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Propulsion and Power Research

Propulsion and Power Research 2013;2(4):254–262

2212-540X & 2013 Nhttp://dx.doi.org/10.10

nCorresponding aut

E-mail addressesz_sh.eslami@yahoo.co

Peer review under rand Astronautics, Chin

ORGINAL ARTICLE

Non-Newtonian fluid flow in an axisymmetricchannel with porous wall

M. Hosseinia,n, Z. Sheikholeslamib, D.D. Ganjic

aDepartment of Mechanical Engineering, Islamic Azad University, Qaemshahr Branch, Qaemshahr, Mazandaran, IranbDepartment of Civil Engineering, University of Mazandaran, Babolsar, IrancDepartment of Mechanical Engineering, Babol University of Technology, Babol, Iran

Received 13 August 2013; accepted 14 October 2013Available online 9 December 2013

KEYWORDS

Non-Newtonian fluid;Axisymmetric channel;Porous media;Optimal HomotopyAsymptotic Method(OHAM);Heat transfer

ational Laboratory f16/j.jppr.2013.10.00

hor.

: m.Hosseini54545m (Z. Sheikholesla

esponsibility of Natia.

Abstract In the present article Optimal Homotopy Asymptotic Method (OHAM) is used toobtain the solutions of momentum and heat transfer equations of non-Newtonian fluid flow inan axisymmetric channel with porous wall for turbine cooling applications. Numerical methodis used for validity of this analytical method and excellent agreement is observed between thesolutions obtained from OHAM and numerical results. Trusting to this validity, effects of someother parameters are discussed. The results show that Nusselt number increases with increaseof Reynolds number, Prandtl number and power law index.& 2013 National Laboratory for Aeronautics and Astronautics. Production and hosting by Elsevier B.V.

All rights reserved.

1. Introduction

The problem of non-Newtonian fluid flow has been undera lot of attention in recent years because of its variousapplications in different fields of engineering specially the

or Aeronautics and Astronautics. Pro1

4@gmail.com (M. Hosseini),mi).

onal Laboratory for Aeronautics

interest in heat transfer problems of non-Newtonian fluidflow, such as hot rolling, lubrication, cooling problems anddrag reduction. Deburge and Han [1] studied a problemconcerning heat transfer in channel flow. Natural convec-tion of a non-Newtonian copper-water nanofluid wasinvestigated by Domairry et al. [2]. They conclude that asthe nanoparticle volume fraction increases, the momentumboundary layer thickness increases, whereas the thermalboundary layer thickness decreases. Sheikholeslami et al.[3] studied the problem of natural convection between acircular enclosure and a sinusoidal cylinder. They con-cluded that streamlines, isotherms, and the number, size andformation of the cells inside the enclosure strongly dependon the Rayleigh number, values of amplitude and the

duction and hosting by Elsevier B.V. All rights reserved.

Figure 1 Schematic diagram of the physical system.

Nomenclature

A, B symmetric kinematic matricesC specific heatCn blade-wall temperature coefficientsδvm=δxn velocity gardientsδam=δxn acceleration gradientsxk general coordinatesf velocity functionk fluid thermal conductivityn power law index in temperature distributionRe injection Reynolds numberKr rotation parameterp fluid pressurePr Prandtl number

T temperatureqnðηÞ temperature functionV injection velocityur ; uz velocity components in r, z directions, respectively

Greek letters

ϕk viscosity coefficientsφ dissipation functionη dimension less coordinates inzdirectionρ fluid densityτij stress tensor componentψ stream function

Non-Newtonian fluid flow in an axisymmetric channel with porous wall 255

number of undulations of the enclosure. Sheikholeslami et al.[4] performed a numerical study to investigate natural convec-tion in a square cavity with curve boundaries filled withCu-water nanofluid. Their results proved that the change ofinclination angle has a significant impact on the thermal andhydrodynamic flow fields. Recently several authors investigatedabout natural convection heat transfer [5–22].

