRP/UGC-SERO), Hyderabad, Government of India. The authors are thankful to Shri. S. Natanagopal,

Secretary, Madura College Board and Dr. R. Murali, Principal, The Madura College (Autonomous),

Madurai, Tamil Nadu, India for their constant encouragement.

6. References

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value problems in a porous catalyst particles. International Journal of Mathematical Archive, 3 (3),

810-821.

2. Ananthaswamy, V., and Rajendran, L. 2013. Analytical solution of non-isothermal

diffusion-reaction processes and effectiveness factors. ISRN Physical Chemistry, 2013, Artricle ID

487240, 1-14.

3. Anitha, S., Subbiah, A., Subramaniam, S., and Rajendran, L. 2011. Analytical solution of

amperometric enzymatic reactions based on Homotopy perturbation method. Electrochimica

Acta, 56, 3345-3352.

4. Ariel, P.D. 2010. Alternative approaches to construction of Homotopy perturbation algorithms.

Nonlinear Science Letters. A., 1, 43-52.

5. Bender, C., and Orszag, S.A. 1978. Advanced Mathematical Methods for Scientists and Engineers.

McGraw-Hill.

6. Cebeci, T., and Bradshaw, P. 1884. Physical and Computational Aspects of Convective Heat

Transfer, Springer-Verlag. New York.

7. Ghori, Q.K., Ahmed, M., and Siddiqui, A.M. 2007. Application of Homotopy perturbation method

to squeezing flow of a Newtonian fluid. International Journal of Nonlinear Science and Numerical

Simulation, 8, 179-184.

8. Guttamann, A.J. 1989. Asymptotic analysis of power-series expansions, in: C. Domb, J.K.

Lebowitz (Editions). Phase Transitions and Critical Phenomena, Academic Press, New York,

1–234.

9. He, J.H. 1999. Homotopy perturbation technique. Computational Methods and Applied

Mechanical Engineering, 178, 257-262.

10. He, J.H. 2003. Homotopy perturbation method: a new nonlinear analytical technique. Applied

Mathematics and Computing, 135, 73-79.

11. He, J.H. 2003. A simple perturbation approach to Blasius equation. Applied Mathematics and





575

Computing, 140, 217-222.

12. Hunter, D.L., and Baker, G.A. 1979. Methods of series analysis III: Integral

approximant methods. Physics. Review. B 19, 3808–3821.

13. Lawrence, P.S., and Rao, B.N. 1992. Interation of boundary layer and free stream flows over an

inclined wall. Journal of Physics D: Applied Physics, 25, 559-561.

14. Li, S.J., and Liu, Y.X. 2006. An improved approach to nonlinear dynamical system identification

using PID neural networks. International Journal of Nonlinear Science and Numerical Simulation,

7, 177-182.

15. Loghambal, S., and Rajendran, L. 2010. Mathematical modeling of diffusion and kinetics of

amperometric immobilized enzyme electrodes. Electrochima Acta, 55, 5230-5238.

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variation and energy dissipation, Quaestion. Math. 24, 93–104.

17. Makinde, O.D. 2004. Exothermic explosions in a slab: a case study of series summation technique.

International Communication of Heat Mass Transfer, 31, 1227–1231.

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summation technique. Mech. Res. Commun. 32, 191–195.

19. Makinde, O.D. 2006. Laminar falling liquid film with variable viscosity along an inclined heated

plate, Applied Mathematics and Computing, 175, 80-88.

20. Makinde, O.D. 2007. Hermite-Pade’ approximation approach to steady flow of a liquid film with

adiabatic free surface along an inclined heat plate. Physica A 381, 1-7.

21. Massoudi, M., and Christie, I. 1995. Effects of variable viscosity and viscous dissipation on the

flow of a third grade fluid in a pipe. International Journal of Nonlinear Mechanical. 30, 687–699.

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biosensors and system of non-linear equations – Homotopy perturbation approach. Journal of

Electro Analytical Chemistry, 644, 50-59.

23. Mousa, M.M., Ragab, S.F., and Nturforsch, Z. 2008. Application of the Homotopy perturbation

method to linear and nonlinear Schrödinger equations. Zeitschrift für Naturforschung, 63,

140-144.

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determining frequency-amplitude relation of a nonlinear oscillator with discontinuities.

International Journal of Nonlinear Science and Numerical Simulation, 8 , 243-248.

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inclined heated plate. International Communication of Heat and Mass Transfer, 31, 879–886.

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plates. International Journal of Nonlinear Mechanics, 20, 91–101.

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third-grade fluid in a pipe. International Journal of Nonlinear Mechanics, 37, 187–195.

