adomian decomposition method applied to study nonlinear...

Applied Mathematical Sciences, Vol. 6, 2012, no. 98, 4889 - 4909

Adomian Decomposition Method Applied to Study

Nonlinear Equations Arising in non-Newtonian flows

A. M. Siddiqui(a), M. Hameed(b)

(a)Department of Mathematics, Pennsylvania State University, York campus, York, PA 17403 – 3398, USA.

(b)Division of Mathematics & Computer Science, USCUPSTATE 800 University Way, Spartanburg, 29303, USA

B. M. Siddiqui(c), B. S. Babcock(a)

(c)Department of Computer Science, COMSATS Institute of Information Technology, Park Road, Chak Shahzad, Islamabad, Pakistan.

Abstract Adomian’s decomposition method (ADM) is employed to solve nonlinear differential equations which arise in non-Newtonian fluid dynamics. We study basic pipe flow problems of a third grade and 6-constant Oldroyd non-Newtonian fluids. The technique of Adomian decomposition is applied with elegant results. The solutions obtained show the reliability and efficiency of this analytical method. Numerical solutions are also obtained by solving nonlinear ordinary differential equations using Chebyshev spectral method. We present a comparative study of the analytical and numerical solutions. The analytical results agree well with the numerical results, which reveal the effectiveness and convenience of the Adomian decomposition method. Keywords: Decomposition Method; Spectral method; Third grade fluid; Oldroyd fluid; Couette flow; Poiseuille flow

4890 A. M. Siddiqui et al 1. Introduction

In recent years, there has been great interest in the application of a recently developed decomposition method, due to Adomian [1, 2], in solving nonlinear ordinary and partial differential equations. A considerable amount of research work has been invested in applying this method to a wide class of linear and nonlinear ordinary differential equations, partial differential equations and integral equations. The ADM has been successfully applied to solve nonlinear equations in studying many interesting problems arising in applied sciences and engineering [3, 4, 6, 7, 9, 10, 11], and is usually characterized by its higher degree of accuracy. It provides analytical solution in the form of an infinite series in which each term can be easily determined. The ADM has been proved to be an effective and reliable method for handling differential equations, linear or nonlinear. Unlike the traditional methods, the ADM needs no discretization, linearization, spatial transformation or perturbation, thus has some significant advantages over the other traditional analytical techniques. The ADM provides an analytical solution in the form of an infinite convergent power series. The rapid convergence of the series solution obtained by ADM has been discussed by Cherrualt et al. [5] and provides insight into the characteristics and behavior of the solution as in the case with the closed form solution.

Non-Newtonian fluids are of great interest to researchers because of their

practical importance in engineering and industry. The classical Navier-Stokes equations have been proved inadequate to describe and capture the characteristics of complex rheological fluids as well as polymer solutions [12]. These kinds of fluids are generally known as non-Newtonian fluids. Most of the biological and industrial fluids are non-Newtonian in nature. Few examples of such fluids are blood, tomato ketchup, honey, mud, plastics and polymer solutions. The inadequacy of the classical theories to describe these complex fluids has led to the development of different new theories to study non-Newtonian fluids. There are different models which have been proposed to describe the non-Newtonian flow behavior. Among these, the fluids of differential type [12, 13] have received considerable attention. Fluid of a third grade is a subclass of fluids of differential type, which has been studied successfully in various types of flow situations [14, 15] and is known to capture the non-Newtonian affects such as shear thinning or shear thinking as well as normal stresses. The constitutive equations for the third grade fluid, as in the case of most non-Newtonian fluids, give rise to complicated and highly nonlinear model equations. This intrinsic nonlinearity due to the presence of normal stresses makes the problem difficult to solve analytically even in simple geometries. In the present work, we successfully apply for the first time the Adomian decomposition method to solve the problems modeling the flow of a third grade fluid

Adomian decomposition method 4891 and 6-constant Oldroyd fluid in an axisymmetric tube. The decomposition method is seen to provide a direct scheme for solving these nonlinear problems without any need for linearization or any restrictive assumptions. Moreover, the method is found to greatly reduce the size of computational work while maintaining high accuracy. The formulas for Adomian polynomials for many forms of nonlinear terms in differential equations show their effectiveness in our problems as they do in a variety of applications in other fields. We have also investigated this problem numerically and obtained solutions using higher order Chebyshev spectral collocation method. Spectral methods have been successfully used in finding the numerical solution of various problems and in studying computational fluid dynamics problems. Spectral methods are proved to offer a superior intrinsic accuracy for derivative calculations [16]. We present numerical solution for both cases and compare analytical solutions obtained by Adomian decomposition method with numerical solutions. We present a comparative study and find that ADM solutions compare well with the numerical solutions therefore prove that the ADM is powerful and effective which can easily handle wide class of nonlinear problems. 2. Basic Equations The basic equations governing the flow of an incompressible fluid neglecting the thermal effects are:

