Commun Nonlinear Sci Numer Simulat 17 (2012) 35–44

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Commun Nonlinear Sci Numer Simulat

journal homepage: www.elsevier .com/locate /cnsns

Spectral approximation for a nonlinear partial differential equationarising in thin film flow of a non-Newtonian fluid

F. Talay Akyildiz a,⇑, Dennis A. Siginer a,b, Huseyin Kaplan c

a Department of Mathematics, Petroleum Institute, Abu Dhabi, United Arab Emiratesb Department of Mechanical Engineering, Petroleum Institute, Abu Dhabi, United Arab Emiratesc Nigde University, Faculty of Arts and Sciences, Department of Mathematics, Nigde, Turkey

a r t i c l e i n f o a b s t r a c t

Article history:Received 17 October 2010Received in revised form 1 March 2011Accepted 7 April 2011Available online 21 April 2011

Keywords:Third-grade fluidFourier–Galerkin methodFilm flowDrainage rate

1007-5704/$ - see front matter � 2011 Elsevier B.Vdoi:10.1016/j.cnsns.2011.04.009

⇑ Corresponding author.E-mail address: fakyildiz@pi.ac.ae (F.T. Akyildiz).

Start-up thin film flow of fluids of grade three over a vertical longitudinally oscillating solidwall in a porous medium is investigated. The governing non-linear partial differentialequation representing the momentum balance is solved by the Fourier–Galerkin approxi-mation. The effect of the porosity, material constants as well as oscillations on the drainagerate and flow enhancement is explored and clarified.

� 2011 Elsevier B.V. All rights reserved.

1. Introduction

There are few non-linear problems in mechanics amenable to a detailed numerical study. A simple and interesting non-linear problem arises in the case of film flow of non-Newtonian fluids. This problem also provides a vehicle for evaluating theusefulness and applicability of several standard procedures in numerical analysis to non-linear problems.

Thin film flow of non-Newtonian fluids has been studied by several authors, Denson [1], Waters and Keeley [2] and re-cently by Hayat et al. [3] among others. In particular the start-up phase of the drainage of the Oldroyd-B fluid, a viscoelasticfluid with constant viscosity, was investigated by Waters and Keeley [2] to determine that elasticity has a profound effect onthe start-up and subsequent drainage profiles. The superposition of oscillatory motion on steady flows has been of muchinterest over the last few decades as they may give rise to flow rate enhancement and can be used to control the flow rateas well. In particular pulsatile pipe flow, in which the liquid is driven by a pressure gradient oscillating about a non-zeromean, and the flow driven by a combination of a steady pressure gradient and oscillating boundaries have been investigatedby several authors, Siginer [4–6] among them. In the same vein the drainage of a thin liquid film down a vertical surfaceoscillating either in its own plane or orthogonally to the flow direction was studied first by Goshawk et al. [7] for Newtonianand linearly viscoelastic fluids. Goshawk and Waters [8] and Akyildiz [9] investigated the same drainage and flow enhance-ment problem with oscillating vertical wall for non-linear viscoelastic fluids, the Oldroyd four constant fluid and the fluid ofgrade three, respectively, both exhibiting shear rate dependent viscosity and elastic effects. Analytic solutions for Newtonianand linearly viscoelastic liquids predict that the disturbance at the solid surface has no effect on the drainage of these liquids.On the other hand, numerical solution for the flow of generalized Newtonian liquids indicates that the drainage of shear-thinning liquids may be significantly enhanced by the oscillation. Similarly, ‘negative enhancement’ or retardation is

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36 F.T. Akyildiz et al. / Commun Nonlinear Sci Numer Simulat 17 (2012) 35–44

predicted in the oscillatory drainage of shear-thickening liquids. However the findings in the literature so far seem to indicatethat in contrast to its influence in shaping the velocity profile in the start-up phase of the flow a purely elastic liquid with con-stant viscosity does not show any flow rate enhancement in response to longitudinal or orthogonal boundary oscillation.