In the heart of all different engineering sciences, every-thing shows itself in the mathematical relation that most ofthese problems and phenomena are modeled by ordinary orpartial differential equations. In most cases, scientific pro-blems are inherently of nonlinearity that does not admitanalytical solution, so these equations should be solved usingspecial techniques. Some of them are solved using numericaltechniques [23] and some are solved using the analyticalmethod of perturbation [24]. In the numerical method,stability and convergence should be considered so as toavoid divergence or inappropriate results. In the analyticalperturbation method, the small parameter is exerted to theequation. Since there are some limitations with the commonperturbation method, and also because the basis of thecommon perturbation method is upon the existence of asmall parameter, developing the method for different appli-cations is very difficult. Therefore, some different methodshave recently introduced some ways to eliminate the smallparameter, such as the Homotopy Perturbation Method [25–27],Differential Transformation Method [28] and HomotopyAnalysis Method [29] and some researchers applied thesemethods for engineering problems [30–32]. Optimal Homo-topy Asymptotic Method (OHAM) is the other method andstronger method for solving nonlinear problems withoutdepending to the small parameter. This method is developedand examined by some authors [33–35].

In this study, OHAM is applied to find the approximatesolutions of nonlinear differential equations governing non-Newtonian fluid flow in an axisymmetric channel with aporous wall for turbine cooling applications and have madea comparison with the numerical solution. Results are givenfor the velocity and temperature for various values ofReynolds number, Prandtl number and power law index.

2. Mathematical formulation

2.1. Flow analysis

This study is concerned with simultaneous developmentof flow and heat transfer for non-Newtonian viscoelasticfluid flow on the turbine disc for cooling purposes. Theproblem to be considered is depicted schematically inFigure 1. The r-axis is parallel to the surface of disk andthe z-axis is normal to it. The porous disc of the channel isat z¼þL. The wall that coincides with the r-axis is heatedexternally and from the other perforated wall non-Newtonian fluid is injected uniformly in order to cool theheated wall.

As can observed in Figure 1 the cooling problem ofthe disk can be considered as a stagnation point flowwith injection. For a steady, ax symmetric, non-Newtonianfluid flow the following equations can be written incylindrical coordinates.

The continuity equation:

∂ðrurÞ∂r

þ ∂ðruzÞ∂z

¼ 0 ð1Þ

And the momentum equations:

ur∂ðurÞ∂r

þ uz∂ðurÞ∂z

¼ � 1ρ

∂P∂r

þ 1ρ

∂τrr∂r

þ 1rðτrr�τθθÞ þ

∂τrz∂z

� �ð2Þ

M. Hosseini et al.256

ur∂ðuzÞ∂r

þ uz∂ðuzÞ∂z

¼

� 1ρ

∂P∂z

þ 1ρ

∂τzr∂r

þ 1rτrz þ

∂τzz∂z

� �ð3Þ

Here τrr; τrz; τzr; τzz are the components of stress matrix. Theanalytical model under consideration leads to the followingboundary conditions:

@z¼ 0 ur ¼ uz ¼ 0 ð4Þ@z¼ L ur ¼ 0; uz ¼ �V ð5ÞHere ur; uz are the velocity components in the r and z directionsand V is the injection velocity; ρ;P are the density, pressure.For particular class of viscoelastic and viscoinelastic fluidsRivlin [36] showed that if the stress components τij at a pointxkðk¼ 1; 2; 3Þ and time t are assumed to be polynominals inthe velocity gradient δvm

δxnðm; n¼ 1; 2; 3Þ and the acceleration

gradients δamδxn

ðm; n¼ 1; 2; 3Þ, and if in addition the medium is

assumed to be isontropic the stress matrix can be expressed inthe form

Jτij J ¼ ϕ0I þ ϕ1Aþ ϕ2Bþ ϕ3A2 þ… ð6Þ

Here I is the unit matrices, A and B are symmetric kinematicmatrixes defined by:

A¼ δviδxj

þ δvjδxi

��������;