0v∇⋅ =% %

, (1)

dv f Tdt

ρ ρ= +∇⋅%%%%

. (2)

Here ρ denotes the fluid density, v%

the velocity vector and f%

the body force vector

per unit mass. The operator ddt

denotes the material time derivative and T%

is the

stress tensor which for a third grade fluid is given by T pI S= − +

% % %, (3)

where I%

is the unit tensor and extra stress S%

is defined as 2 2

1 1 2 2 1 1 3 2 1 2 2 1 3 1 1( ) ( )S A A A A A A A A trA Aμ α α β β β= + + + + + +% % % % % % % % % %%

, (4) μ is the coefficient of the shear viscosity, 1 2 1 2, , ,α α β β and 3β material constants and p represents the pressure. The tensors 1 2,A A

% % and 3A

% are given by

1 (grad ) (grad )TA v v= +% % %

, (5)

2 1 1 1(grad ) (grad )TdA A A v v Adt

= + +% % %% %

, (6)

3 2 2 2(grad ) (grad )TdA A A v v Adt

= + +% % %% %

. (7)

4892 A. M. Siddiqui et al The constitutive equation of a 6-constant Oldroyd fluid is given by

T pI S= − +% % %

, (8) wherein the extra stress S

% can be written as

[ ]3 51 1 1 1 1( ) ( )

2 2DSS SA A S tr S A ADt

λ λλ+ + + +%% % % %% % % %

211 2 4 1( )DAA A

Dtμ λ λ= + +%% %

, (9)

where 1 2 3 4, , , ,μ λ λ λ λ and 5λ are material constants. The convected derivative DDt

is

defined as

,TDS dS LS SLDt dt

= − −% %% %% %

(10)

and the first Rivlin–Ericksen tensor 1A%

can also be written as 1 ,TA L L= +

% % % (11)

where (grad ).L v=

% % (12)

3. Problem Formulation We consider steady, isothermal, laminar, fully developed flow of a non-Newtonian incompressible fluid down the axis of a long pipe of circular cross section with diameter 2R. Choosing cylindrical coordinates ( , , )r zθ , we take the axis of symmetry the pipe along the z-axis. For a unidirectional axisymmetric fully developed flow we choose the velocity filed and the extra stress, respectively, as

(0,0, ( ))v u r=%

, (13) ( )S S r=

% %. (14)

Thus the equation of continuity (1) is satisfied identically whereas the equation of motion (2), in the absence of body forces, yields

10 ( )rrp d r Sr r dr∂

= − +∂

, (15)

10 ( )rzp d r Sz r dr∂

= − +∂

. (16)

On differentiating (15) with respect to z and (16) with respect to r, we obtain

2 10 ( )rrp d d rS

z r dz r dr∂ ⎡ ⎤= − + ⎢ ⎥∂ ∂ ⎣ ⎦

, (17)

2 10 ( )rzp d d r S

r z dr r dr∂ ⎡ ⎤= − + ⎢ ⎥∂ ∂ ⎣ ⎦

. (18)

From equations (17), (18) and (14) we conclude that

Adomian decomposition method 4893

1 ( ) constant.rzd r S

r dr= (19)

Now if we substitute (19) into (16) we arrive at

constant.p dpz dz∂

= =∂

(20)

Substituting (20) into (16) and integrating we conclude that

1

2rzCr dpS

dz r= + , (21)

where C1 is a constant of Integration and dpdz

is the constant pressure gradient. In

order to determine the value of the constant C1 we use the symmetry condition (0) 0rzS = , (22)

which yields 1 0C = . (23)

Thus, equation (21) reduces to

2rzr dpS

dz= . (24)

In the next section, we derive the resulting differential equations for the pipe flow under consideration for both fluids. Namely, the third grade fluid and the 6-constant Oldroyd fluid. Third grade Fluid On making use of assumptions (13) and (14) in the constitutive equation for third grade fluid (4), we obtain the expression for stress component rzS , which after substituting into (24), yields a nonlinear governing equation given as

32 02

du du r dpdr dr dz

βμ μ

⎛ ⎞+ − =⎜ ⎟⎝ ⎠

, (25)

where 2 3β β β= + . (26)

The boundary condition for the pipe flow problem is the no slip condition at the walls

( ) 0u r = at r R= . (27) Here we would like to point out that the β represents the non-Newtonian contribution to the flow. If we set 0β = in (25), the equation for the famous Hagen-Poiseuille flow is recovered.