In this paper the start-up phase of the drainage of a viscoelastic fluid, that of the fluid of grade three, down a vertical lon-gitudinally oscillating wall in a porous medium is considered. The modified constitutive equation for this class of fluids forporous media flow has been used to investigate other types of non-Newtonian flows in porous media, for instance to derivean exact analytical solution for magneto-hydrodynamic Poiseuille flow and an approximate solution via homotopy analysisfor Poiseuille flow in porous media with temperature dependent viscosity, Hayat et al. [10] and Ellahi and Afzal [11], respec-tively. However the flow of grade fluids over a vertical oscillating wall in a porous medium has not been investigated so far.The start-up phase of the flow is the subject of the present work. The momentum balance is solved numerically via the Fou-rier–Galerkin method. The effects of the oscillation, material parameters and porosity on the drainage rate are investigated.

2. Mathematical formulation and preliminary results on Fourier approximation

Consider a thin liquid film draining down a flat vertical porous surface defined by Cartesian coordinates (x,y,z). The x axispoints vertically downwards, the solid surface lies in the plane y = 0 with the thickness of the liquid film measured in thepositive y direction, and the z axis is positioned perpendicular to the gravitational force completing a set of right-handedaxes. The vertical surface oscillates with a given amplitude and frequency in the x-direction such that the flow field istwo dimensional and the velocity of the liquid at time t is given by

u ¼ ½uðx; y; tÞ;vðx; y; tÞ;0� ð2:1Þ

Since, by its very nature the drainage of liquid films is slow, inertial stresses are negligible when compared to viscous andgravitational forces and are therefore omitted. In addition surface-tension forces can also be neglected as liquid films are thinand the curvature of the free surface is small. Further, if it is assumed that the flow is locally parallel to the plane y = 0, thevelocity field reduces to

u ¼ ½uðx; y; tÞ;0� ð2:2Þ

This velocity as defined in (2.2) satisfies the incompressibility condition. The linear momentum balance in the porous med-ium is given by Tan and Masuoka [12] based on local volume averaging concepts of Vafai and Tien [13] and Khuzhayorovet al. [14],

qdudt¼ r � T þ r þ qF ð2:3Þ

where q, d/dt, T, r, F represent the density, the material derivative, the Cauchy stress tensor, the Darcy resistance in porousspace and the body force per unit mass, respectively. The Cauchy stress tensor of an incompressible third grade fluid whichsatisfies material objectivity and coordinate independence constraints of continuum mechanics but not the constraints im-posed by thermodynamics has the well known form

T ¼ �pI þ lA1 þ a1A2 þ a2A21 þ b1A3 þ b2ðA1A2 þ A2A1Þ þ b3ðtrA2

1ÞA1 ð2:4Þ

p, I, l, ai (i = 1,2), bi (i = 1, . . . ,3) and Ai (i = 1, . . . ,3) stand for the pressure, the identity tensor, constant zero-shear viscosity,the material constants and the Rivlin-Ericksen tensors defined through the following recurrence relations:

A1 ¼ ðgrad uÞ þ ðgrad uÞT

An ¼dAn�1

dtþ An�1ðgrad uÞ þ ðgrad uÞT An�1; n > 1

Thermodynamics requires that Helmholtz free energy be a minimum at rest and the Clausius–Duhem inequality is satisfied.Application of these principles impose on the constitutive constants ai (i = 1,2) and b(i = 1, . . . ,3) the following restrictions,Fosdick and Rajagopal [15],

l P 0; a1 P 0; b1 ¼ b2 ¼ 0; b3 P 0ja1 þ a2j 6

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi24lb3

p

reducing the thermodynamics compliant form of (2.4) to (2.5),

T ¼ �pI þ lA1 þ a1A2 þ a2A21 þ b3ðtrA2

1ÞA1 ð2:5Þ

The phenomenological constitutive law (2.6) based on the homogenization theory for determining the macroscopic behaviorof viscoelastic fluid flow in porous media at low Reynolds and Deborah numbers has been proposed for the flow of Oldroyd-Bfluids in porous media, Alishayev and Mirzadzhanzadeh [16], Khuzhayorov et al. [14]. Homogenization techniques allowdetermination of an equivalent macroscopic behavior by scaling up the description at the heterogeneous scale.