��������

B¼ δaiδxj

þ δajδxi

þ 2δvmδxi

δvmδxj

��������

�������� ð7Þ

And ϕkðk ¼ 0; 1; 2; 3Þ are polynomials in the invariants ofA, B, A2. This study is restricted to second order fluids forwhich ϕkðk¼ 0; 1; 2; 3Þ are constant and ϕkðk ¼ 4; 5; :::Þ arezero. So that the stress components are as follows:

τrr ¼ ϕ1Arr þ ϕ2Arr2 þ ϕ3Brr ð8Þ

τzz ¼ ϕ1Azz þ ϕ2Azz2 þ ϕ3Bzz ð9Þ

τθθ ¼ ϕ1Aθθ þ ϕ2Aθθ2 þ ϕ3Bθθ ð10Þ

τrz ¼ ϕ1Arz þ ϕ2Arz2 þ ϕ3Brz ð11Þ

For the solution of the problem depicted in Figure 1 inthe case of axially symmetric flow it is convenient to definea stream function so that the continuity equation is satisfied:

ψ ¼ Vr2f ðηÞ ð12Þwhere η¼ z=L and the velocity components can be derived as:

ur ¼Vr

Lf0 ðηÞ ð13Þ

uz ¼ �2Vf ðηÞ ð14Þ

Using Eqs. (12)–(14) the equations of motion reduce to:

f02�2f f

00 ¼ � L2

ρV2r

∂P∂r

þ ϕ1

ρVLf000

þ ϕ2

ρL2ðf 00 2�2f

0f000 Þ þ ϕ3

ρL2ðf 00 2�2f f ivÞ ð15Þ

4f f0 ¼ � L2

ρV2

∂P∂z

�2ϕ1

ρVLf00

þ2ϕ2

ρL214f

0f00 þ r2

Lf00f000

� �

þ4ϕ3

ρL211f

0f00 þ f f

000 þ r2

Lf00f000

� �ð16Þ

The pressure term can be eliminated by differentiatingEq. (15) with respect to z and Eq. (16) with respect to r andsubtracting the resulting equations. This gives the followingequations:

�2f f000 ¼ f iv

Re�K1 ð4f 00 f 000 þ 2f

0f ivÞ

�K2 ð4f 00 f 000 þ 2f0f iv þ 2f f vÞ ð17Þ

where K1 ¼ Φ2

ρL2; K2 ¼ Φ3

ρL2is the injection Reynolds

number.For K2 ¼ 0, the equation turned to:

f iv þ 2Re f f000 �K1 Re ð4f 00 f 000 þ 2f

0f ivÞ ¼ 0 ð18Þ

The boundary conditions are:

f ð0Þ ¼ 0; f0 ð0Þ ¼ 0;

f ð1Þ ¼ 1; f0 ð1Þ ¼ 0: ð19Þ

2.2. Heat transfer analysis

The energy equation for the present problem with viscousdissipation in non-dimensional form is given:

ρC ur∂T∂r

þ uz∂T∂z

� �¼ k∇2T þ φ ð20Þ

φ¼ τrr∂ur∂r

þ τθθurrþ τzz

∂uz∂z

þ τrz∂ur∂z

þ ∂uz∂r

� �ð21Þ

Here ur; uz are the velocity components in the r and zdirections and V is the injection velocity; ρ;P;T ;C; k arethe density, pressure, temperature, specific heat, and heatconduction coefficient of fluid, respectively. φ is thedissipation function.

Non-Newtonian fluid flow in an axisymmetric channel with porous wall 257

Letting the blade wall (z¼0) temperature distribution be

Tw ¼ T0 þ ∑1

n ¼ 0Cn ðr=LÞn

And assuming the fluid temperature to have the form of [1]

T ¼ T0 þ ∑1

n ¼ 0Cn

r

L

� �nqnðηÞ ð22Þ

where T0 is the temperature of the incoming coolant (z¼L)and neglecting dissipation effect the following equations andboundary conditions are obtained:

q00n�Pr Reðf 0qn�2f q

0nÞ ¼ 0;