4894 A. M. Siddiqui et al 6-Constant Oldroyd Fluid

As before, we use (13) and (14) in the constitutive equation (9) for 6-constant Oldroyd fluid to obtain stress component rzS . Then the expression for rzS is combined with the momentum equation (24) to yield [8] the first order non-linear differential equation

3 2

1 2 02 2

du du r dp du r dpdr dr dz dr dz

α αμ μ

⎛ ⎞ ⎛ ⎞+ − − =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

, (28)

where 1 1 4 4 2 3 5( )( )α λ λ λ λ λ λ= − − + , (29) 2 1 3 3 1 3 5( )( )α λ λ λ λ λ λ= − − + . (30)

The associated no slip boundary condition for the pipe flow is 0u = at r R= . (31)

4. Adomian's Decomposition Method

To give a brief resume of the decomposition method, we consider the following differential equation

,Lu Nu g+ = (32) where L is a linear operator, N a non-linear operator, g the source term and u the function of independent variable y alone. If we apply 1L− to (32), the result can be written as

1 1u f L g L Nu− −= + − , (33) where the function f represents the term arising from the solution of the homogeneous equation

0Lu = (34) which involves the constants of integration. The decomposition method consists of representing the solution u(y) by the decomposition series,

0

( ) ( )nn

u y u y∞

=

=∑ . (35)

The non-linear term Nu(y) from the series of Adomian polynomials is given by

0

( ) .nn

Nu y A∞

=

=∑ (36)

Here, the Adomian polynomials nA are given by 0 0( )A f u= ,


1 1 00

( )dA u f udu

= ,

2 21

2 2 0 020 0

( ) ( )2!ud dA u f u f u

du du= + ,

32 31

3 3 0 1 2 0 02 30 0 0

( ) ( ) ( )3!ud d dA u f u u u f u f u

du du du= + + , (37)

etc. If we define

10u f L g−= + , (38)

and make use of (35) - (36) in (33), then the later equation can be written as

10

0 0( )n n

n nu y u L A

∞ ∞−

= =

= −∑ ∑ , (39)

which gives 1

1 0u L A−= − , 1

2 1u L A−= − , (40) 1

3 2u L A−= − , 1

4 3u L A−= − , etc. Finally, we summarize the result as follows:

0,n

nu u

∞

=

=∑

where 1 2 3, , ,...,u u u are evaluated according to (40). We point out that the convergence of this method has been discussed in [5]. In the sequel, we apply the Adomian decomposition method to two boundary value problems: The first one studies the pressure driven flow of a third grade fluid in a circular pipe described by (25) and (27) and the second one analyses the pressure driven flow of a 6-constant Oldroyd fluid in a circular pipe described by (28) and (31). 4(a) Solution for a Third Grade Fluid

The resulting differential equation (25) along with the boundary condition (27) for this fluid may be written as

4896 A. M. Siddiqui et al

322r dp duLu

dz drβ

μ μ⎛ ⎞= − ⎜ ⎟⎝ ⎠

,

( ) 0u r = at r R= , (41) where

dLdr

= , (42)

is a linear invertible operator.

Applying the inverse operator 1L− on both sides of the eqn. (41), we obtain

3

1 1 12( )2r dp duL Lu r L L

dz drβ

μ μ− − −

⎡ ⎤⎛ ⎞ ⎛ ⎞= − ⎢ ⎥⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠ ⎢ ⎥⎣ ⎦

.

Equivalently, we write

3

1 12( )2r dp duu r C L L

dz drβ

μ μ− −

⎡ ⎤⎛ ⎞ ⎛ ⎞= + − ⎢ ⎥⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠ ⎢ ⎥⎣ ⎦

, (43)

where C is the constant of integration. We decompose u (r) and the non-linear

term3

( ) duNu rdr

⎛ ⎞= ⎜ ⎟⎝ ⎠

, respectively, as follows:

0

0

( ) ( ),

( ) .

nn

nn

u r u r

Nu r A

∞

=

∞

=

=

=

∑

∑ (44)

The first few Adomian polynomials of the non-linear term Nu(r) are given by

30

0

0 11

,

3 ,

duAdrdu duAdr dr

⎛ ⎞= ⎜ ⎟⎝ ⎠⎛ ⎞⎛ ⎞= ⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠

2 2

0 02 12 3 3 ,du dudu duA

dr dr dr dr⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞= +⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠

(45)

2 3

0 3 0 1 2 13 3 6 ,du du du du du duA

dr dr dr dr dr dr⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎛ ⎞⎛ ⎞ ⎛ ⎞= + +⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎝ ⎠⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠ ⎝ ⎠

2 2 2

0 0 0 34 1 2 2 14 3 3 3 6 .du du du dudu du du du duA

dr dr dr dr dr dr dr dr dr⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞= + + +⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

Adomian decomposition method 4897 Now, we use equation (44) in (43) and write the sum as