F.T. Akyildiz et al. / Commun Nonlinear Sci Numer Simulat 17 (2012) 35–44 37

1þ k@

@t

� �rp ¼ �lu

K1þ kr

@

@t

� �u ð2:6Þ

k, kr, u and K are the relaxation and retardation times, and the porosity and permeability of the porous medium, respectively.Tan and Masuoka [12] use the analogy in the interpretation of the pressure gradient in (2.6) as a measure of the resistance toflow in the bulk of the porous medium and the interpretation of the Darcy resistance r in (2.3) to flow in the pore space as ameasure of the resistance to flow offered by the solid matrix and infer r from (2.6) to satisfy the following equation

r ¼ �luK

1þ k@

@t

� ��1

1þ kr@

@t

� �u ð2:7Þ

It should be remarked that the Oldroyd-B constitutive model is a purely elastic model and does not exhibit shear rate depen-dent viscosity whereas fluid of grade three is capable of predicting elastic as well as shear dependent viscosity behavior.Hayat et al. [17] following Tan and Masuoka [12] apply the same type of analogy used to derive (2.7) and suggest the fol-lowing ansatz for the pressure gradient and equivalently for the resistance to the flow in the x direction for the fluid of gradethree

@p@x¼ rx ¼ �

uK

lþ a1@

@tþ 2b3

@u@y

� �2" #

u ð2:8Þ

Hence, the equation of motion of the fluid of grade three in the porous medium (2.9) is derived by substituting the consti-tutive law (2.5) and the ansatz (2.8) in the linear momentum balance in the pore space (2.3),

q@u@t¼ l @

2u@y2 þ a1

@3u@y2@t

þ 6b3@u@y

� �2@2u@y2 �

uK

lþ a1@@t

þ2b3@u@y

� �24 35uþ qg ð2:9Þ

New non-dimensional variables are introduced using the length and time scales (v2/g)1/3 and (v/g2)1/3, respectively, withv = l/q the kinematic viscosity,

Y ¼ yðg=v2Þ1=3; H ¼ hðx; tÞyðg=v2Þ1=3

; T ¼ tðg2=vÞ1=3

U ¼ u=ðvgÞ1=3; W ¼ u

Kðv2=gÞ2=3

; K ¼ a1=ðv2=gÞ2=3

C ¼ b3ðvgÞðg=v2Þ4=3ð1=qgÞ; X ¼ xðg=v2Þ1=3

ð2:10Þ

Eq. (2.9) reads in terms of the dimensionless variables and parameters (2.10),

@U@T¼ @2U

@Y2 þK@3U

@Y2@Tþ 2C

@U@Y

� �2@2U

@Y2 �WU

!�WU þ 1

" #ð1þWKÞ�1 ð2:11Þ

No slip condition at the wall Y = 0 and zero shear condition on the free surface of the liquid are imposed,

Uð0; tÞ ¼ A sinðWTÞ; T P 0;@U@Y¼ 0 on Y ¼ H ð2:12Þ

The liquid is at rest at t = 0,

UðY ;0Þ ¼ 0; ð2:13Þ

To calculate the shape of the film profile at a given time T the thickness H is allowed to vary with X, while assuming the flowis still locally parallel. Combining the material derivative at the free surface

VðX;H; TÞ ¼ @H@Tþ UðX;H; TÞ @H

@X0ð2:14Þ

with the equation of continuity yields a differential equation in H,

� @H@X

Z H

0

@U@H

dY þ U� �

¼ @H@T

ð2:15Þ

Introducing the flow rate Q(H,T) per unit width across the film thickness H,

QðH; TÞ ¼Z H

0UðX;Y; TÞdY ð2:16Þ

differentiating Q with respect to H and substituting the result into (2.15) and integrating gives,