Appendix A. Basic concepts of the Homotopy perturbation method

To explain this method, let us consider the following function:

r ,0)()( rfuDo (A.1)

with the boundary conditions of





576

r ,0) ,(

n

uuBo (A.2)

where oD is a general differential operator, oB is a boundary operator, )(rf is a known analytical

function and is the boundary of the domain . In general, the operator oD can be divided into a

linear part L and a non-linear part N . Equation (A. 1) can therefore be written as

0)()()( rfuNuL (A.3)

By the Homotopy technique, we construct a Homotopy ]1,0[:),( prv that satisfies

.0)]()([)]()()[1(),( 0 rfvDpuLvLppvH o (A.4)

.0)]()([)()()(),( 00 rfvNpupLuLvLpvH (A.5)

where p[0, 1] is an embedding parameter, and 0u is an initial approximation of Eq. (A. 1) that satisfies

the boundary conditions. From Eq. (A. 4) and Eq. (A. 5), we have

0)()()0,( 0 uLvLvH (A.6)

0)()()1,( rfvDvH o (A.7)

When p=0, Eq. (A. 4) and Eq. (A. 5) become linear equations. When p =1, they become non-linear

equations. The process of changing p from zero to unity is that of 0)()( 0 uLvL to 0)()( rfvDo .

We first use the embedding parameter p as a “small parameter” and assume that the solutions of Eq. (A.

4) and Eq. (A. 5) can be written as a power series in p :

...22

10 vppvvv (A.8)

Setting 1p results in the approximate solution of Eq. (A. 1):

...lim 2101

vvvvup

(A.9)

This is the basic idea of the HPM.

Appendix. B Solution of non-linear equations (6) and (7) using HPM

In this Appendix, we indicate how Eqns. (8) and (9) in this paper is derived. To find the solution of Eqns.(6)

and (7), when small, Eqns.(6) and (7) reduces to

02

1)1(2

2

2

2

y

dy

d (B.1)

02

1)1(2

y

dy

du (B.2)

We construct the Homotopy as follows





577

02

1)1()1()1(2

2

2

22

2

2

y

dy

dpy

dy

dp (B.3)

02

1)1()1()1(2

y

dy

dupy

dy

dup (B.4)

The analytical solution of (B.1) and (B.2) is

..........22

10 pp (B.5)

..........22

10 uppuuu (B.6)

Substituting (B.5) into (B.3) and (B.6) into (B.4) we get

0

2

)..........()..........(1)1(

)..........(

)1()..........(

)1(

22

210

22

102

2

22

102

2

2

22

102

ppppy

dy

ppd

p

ydy

ppdp

(B.7)

0

2

)..........()..........(1)1(

)..........(

)1()..........(

)1(

22

210

22

10

22

10

22

10

ppppy

dy

uppuud

p

ydy

uppuudp

(B.8)

Comparing the coefficients of like powers of p in (B.7) and (B.8) we get

0)1(: 2

2

02

0 ydy

dp

(B.9)

0)1(: 00 ydy

dup (B.10)

0)1(2

)1(: 20

22

02

2

12

1

yydy

dp (B.11)

0)1(2

)1(: 20

2

011 uyuy

dy

dup

(B.12)

The initial approximations is as follows

,0)0(0)1(',0)0( 000 uand (B.13)

....3,2,1 0,(0)0)1(')0( iuand iii (B.14)

Solving (B.9), (B.10), (B.11) and (B.12) and using the boundary conditions (B.13) and (B.14) we obtain





578

the following results:

40 )1(1

12y

(B.15)

28

)1(

132

)1(

20

)1(

28856

)1(

12

)1(

721330560

101

2016

11 812523842232

1

yyyyy (B.16)

2

2

0

yyu (B.17)

610222

621 )1(10)1(31530

8640120

71)1(36

72yyxyyyyu

(B.18)

According to the HPM, we can conclude that

)(lim 101

yp

(B.19)

)(lim 101

uuyyup

(B.20)

After putting (B.15) and (B.16) into (B.19) and (B.17), (B.18) into (B.20) we obtain the solutions in the

text (7) and (8) respectively.

Appendix C. Nomenclature

Symbol Meaning

u Axial Velocity

T Absolute temperature

0T Incline plane wall temperature

k Thermal conductivity

Liquid film thickness

Inclination angle

Fluid density

g Gravitational acceleration

y Vertical distance

Dynamic viscosity

0 Fluid Velocity

Strength of dependency between and T

Dimensionless temperature

y Dimensionless vertical distance

Brinkmann number

Variable viscosity parameter

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