1 1

0 n=0

2( ) .2n n

n

r dpu r C L L Adz

βμ μ

∞ ∞− −

=

⎛ ⎞= + −⎜ ⎟

⎝ ⎠∑ ∑ (46)

We identify the zeroth component as

10 ( )

2r dpu r C L

dzμ− ⎛ ⎞

= + ⎜ ⎟⎝ ⎠

, (47)

and the remaining components as the recurrence relation,

11

02 , 0.n n

nu L A nβ

μ

∞−

+=

= − ≥∑ (48)

From equation (48), we write the first few components as follows: 1

1 02( ) [ ]u r L Aβμ

−= − ,

12 1

2( ) [ ]u r L Aβμ

−= − , (49)

13 2

2( ) [ ]u r L Aβμ

−= − ,

14 3

2( ) [ ]u r L Aβμ

−= −

M

The corresponding boundary condition (no slip condition), after making use of (44), becomes

( ) 0 at , 0.nu r r R n= = ≥ (50) Consequently, we obtain

2 2

01

2 2 2dp r Rudzμ

⎛ ⎞⎛ ⎞= −⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

, (51)

3 4 4

12 1( ) ,

2 4 4dp r Ru rdz

βμ μ

⎡ ⎤⎛ ⎞⎛ ⎞= − −⎜ ⎟⎜ ⎟ ⎢ ⎥

⎝ ⎠⎝ ⎠ ⎣ ⎦ (52)

2 5 6 6

22 1( ) 3 ,

2 6 6dp r Ru rdz

βμ μ

⎡ ⎤⎛ ⎞ ⎛ ⎞= −⎜ ⎟ ⎜ ⎟ ⎢ ⎥

⎝ ⎠ ⎝ ⎠ ⎣ ⎦ (53)

3 7 8 8

32 1( ) ,

2 8 8dp r Ru r Rdz

βμ μ

⎡ ⎤⎛ ⎞ ⎛ ⎞= − −⎜ ⎟ ⎜ ⎟ ⎢ ⎥

⎝ ⎠ ⎝ ⎠ ⎣ ⎦ (54)

4 9 10 10

42 1( ) 55 .

2 10 10dp r Ru rdz

βμ μ

⎡ ⎤⎛ ⎞ ⎛ ⎞= −⎜ ⎟ ⎜ ⎟ ⎢ ⎥

⎝ ⎠ ⎝ ⎠ ⎣ ⎦ (55)

Hence the approximate solution u(r) up to order four in a series form is given by


0 1 2 3 4( ) ( ) ( ) ( ) ( ) ( )u r u r u r u r u r u r= + + + + +L and summarizing the result, we write the fourth order approximate solution for the velocity field u(r) in the form

32 2 4 41 2 1( )

2 2 2 2 4 4dp r R dp r Ru rdz dz

βμ μ μ

⎛ ⎞ ⎡ ⎤⎛ ⎞ ⎛ ⎞⎛ ⎞= − − −⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎢ ⎥⎝ ⎠ ⎝ ⎠⎝ ⎠⎝ ⎠ ⎣ ⎦

2 5 3 76 6 8 82 1 2 13 12

2 6 6 2 8 8dp r R dp r Rdz dz

β βμ μ μ μ

⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞+ − − −⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠

4 9 10 102 155 .

2 10 10dp r Rdz

βμ μ

⎛ ⎞⎛ ⎞ ⎛ ⎞+ −⎜ ⎟⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠ (56)

The other quantities of practical interest in a circular pipe flow of radius R can be computed easily. The volume flow rate Q is given by the formula

0

2 ( ) .R

Q u r rdrπ= ∫ (57)

By substituting the velocity field from equation (56) into (57) and performing the integration, we obtain the following expression for Q,

41

2 4dp RQdz

πμ

⎛ ⎞⎛ ⎞= − ⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠

3 2 56 82 1 2 13

2 6 2 8dp R dp Rdz dz

β π β πμ μ μ μ

⎛ ⎞ ⎛ ⎞⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞+ −⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠

(58)

3 7 4 910 122 1 2 112 55


β π β πμ μ μ μ

⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞+ − +⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠L .

The Hagen–Poiseuille equation [6] follows from equation (25) by setting the parameter 0β = and (56) gives the corresponding solution, namely

2 21( ) .2 2 2

dp r Ru rdzμ

⎛ ⎞= −⎜ ⎟

⎝ ⎠

The average velocity ( )u r for the circular geometry under consideration is given by the formula

2( ) Qu rRπ

= (60)

and after making use of equation (58) for Q, the above expression takes the form,

32 41 2 1( )

2 4 2 6dp R dp Ru rdz dz

βμ μ μ

⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞⎛ ⎞= − +⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟

⎝ ⎠ ⎝ ⎠⎝ ⎠⎝ ⎠ ⎝ ⎠


2 5 3 76 82 1 2 13 12


β βμ μ μ μ

⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞− +⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠ (61)

4 9 102 155

2 12dp Rdz

βμ μ

⎛ ⎞⎛ ⎞ ⎛ ⎞− +⎜ ⎟⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠L .

On setting the parameter 0β = , one can deduce the Newtonian average velocity

21( )

2 4dp Ru rdzμ

⎛ ⎞⎛ ⎞= − ⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠. (62)

For the sake of completeness, we list here the normal stress components of the constitutive equation for third grade fluid (4) by making use of the assumptions (13) and (14). These are

2

1 2( )(2 )rr

du rSdr

α α ⎛ ⎞= + ⎜ ⎟⎝ ⎠

, (63)

2

2( )

zzdu rS

drα ⎛ ⎞= ⎜ ⎟

⎝ ⎠. (64)

The expression for normal stress difference turns out to be 2

1( )2rr zz

du rS Sdr

α ⎛ ⎞− = ⎜ ⎟⎝ ⎠

,

22 42

2 2 412

1 2 1 2 11 32 2 2rr zz

dp dp dpS S r r rdz dz dz

α β βμ μ μ μ μ

⎡ ⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎛ ⎞− = − +⎢⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎢⎣

23 6 4 8

6 82 1 2 112 552 2

dp dpr rdz dz

β βμ μ μ μ

⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞− + + ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎥⎦L . (65)

By setting the parameter 1α equal to zero, the difference becomes 0rr zzS S− = , (66)

which agrees with that of the Newtonian case. 4(b) Solution for 6-Constant Oldroyd Fluid The governing equation (28) for pipe of 6-constant Oldroyd fluid may be written in operator form as

3 2

212 2

rr dp du dp duLudz dr dz dr

ααμ μ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎛ ⎞= − +⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠

, (67)

where


.dLdr

=

The corresponding boundary condition associated with pipe flow is the no slip condition

u = 0 and r = R. (68) As before, we apply the inverse operator L-1 on both sides, to get

3 221 1 1 1

1 ( )2 2r dp du dp duL L u r L L L r

dz dr dz drαα

μ μ− − − −

⎡ ⎤ ⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎛ ⎞= − +⎢ ⎥ ⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠⎝ ⎠ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

,

which may be written as 3 2

1 1 121( ) ,

2 2r dp du dp duu r C L L L r

dz dr dz drαα

μ μ− − −

⎡ ⎤ ⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎛ ⎞= + − +⎢ ⎥ ⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠⎝ ⎠ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

(69)

where C is the constant of integration. Substituting the decomposition series

0( ) ( )n

nu r u r

∞

=

=∑ ,

0 0

( ) n nn n

Nu r A B∞ ∞

= =

= +∑ ∑ , (70)

into both sides of (69), we obtain

1 1 121

0 0 0( )

2 2n n nn n n

r dp dpu r C L L A L r Bdz dz

ααμ μ

∞ ∞ ∞− − −

= = =

⎛ ⎞ ⎡ ⎤⎛ ⎞= + − +⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦⎝ ⎠∑ ∑ ∑ , (71)

where

3 2

,n ndu duA Bdr dr

⎛ ⎞ ⎛ ⎞= =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

. (72)

We have already generated the Adomian polynomials for An in our previous problem. The Adomian polynomials for Bn, obtained with the help of eqn. (37), are

2

00

duBdr

⎛ ⎞= ⎜ ⎟⎝ ⎠

,

0 11 2 ,du duB

dr dr⎛ ⎞⎛ ⎞= ⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠

2

0 2 12 2 ,du du duB

dr dr dr⎛ ⎞⎛ ⎞ ⎛ ⎞= +⎜ ⎟ ⎜ ⎟⎜ ⎟

⎝ ⎠ ⎝ ⎠⎝ ⎠ (73)

0 3 1 23 2 2 ,du du du duB

dr dr dr dr⎛ ⎞⎛ ⎞ ⎛ ⎞⎛ ⎞= + ⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠⎝ ⎠⎝ ⎠


20 34 1 2

4 2 2 ,

du dudu du duBdr dr dr dr dr

⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞= + +⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠

M

The rest of the polynomials, if desired, can be constructed in a similar manner. We identify the zeroth component by

10 ( )

2r dpu r C L

dzμ− ⎛ ⎞

= + ⎜ ⎟⎝ ⎠

, (74)

and the remaining components un+1(r) by the recurrence relation 1 12

1 10 0

( ) , 02n n n

n n

dpu r L A L r B ndz

αα

μ

∞ ∞− −

+= =

⎡ ⎤⎛ ⎞= − + ≥⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦∑ ∑ . (75)