ðn¼ 0; 2; 3; 4;…Þ ð23Þqnð0Þ ¼ 1; qnð1Þ ¼ 0: ð24Þ

3. Fundamentals of Optimal HomotopyAsymptotic Method

Following differential equation is considered:

LðuðtÞÞ þ NðuðtÞÞ þ gðtÞ ¼ 0; BðuÞ ¼ 0 ð25Þwhere L is a linear operator, τ is an independent variable, uðtÞ isan unknown function, gðtÞ is a known function, NðuðtÞÞ isa nonlinear operator and B is a boundary operator. By means ofOHAM one first constructs a set of equations:

ð1�pÞ½Lðϕðτ; pÞÞ þ gðτÞ��HðpÞ½Lðϕðτ; pÞÞþgðτÞ þ Nðϕðτ; pÞÞ� ¼ 0

Bðϕðτ; pÞÞ ¼ 0; ð26Þ

where pA ½0; 1� is an embedding parameter, HðpÞ denotes anonzero auxiliary function for pa0 and Hð0Þ ¼ 0, ϕðτ; pÞ isan unknown function. Obviously, when p¼ 0 and p¼ 1, itholds that:

ϕ ðτ; 0Þ ¼ u0ðτÞ; ϕ ðτ; 1Þ ¼ uðτÞ: ð27Þ

Thus, as p increases from 0 to 1, the solution ϕ ðτ; pÞvaries from u0ðτÞ to the solution uðτÞ, where u0ðτÞ isobtained from Eq. (2) for p¼ 0:

Lðu0ðτÞÞ þ gðτÞ ¼ 0;Bðu0Þ ¼ 0: ð28Þ

We choose the auxiliary function HðpÞ in the form:

HðpÞ ¼ pC1 þ p2C2 þ… ð29Þ

where C1,C2 ,… are constants which can be determinedlater.

Expanding ϕðτ; pÞ in a series with respect to p, one has:

ϕðτ; p;CiÞ ¼ u0ðτÞ þ ∑kZ 1

ukðτ;CiÞpk;i¼ 1; 2;… ð30Þ

Substituting Eq. (30) into Eq. (26), collecting the samepowers of p, and equating each coefficient of p to zero, we

obtain set of differential equation with boundary conditions.Solving differential equations by boundary conditionsu0ðτÞ; u1ðτ;C1Þ; u2ðτ;C2Þ; … are obtained. Generallyspeaking, the solution of Eq. (1) can be determinedapproximately in the form:

~uðmÞ ¼ u0ðτÞ þ ∑m

k ¼ 1ukðτ;CiÞ: ð31Þ

Note that the last coefficient Cm can be function of τ.Substituting Eq. (31) into Eq. (25), there results thefollowing residual:

Rðτ;CiÞ ¼ Lð ~uðmÞðτ;CiÞÞ þ gðτÞ þ Nð ~uðmÞðτ;CiÞÞ: ð32ÞIf Rðτ;CiÞ ¼ 0 then ~uðmÞðτ;CiÞ happens to be the exactsolution. Generally such a case will not arise for nonlinearproblems, but we can minimize the functional:

JðC1;C2;…;CnÞ ¼Z b

aR2ðτ;C1;C2;…;CmÞ dτ; ð33Þ

where a and b are two values, depending on the givenproblem. The unknown constants Ciði¼ 1; 2;…;mÞ can beidentified from the conditions:

∂J∂ C1

¼ ∂J∂ C2

¼…¼ 0: ð34Þ

With these constants, the approximate solution (of order m)(Eq. (31)) is well determined. It can be observed that themethod proposed in this work generalizes these two methodsusing the special (more general) auxiliary function HðpÞ.