XðH; TÞ � X0ðHÞ ¼Z T

0

@Q@H

ds ð2:17Þ

38 F.T. Akyildiz et al. / Commun Nonlinear Sci Numer Simulat 17 (2012) 35–44

where X0(H) = X(H,0) is the initial profile and can be chosen to represent any suitable initial shape. Eq. (2.17) effectivelydetermines the position of the free surface X as a function of H and T. Next some useful mathematical notation is introduced.Denote the inner product in L2ð0;HÞ by

ðF;GÞ ¼Z H

0FðYÞGðYÞdY ð2:18Þ

If F 2 L2ð0;HÞ , then Fourier sine series is defined as

FðYÞ ¼X1k¼1

bF sðkÞ sinððð2k� 1Þp=2HÞYÞ ð2:19Þ

bF ðkÞ ¼ 2H

Z p

0sin

ð2k� 1ÞpY2H

� �FðYÞdY k ¼ 1;2; . . . ð2:20Þ

Similarly Fourier cosine series is defined as

FðyÞ ¼bF cð0Þ

2þX1k¼1

bF cðkÞ cosð2k� 1Þp

2H

� �Y

� �ð2:21Þ

bF cðkÞ ¼2H

Z H

0cos

ð2k� 1ÞpY2H

� �f ðYÞdY k ¼ 1;2; . . . ð2:22Þ

Denote by kFkHm the Sobolev norm, given by

kFk2Hm ¼

P1k¼1

1þ 2k�12

�� ��2� �mjbF sðkÞj2

jbF cð0Þj22 þ

P1k¼1

1þ 2k�12

�� ��2� �mjbF cðkÞj2

8>>><>>>: ð2:23Þ

The space of periodic Sobolev functions on the interval [0,H] is defined as the closure of the space of smooth periodicfunctions with respect to the Hm norm, and will be denoted by Hm. In particular, the space L2ð0;HÞ with norm denoted bykkL2 is recovered for m = 0. We now define subspaces L2ð0;HÞ of spanned by the set

D�N ¼ffiffiffi2p

sinðð2k� 1ÞpY=2HÞffiffiffiffiHp ; 1 6 k 6 N

( )

D��N ¼ffiffiffi2p

cosðð2k� 1ÞpY=2HÞffiffiffiffiHp ; 1 6 k 6 N

( ) ð2:24Þ

The operators PN and P�N denote orthogonal, self-adjoint, projections of L2 onto D�N and D��N defined respectively by

PNFðYÞ ¼XN

k¼1

sinðð2k� 1ÞpY=2HÞbF sðkÞ

P�NFðYÞ ¼bF cð0Þ

2þXN

k¼1

cosðð2k� 1ÞpY=2HÞbF cðkÞð2:25Þ

For, F 2 Hm the estimates

kF � PNFkL2 6 CpN�mk@my FkL2

kF � PNFkHn 6 CpNn�mk@my FkL2

ð2:26Þ

hold for an appropriate constant Cp and a positive integer n. The reader is referred to Canuto and Hussaini [18] for the proofof these inequalities.

The space of continuous functions from the interval [0,T] onto the space Hn is denoted by C([0,T],Hn). Similarly, we alsoconsider the space Cð½0; T�;D�NÞ , where the topology on the finite-dimensional space D�N can be given by any norm. Finallynote the inverse inequality

k@my ukL2 6 NmkukL2 ð2:27Þ

that holds for integers m > 0 and u 2 D�N . A proof of this estimate can also be found in [10]. We will make use of the Sobolevlemma, which guarantees the existence of a constant c such that

Fig. 1. Effect of the variation of the material parameter C (dimensionless form of b3) in time on the velocity for W = 1, K = 1, A = 1, H = 1 and W = 1.