From (75), the first few un+1(r) are

[ ]1 121 1 0 0( ) ,

2dpu r L A L r Bdz

ααμ

− − ⎡ ⎤⎛ ⎞= − + ⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

[ ]1 122 1 1 1( ) ,

2dpu r L A L r Bdz

ααμ

− − ⎡ ⎤⎛ ⎞= − + ⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

[ ]1 123 1 2 2( ) ,

2dpu r L A L r Bdz

ααμ

− − ⎡ ⎤⎛ ⎞= − + ⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

[ ]1 124 1 3 3( ) ,

2

dpu r L A L r Bdz

ααμ

− − ⎡ ⎤⎛ ⎞= − + ⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦M

(76)

Substituting the decomposition series (70), into the boundary condition (44), we get ( ) 0nu r = at , 0.r R n= ≥ (77)

Using the boundary condition (77), we obtain from (74) and (75) 2 2

01( )

2 2 2dp r Ru rdzμ

⎡ ⎤⎛ ⎞= −⎜ ⎟ ⎢ ⎥⎝ ⎠ ⎣ ⎦

, (78)

3 4 4

1 1 21( ) ( ) ,

2 4 4dp r Ru rdz

α αμ

⎡ ⎤⎛ ⎞= − − −⎜ ⎟ ⎢ ⎥

⎝ ⎠ ⎣ ⎦

( )( )5 6 6

2 1 2 1 21( ) 3 2 ,

2 6 6dp r Ru rdz

α α α αμ

⎡ ⎤⎛ ⎞= − − −⎜ ⎟ ⎢ ⎥

⎝ ⎠ ⎣ ⎦

( ) ( )( )7 8 8

3 1 2 1 2 1 21( ) 2 6 5

2 8 8dp r Ru rdz

α α α α α αμ

⎡ ⎤⎛ ⎞= − − − − −⎜ ⎟ ⎢ ⎥

⎝ ⎠ ⎣ ⎦,


( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

4 1 2 1 2 1 2 1 2 1 2

9 10 102

1 2 1 2 1 1 2

( ) 2 6 5 3 2

1 3 2 6 2 .2 10 10

u r

dp r Rdz

α α α α α α α α α α

α α α α α α αμ

= − − − − − + −⎡⎣

⎡ ⎤⎛ ⎞⎤− − + − −⎜ ⎟ ⎢ ⎥⎦ ⎝ ⎠ ⎣ ⎦

Finally, the fourth order approximate solution for the velocity profile is given by 32 2 4 4

1 21 1( )

2 2 2 2 4 4dp r R dp r Rudz dz

α αμ μ

⎡ ⎤ ⎛ ⎞⎛ ⎞ ⎛ ⎞= − − − −⎜ ⎟⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎝ ⎠

( ) ( )5 6 6

1 2 1 213 2

2 6 6dp r Rdz

α α α αμ

⎛ ⎞⎛ ⎞+ − − −⎜ ⎟⎜ ⎟

⎝ ⎠ ⎝ ⎠

( ) ( ) ( )7 8 8

1 2 1 2 1 212 6 5

2 8 8dp r Rdz

α α α α α αμ

⎛ ⎞⎛ ⎞− − − − −⎜ ⎟⎜ ⎟

⎝ ⎠ ⎝ ⎠

( ) ( ) ( )( ) ( )

( ) ( ) ( )

1 2 1 2 1 2 1 2 1 2

9 10 102

1 2 1 2 1 1 2

2 6 5 3 2

1 3 2 6 2 .2 10 10

dp r Rdz

α α α α α α α α α α

α α α α α α αμ

− − − − − + −⎡⎣

⎡ ⎤⎛ ⎞⎤− − + − −⎜ ⎟ ⎢ ⎥⎦ ⎝ ⎠ ⎣ ⎦

(79)

In view of (79), expressions for practically important quantities, namely the volume flow rate Q, the average velocity ( )u r and normal stress difference rr zzS S− can be computed as in the preceding problem. 5. Numerical Method

In this section, we present numerical methods used to find the solution of the problem of the fully developed pipe flow for the third grade non-Newtonian fluid as well as for 6-constant Oldroyd fluid. We will solve non-dimensionalize governing equations for these problems. Our interest is to compare the analytical solutions obtained by Adomian decomposition method with numerical solutions. We find numerical solutions using Chebyshev spectral collocation method. Chebyshev polynomials have been successfully employed in finding numerical solution of various boundary value problems and in the study of computational fluid dynamics [16]. In solving ordinary differential equations, linear or nonlinear, to high accuracy on a simple domain, and if the data defining the problem are smooth, then spectral methods are usually the best tool. They can often achieve high order of accuracy (ten digits) of accuracy as compared with the finite difference or finite element methods.