4. Solution with Optimal HomotopyAsymptotic Method

In this section, OHAM is applied to nonlinear ordinarydifferential Eqs. (18) and (23). According to the OHAM,applying Eqs. (26) to (18) and (23):

f iv þ 2Ref f000 �K1Reð4f 00 f 000 þ 2f

0f ivÞ ¼ 0;

qn00 �Pr Reðf 0qn�2f qn

0 Þ ¼ 0;

ðn¼ 0; 2; 3; 4;…Þð35Þ

where primes denote differentiation with respect to η. Weconsider f ; qn; H1ðpÞ and H2ðpÞ as following:f ¼ f 0 þ pf 1 þ p2f 2;

qn ¼ qn0 þ pqn1 þ p2qn2 ;

H1ðpÞ ¼ pC11 þ p2C12;

H2ðpÞ ¼ pC21 þ p2C22: ð36Þ

Substituting f ; qn; H1ðpÞ and H2ðpÞ from Eq. (36) intoEq. (35) and some simplification and rearranging based on

Table 1 Comparison between numerical results and OHAMsolution at Re¼ 0:5; K1 ¼ 0:01; Pr ¼ 1; n¼ 0.

η f qn

Error Error

0 0 00.1 0.0000053 0.000000140.2 0.0000091 0.000000130.3 0.0000104 0.000000110.4 0.0000090 0.00000050.5 0.0000058 0.00000170.6 0.0000021 0.00000360.7 0.00000045 0.00000520.8 0.0000012 0.00000520.9 0.00000057 0.00000261 0 0

M. Hosseini et al.258

powers of p-terms, we have:

p0 : f IV ¼ 0;

qn00 ¼ 0;

f 0ð0Þ ¼ 0; f 00ð0Þ ¼ 0;

f 0ð1Þ ¼ 1; f 00ð1Þ ¼ 0:

qn0ð0Þ ¼ 1; qn0 ð1Þ ¼ 0:

ð37Þ

p1 : f 1IV þ C11 f 0

IV þ 2C11 Re f 0f 0000

�4C11 K1Re f 000f 0

000

�2C11 K1Re f 0IV f 0

0 � f 0IV ¼ 0;

2 C21 Re Pr qn00

�C21 Re Pr n f 00 qn0 þ qn1

00

þ C21 qn000 � qn0

00 ¼ 0;

f 1ð0Þ ¼ 0; f 10ð0Þ ¼ 0;

f 1ð1Þ ¼ 0; f 10ð1Þ ¼ 0:

qn1ð0Þ ¼ 0; qn1 ð1Þ ¼ 0;

⋮

ð38Þ

Solving Eqs. (37) and (38) with boundary conditions:

f 0ðηÞ ¼ �2η3 þ 3η2;qn0 ðηÞ ¼ �ηþ 1; ð39Þ

f 1ðηÞ ¼ C11 Reð�0:05714285714η7 þ 0:2η6

þ4:8nη5�12nη4�0:5142857144η3

þ9:6nη3Þ;qn1 ðηÞ ¼ C21 Pr Reð0:3nη5�nη4 þ nη3

þη2�0:3nη�ηÞ;⋮ ð40Þ

The terms f 2ðηÞ and qn2 ðηÞ are mentioned graphicallybecause they are too long. Therefore final expression forf ðηÞ and qnðηÞ are:f ðηÞ ¼ f 0ðηÞ þ f 1ðηÞ þ f 2ðηÞ;qnðηÞ ¼ qn0ðηÞ þ qn1 ðηÞ þ qn2 ðηÞ: ð41Þ

From Eq. (32) by Substituting f ðηÞ and qnðηÞ into Eq. (41),R1ðη;C11;C12Þ and R2ðη;C21;C22Þ are obtained and J1 andJ2 are obtained in the flowing manner:

J1ðC11;C12Þ ¼Z 1

0R1

2ðη;C11;C12Þ dη;

J2ðC21;C22Þ ¼Z 1

0R2

2ðη;C21;C22Þ dη:

The constants C11; C12;C21 and C22 obtain from Eq. (42).In the particular cases:

Pr¼ 1;K1 ¼ 0:01; n¼ 0 and Re¼ 0:5

C11 ¼ 1:09935;C12 ¼ � 0:0167296;

C21 ¼ 1:08018;C22 ¼ � 0:0572104:ð42Þ

By substituting Eqs. (42) into (41), an expression for f ðηÞand qnðηÞ are obtained.