Fig. 2. Effect of the porous medium parameter W (dimensionless form of the porosity /) on the velocity profiles for C = 10, K = 1, A = 1, H = 1 and W = 1.

F.T. Akyildiz et al. / Commun Nonlinear Sci Numer Simulat 17 (2012) 35–44 39

Fig. 3. Effect of the amplitude A of oscillations on the velocity profiles for W = 1, K = 1, H = 1, and W = 1.

40 F.T. Akyildiz et al. / Commun Nonlinear Sci Numer Simulat 17 (2012) 35–44

supyjf ðyÞj 6 ckfkH1 ð2:28Þ

We now note that exactly the same estimates hold for P�N . In the following, it will always be assumed that a solution of theproblem (2.25)–(2.28) exists on some time interval [0,T], and with a certain amount of spatial regularity. In particular, wesuppose that a solution exists in the C([0,T],H1) space for some T > 0. With these preliminaries in place, we are now set totackle the problem of defining a suitable spectral projection of (2.9)–(2.13) and proving its convergence.

3. The Fourier–Galerkin Method

fekðyÞ; k 2 Ng ¼ffiffi2p

sinðð2k�1ÞpY=2HÞffiffiffiHp ; k 2 N

n ois chosen to be an orthonormal basis of the Hilbert space L2

0½0;H� . The subspace

of this space is spanned by DN ¼ffiffi2p

sinðð2k�1ÞpY=2HÞffiffiffiHp ; 1 6 k 6 N

n o. Fourier–Galerkin approximation of (2.11) and (2.13) is to find

the functions UN(T) 2 DN for all 0 6 T 6 f, such that

R H0 @T UNðY; TÞxðYÞdY � 1

ð1þWKÞR H

0 ð@2Y UNðY; TÞ þK@2

Y@T UNðY ; TÞ þ 1þ 2Cð@Y UNðY; TÞÞ2 þ ð@2Y UNðY; TÞ

�WUNðY; TÞÞ �WUNðY; TÞÞxðYÞdY ¼ 0R H0 UNðY ;0ÞxðYÞdY ¼ 0

ð3:1Þ

for "x 2 DN and T 2 [0,f], since for each T, UN(.,T) have the form

UNðY; TÞ ¼ A sinðWTÞ þXN

k¼1

bUNðk; TÞffiffiffi2p

sinðð2k� 1ÞpY=2HÞffiffiffiffiHp ð3:2Þ

taking x ¼ffiffiffi2H

qsinðð2k� 1ÞpY=2HÞ; 1 6 k 6 N in (3.1), substituting the Fourier expansion (3.2) in (3.1) and integrating for

each basis function, the following system of non-linear differential equations for the Fourier coefficients bUN are derived,

14H2 ð4H2 þ n2

1p2KÞd

bUNðn; TÞdT

þ p2n21

4H2ð1þWKÞbUNðn; TÞ � 2

ffiffiffi2p

pKHffiffiffiffiHpð1þWKÞ

þ 12CIJK2p4

ffiffiffi2p

8H11=2ð1þWKÞ

�XN

i;j;k¼1

CijkbUNði; TÞbUNðj; TÞbUNðk; TÞbUNðn; TÞ þ Fðn; TÞ

� 1ð1þWKÞ

bUNðn; TÞ þ CWIJp2

H3

XN

i;j;k¼1

Ci;j;kbUNði; TÞbUNðj; TÞbUNðk; TÞbUNðn; TÞ

!¼ 0 ð3:3Þ

F.T. Akyildiz et al. / Commun Nonlinear Sci Numer Simulat 17 (2012) 35–44 41

bUNðn;0Þ ¼ 0 ð3:4Þ

where 1 6 n 6 N, n1 = 2n � 1, I = 2i � 1, J = 2j � 1, K = 2k � 1,

Fðn; TÞ ¼ 2

ffiffiffiffi2H

rðA cosðWTÞð1þWKÞ � 1þ AW sinðWTÞÞ

pð2nþ 2nWK� 1�WKÞ

Cijk ¼Z H

0cosðIpY=2HÞ cosðJpY=2HÞ sinðKpY=2HÞ sinðnpY=2HÞdY

ð3:5Þ

This is a non-linear system of ordinary differential equations for the functions fbUNðk; tÞgNk¼1 . Using standard existence the-