We consider the flow of a non-Newtonian third grade fluid in a circular pipe. The governing equation was given in (25) as


32 0

2du du r dpdr dr dz

βμ μ

⎛ ⎞+ − =⎜ ⎟⎝ ⎠

. (80)

When finding the analytical solution, we solved the above first order equation subject to the no slip boundary condition at the walls of the pipe. To facilitate the numerical solution, we differentiate the above equation with respect to r, keeping in mind that the pressure does not vary with r, and obtain a second order differential equation and solve it subject to the no slip at the walls and symmetry condition at the centerline. The system we solve numerically is given below:

22 2

2 2

6 1 02

d u du d u dpdr dr dr dz

βμ μ⎛ ⎞+ − =⎜ ⎟⎝ ⎠

(81)

with boundary conditions given as '( ) 0 at 0,( ) 0 at ,

u r ru r r R

= == =

(82)

where the prime denotes the derivative with respect to r.

First, we non-dimensionalize the equation and the boundary conditions by introducing the following non-dimensional parameters:

2

2* , * , * , * , p * .u r z U R pu r zU R R R U

βλμ μ

= = = = = (83)

The non-dimensional equation (after dropping *) together with the boundary conditions yield

22 2

2 2

16 02

d u du d u dpdr dr dr dz

λ ⎛ ⎞+ − =⎜ ⎟⎝ ⎠

, (84)

'( ) 0 at 0,( ) 0 at 1.

u r ru r r

= == =

(85)

To solve equation (84) subject to boundary conditions (85), we employ spectral collocation method of Chebyshev type. We look for an approximate solution Nφ which is a global Chebyshev polynomial of degree N defined on the interval [-1, 1] by 1( ) cos , cosNT y N yθ θ −= = . (86) We discretize the interval by using collocation points to define the Chebyshev nodes in [-1, 1], namely

cos , 0,1,...jjy j NNπ⎛ ⎞= =⎜ ⎟

⎝ ⎠.

The function ( )yφ is approximated by an interpolating polynomial which is constructed in terms of the values of φ at each of the collocation points by employing a truncated Chebyshev series of the form


0

( ) ( ) ( ), 0,1,..., ,N

N j k k jk

y y T y j Nφ φ φ=

= = =∑ % (87)

where kφ% represents the series coefficients. The derivatives of the functions at the collocation points are given by

0

( ) ,N

Nj jk k

ky Ddy

φ φ=

∂=∑ % (88)

where D represents the derivative matrix [17] and is given by

( 1)

; 0,1,..., .

j kc jD j k k Njk c y yk j k

+−= ≠ =

− (89)

The higher order derivatives are computed as simply multiple powers of D, i.e.,

0

( ) , 0,1,..., ,i N

iNj jk ki

ky D k N

dyφ

φ=

∂= =∑ % (90)

where i is the order of the derivative. As described above, the Chebyshev polynomials are defined on the finite

interval [-1,1]. Therefore to apply Chebyshev spectral method to our equation, we make a suitable linear transformation and transform the physical domain [0,1] to a computational domain [-1, 1]. We sample the unknown function u at the Chebyshev points to obtain the data vector 0 1[ ( ), ( ),...... ( )] .T

Nu u y u y u y= The next step is to find a Chebyshev polynomial φ of degree N that interpolates the data, i.e.,

( ) , 0,1,...j jy u j Nφ = = , and obtain the spectral derivative vector ''u by differentiating φ and evaluating at the grid points, i.e.,

'' ''( ), 0,1,...,j ju t for j Nφ= = . This transforms the nonlinear differential equation into a nonlinear algebraic equations which can be solved by Newton’s iterative method.

Using the numerical method described in this section, we first consider flow

of a third grade fluid in a pipe, which is modeled by the non-dimensional equation (84) with boundary conditions given in (85). A comparison of numerical solutions and the ADM solution is presented in the graphs to show the validity of the analytical solutions. The results are presented for different values of non-Newtonian parameter and imposed pressure gradient. The numerical results are found be in good agreement with the analytical results obtained by the Adomian’s decomposition method.


Fig.1 Radial velocity field u for third grade pipe flow, (a) Comparison numerical and ADM solution for λ =0.1, dpdx

=-1.5,

(b) Comparison numerical and ADM solution for λ =0.2 dpdx

=-1.5 .

Fig.2 (a) Effect of non-Newtonian parameter λ on radial velocity for third grade pipe flow for fixed pressure gradient.

(b) Effect of imposed pressure gradient dp/dx on radial velocity for third grade pipe flow for fixed λ .