5. Results and discussion

The objective of the present study is to apply OptimalHomotopy Asymptotic Method to obtain an explicit analy-tic solution of heat transfer equation of a non-Newtonianfluid flow in an axisymmetric channel with a porous wallfor turbine cooling applications (Figure 1).

Validity of Optimal Homotopy Asymptotic Method isshown in Table 1 and Figure 2. Excellent agreementbetween numerical solution obtained by four-order Rung-Kutte method and analytical solution is obvious in Table 1and this figure too. In Table 1, error is introduced asfollows: Error¼ jf ðηÞNM� f ðηÞOHAMj.

This accuracy gives high confidence to us about validityof this problem and reveals an excellent agreement ofengineering accuracy. This investigation is completed bydepicting the effects of some important parameters toevaluate how these parameters influence on this fluid. Thevelocities f ðηÞ and f

0 ðηÞ, and temperature qnðηÞ profiles forvarious parameters Re; Pr; K1 and n are illustrated inFigures 3–5.

As seem in Figure 3 for constant value of K1 velocityprofile increases as Reynolds number increases. At lowReynolds numbers the velocity profile exhibit center linesymmetry indicating a poisuille flow for non-Newtonianfluids. At higher Reynolds numbers the maximum velocitypoint is shift to the solid wall where shear stress becomeslarger as the Reynolds number grows. Since f

00 ð0Þ ismeasure of friction force, it is advisable to use viscoinelasticfluids as a coolant fluid at least for industrial gas turbineengines. Figure 4 shows the variation f

00 ð0Þ for different

Figure 2 Comparison between the solutions via OHAM andnumerical solution for (a) f ðηÞ, (b) f

0 ðηÞ and (c) qnðηÞ, whenK1 ¼ 0:01; Pr ¼ 1; n¼ 2 and Re¼ 1.

Figure 3 Velocity component profile (a) f , (b) f 0 for variable Re atK1 ¼ 0:01.

Figure 4 Skin friction under the effect of K1 and Re.

Non-Newtonian fluid flow in an axisymmetric channel with porous wall 259

values of K1 and Reynolds number. One can observe thatfor constant values of K1, f

00 ð0Þ increase with increase ofvalue of Re, especially at high Reynolds numbers.

Figure 5 Temperature profile ðqnÞ (a) for variable n at K1 ¼ 0:01;Re¼ 1 and (b) at K1 ¼ 0:01.

Figure 6 Nusselt number (a) for variable n at K1 ¼ 0:01; Re¼ 1and (b) for variable Re at K1 ¼ 0:01; n¼ 0.

M. Hosseini et al.260

In Figure 5 temperature profile for different values ofpower law index (n) and Reynolds number are shown. Itshows that for constant value of η temperature increases ifpower law index decreases. Also it shows that increasingReynolds number leads to increase the curve of temperatureprofile and decrease of qnðηÞ values. Figure 6 illustrates howNusselt number (Nu¼ �qn

0 ð0Þ) can be changed withvarious amount of Re; Pr; n. It shows that Nusselt numberincreases with increase of Reynolds number, Prandtlnumber and power law index.

6. Conclusion

In this paper, OHAM is applied to solve the problem ofnon-Newtonian fluid flow in an axisymmetric channel witha porous wall for turbine cooling applications. Complemen-tary numerical solutions were obtained via fourth gradeorder Rung-Kutta and very excellent agreement between the

solutions obtained from OHAM. The results show that theincrement in the Reynolds number has similar effects onvelocity components, both of them increases with increaseof Reynolds number. At higher Reynolds numbers themaximum velocity point is shift to the solid wall whereshear stress becomes larger as the Reynolds number grows.Increment in skin friction is the effect of Reynolds numberincrement, especially at high Reynolds numbers. IncreasingReynolds number, Prandtl number and power law indexleads to increase in Nusselt number.

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