ory it is easy to show the existence of a unique solution on some time interval [0,TN), where TN possibly may be equal to T.The proof of this argument and error analysis is not the subject of the present work and will be given in an upcoming pub-lication Akyildiz and Siginer [19]. For the numerical solution of the system of non-linear first order differential equationsvariable- tep size Runge–Kutta–Fehlberg method resulting in fifth order accurate solutions is used. It is well known thatRunge–Kutta method and its variants have been used for the numerical solution of highly non-linear chaotic systems. Forexample it has been used successively to visualize the chaotic attractors for the Lorenz and Rossler system, Perc [20]. Otherexamples can be found in Perc and Marhl [21] and in the handbook of Chaos Control [22].

0.5 1 1.5 2 2.52

3

4

5

6

7

8

9

10

Fig. 4. Effect of the material parameter C (dimensionless form of b3) on the drainage rate for W = 0, K = 1, U = 1, H = 1, T = 10 and W = 1.

42 F.T. Akyildiz et al. / Commun Nonlinear Sci Numer Simulat 17 (2012) 35–44

4. Results and discussion

Thin film flow of fluids of grade three over a solid longitudinally oscillating vertical wall in a porous medium has beeninvestigated. The effect of the material parameter, the oscillations and the porous medium parameter on the velocity profilesand in enhancing the drainage rate has been clarified. The influence of the variation of the material parameter b3, of theporosity u and the amplitude of the oscillations on the velocity field is shown in Figs. 1–3, respectively. Fig. 1 exploresthe effect of the dimensionless form C of the material parameter b3 on the velocity profiles. Increasing values of C has adampening effect, increasingly more pronounced in time, on the amplitude of the oscillations of the velocity profiles. Theamplitude and the wave length of the oscillations decrease in time. The effect of the dimensionless porous medium param-eter W is shown in Fig. 2. Increasingly larger values of W trigger increasingly strong dampening of the oscillations of thevelocity profile all other parameters held fixed that is the amplitude of the oscillations decrease with increasing W. Howeverthe wave length stays almost the same. Fig. 3 shows the effect of varying the amplitude of the oscillations on the velocityprofiles. The larger the amplitude the larger is the velocity. The velocity profile reaches a steady state if the wall does notoscillate. The effect of the material parameter C and the porous medium parameter W on the drainage rate is shown in Figs.4 and 5, respectively. Increasing the value of either C or W increases the thickness of the film all other parameters kept fixed.However increasingly thicker film does not necessarily lead to flow rate enhancement. For example we recall from Fig. 2 thatvelocity decreases with increasing porosity all other parameters held fixed. Thus with increasing porosity film thickness

Fig. 5. Effect of the porous medium parameter W (dimensionless form of the porosity /) on the drainage rate for C = 10, K = 1, U = 1, H = 1, T = 10 and W = 1.

0 0.5 1 1.5 2 2.5 3 3.5 40

1

2

3

4

5

6

7

8

9

10

Fig. 6. Effect of the frequency of the oscillations on the drainage rate for C = 10, K = 1, U = 1, H = 1, T = 10 and W = 0.

F.T. Akyildiz et al. / Commun Nonlinear Sci Numer Simulat 17 (2012) 35–44 43

increases but the velocity decreases the combination of which may or may not produce an enhanced flow rate. Fig. 6 exploresthe effect of the frequency of the oscillations on the drainage rate. Film thickness increases with increasing frequency, how-ever the rate of increase in the thickness of the film decreases rapidly with increasing frequency.

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