We now consider the flow of a non-Newtonian 6-constant Oldroyd fluid in a circular pipe of radius R. Model equation for this problem which is derived in earlier section 3 and is given below

3 2

1 2 02 2

du du r dp du r dpdr dr dz dr dz

α αμ μ

⎛ ⎞ ⎛ ⎞+ − − =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

. (91)

When finding the analytical solution, we solved the above first order highly nonlinear equation subject to no-slip boundary condition at the walls of the pipe. As in the third grade case, we differentiate the above equation to facilitate the numerical solution, and get a second order differential equation. We solve the resulting equation subject to the no slip condition at the walls and symmetry condition at the centerline. The equation we solve numerically is give as


2 22 2 2

2 212 2 2

13 02 2d u du d u dp du dp du d u dprdr dr dr dz dr dz dr dr dz

α αα μ μ μ⎛ ⎞ ⎛ ⎞ ⎛ ⎞+ − − − =⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠

, (92)

with boundary conditions given by '( ) 0 at 0,( ) 0 at .

u r ru r r R

= == =

(93)

We non-dimensionalize the equation and the boundary conditions by introducing the following non-dimensional parameters:

2 21 2

1 22 2* , * , * , * , * , p*U Uu r z R pu r zU R R R R U

α αλ λμ μ μ

= = = = = =

Then, the non-dimensional equation (after dropping *) with boundary conditions becomes

2 22 2 2

21 22 2 2

13 02 2d u du d u dp du dp du d u dprdr dr dr dz dr dz dr dr dz

λλ λ⎛ ⎞ ⎛ ⎞ ⎛ ⎞+ − − − =⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠

, (94)

'( ) 0 at 0,( ) 0 at 1.

u r ru r r

= == =

(95)

Fig.3 Radial velocity field u for 6 constant Oldroyd pipe flow (a) Comparison numerical and ADM solution for 1λ =0.1,

2λ =0.2 and fixed pressure gradient dpdx

=-1, (b) Comparison numerical and ADM solution for 1λ =0.3, 2λ =0.4 and same

fixed pressure gradient.


Fig. 4 Radial velocity field u for 6 constant Oldroyd fluid in a pipe (a) Effect of material constant 2λ , with

keeping 1λ =0.1, and fixed pressure gradient (b) Effect of material constant 1λ , with keeping 2λ =0.1, and fixed pressure

gradient.

6 Discussion & Conclusion Numerical and analytical solutions are obtained for the incompressible flow of a non-Newtonian third order fluid and 6-constant Oldroyd fluid through an axi-symmetric tube. Analytical solutions are obtained by the Adomian decomposition method and the numerical solutions are obtained using highly accurate Chebyshev spectral methods. To the best of our knowledge, this is the first attempt to apply this robust and highly effective analytical technique as well as highly accurate Chebyshev spectral method to study the non-Newtonian flows in this geometry. The proposed numerical method offers a superior intrinsic accuracy for the derivative calculations. The numerical results indicate the usefulness of the spectral methods in obtaining accurate solutions to the nonlinear problems arising in non-Newtonian fluid mechanics. As compared to other numerical techniques, such as finite differences, the nonlinearity is not a major complication for spectral methods. It is found that the analytical method used in this work (ADM) is more powerful and easy to apply for obtaining analytic approximate solutions. It provides analytic, verifiable, and rapidly convergent approximations which allow further insight into the characteristics and behavior of the solution as is the case with closed form solutions. Moreover, this method introduces the solutions, without any need for linearization or discretization which produces a rapidly converging series solution, components of which are elegantly computed. We would like to point out that even first few terms of the solution series give a pretty accurate approximation which compares well with the numerical results and proves the validity of this method.


The first set of results represents the solutions for the fully developed Poiseuille flow in a pipe. Fig. 1a,b shows the comparison of numerical solutions with the analytical solutions for the velocity field in the case of a third order fluid. The solutions are compared for different values of non-Newtonian parameter and imposed pressure gradient and we notice a close agreement between the two solutions. Fig. 2 (a) shows the influence of third grade parameter on the velocity profile. It is observed that for a given pressure gradient, the increase in non-Newtonian parameter λ results in the increase of velocity. Fig. 2(b) depicts the velocity profiles for the fixed non-Newtonian parameter 0.2λ = and for different values of the imposed pressure gradients. Fig. 3, 4 shows the results for a different non-Newtonian fluid model known as the Oldroyd 6-constant fluid. In our formulation of the pipe flow only two of the material constants 1λ and 2λ appear. We again observe a close agreement between the two solutions. In fig. 4, we show the effects of these parameters on the flow profile. We find that for a fixed value 1 0.1λ = ,

the increase in the 2λ results in the increase of the radial velocity and vice versa. On the other hand for a fixed 2 0.1λ = , the increase in the value of 1λ results in the decrease of the radial velocity. Analytical expressions for the other physical quantities of interest such as volume flux Q, average velocity u , shear stress distribution and normal stress are also computed and presented for both third grade fluid as well as 6-constant Oldroyd fluid. The velocity filed and other physical quantities for the classical viscous fluid can be recovered from the given expressions as a special case in appropriate limits. References [1] G. Adomian, A review of the decomposition method and some recent results

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