European Journal of Mechanics B/Fluids 29 (2010) 387–394

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European Journal of Mechanics B/Fluids

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

Non-Newtonian fluid flow induced by peristaltic waves in a curved channelN. Ali a,∗, M. Sajid b, Z. Abbas a, T. Javed aa Department of Mathematics, International Islamic University, Islamabad, Pakistanb Theoretical Plasma Physics Division, PINSTECH, P.O. Nilore, Islamabad, Pakistan

a r t i c l e i n f o

Article history:Received 12 February 2010Received in revised form15 April 2010Accepted 20 April 2010Available online 5 May 2010

Keywords:Curved channelPeristalsisNon-Newtonian flowTrapping

a b s t r a c t

This paper is devoted to the study of peristaltic flow of a non-Newtonian fluid in a curved channel. Theconstitutive relationship between stress and shear rate for a non-Newtonian third grade fluid is used.The problem is governed by a set of two nonlinear partial differential equations. These equations arethen transformed into a single nonlinear ordinary differential equation in the stream function underlong wavelength and low Reynolds number assumptions. This nonlinear ordinary differential equationis solved for stream function by the shooting method using Mathematica. The important phenomenonof pumping and trapping is presented graphically and discussed in detail. It is found that for a non-Newtonian third grade fluid an increase in the curvature of the channel helps in reducing the pressurerise over one wavelength in pumping region. This result is in contrast to the previous result obtained forthe pressure rise over one wavelength for a Newtonian fluid. For a Newtonian fluid, the pressure rise overone wavelength increases with an increase in the curvature. The trapping phenomenon is also alteredwith the presence of curvature and as a result the symmetry observed for a bolus of the trapped fluid inthe case of a straight channel is destroyed and splits into two asymmetrical parts for the curved channel.The outer bolus suppresses the inner bolus towards the lower wall. It is also noted that an increase insize and circulation of boluses achieve a maximum for large values of the shear thickening parameter β .Moreover, the size of two boluses in a third grade fluid is larger in comparison with their counterpartsin a Newtonian fluid. Further, the lower trapping limit of the flow rate Θ is also changed in the curvedchannel. In fact the lower trapping limit of the curved channel exceeds that of the straight channel.

© 2010 Elsevier Masson SAS. All rights reserved.

1. Introduction

Peristaltic flow in planar channels and axisymmetric tubes hasbeen the subject of extensive studies in the recent past. These flowsfrequently occur in physiology and industry. Examplesmay includeurine transport from kidney to bladder, movement of chime insmall intestine, vasomotion of small blood vessels and transportof corrosive fluids. The fundamental studies on peristalsis wereperformed by Shapiro et al. [1] using wave frame of reference andFung and Yih [2] using laboratory frame of reference. The initialstudies on peristaltic flows were carried out for Newtonian fluids.However, it is well known that most of the fluids occurring inphysiology and industry are non-Newtonian in nature and havea great impact on the mathematical and physical nature of theproblem. Motivated by this fact, Raju and Devanathan [3] madefirst investigation regarding peristaltic flows of non-Newtonianfluids. They studied peristaltic flow of blood using power-lawfluid. In another paper [4], they used the constitutive equation of

∗ Corresponding author.E-mail address: nasirali_qau@yahoo.com (N. Ali).

0997-7546/$ – see front matter© 2010 Elsevier Masson SAS. All rights reserved.doi:10.1016/j.euromechflu.2010.04.002

a viscoelastic fluid with fading memory to study the peristaltictransport. One can see the literature regarding the peristalticflows of different non-Newtonian fluids [5–16] and the referencestherein.In all the above-mentioned attempts peristaltic flows have been

discussed in two-dimensional channels or axisymmetric tubes.But in reality, the geometry of most physiological conduits andglandular ducts is curved. The effect of curvature seemsmeaningfulin this context. This fact provides a natural motivation to studyperistaltic flow in curved channels. Sato et al. [17] discussed thetwo-dimensional peristaltic flow of a viscous fluid in a curvedchannel in the laboratory frame of reference. They have used longwavelength and low Reynolds number approximations in derivingthe flow equations. Ali et al. [18] performed the analysis done bySato et al. [17] in the wave frame of reference. However, until nowno effort has been made to study peristaltic transport of a non-Newtonian fluid in a curved channel. Thus the aim of this paper isto perform such a study. The constitutive equation of a third gradefluid is used to describe the rheology of the fluid. This fluidmodel iscapable of predicting the shear thickening effects even under longwavelength assumption. A detailed discussion on thermodynamicsand the stability issues of a third grade fluid model is presented

388 N. Ali et al. / European Journal of Mechanics B/Fluids 29 (2010) 387–394

Fig. 1. Schematic diagram of the problem.

by Fosdick and Rajagopal [19] and Dunn and Rajagopal [20]. Thepresentation of paper is as follows:Section 2 presents the formulation of the problem. The method

of solution is given in Section 3. In Section 4, a detailed discussionabout the results is given. The concluding remarks are given inSection 5.

2. Description of the problem

In this section we briefly describe the problem of our interest.With this end in view, we consider a channel of width 2a coiledin a circle with centre O and radius R∗ (Fig. 1). The space insidethe channel is filled with a homogeneous incompressible fluid ofconstant properties. The velocity components in radial (R) and axial(X) directions are V and U , respectively. The fluid in the channelflows due to the contraction and expansion of the flexible walls ofthe channel. The inertial effects are taken small. The wall surfacesare mathematically described as follows:

H(X, t

)= a+ b sin

[2πλ

(X − ct

)], Upper wall

−H(X, t

)= −a− b sin

[2πλ

(X − ct

)]Lower wall.

(1)

Here c is the speed, λ is the wavelength and b is the amplitude.For the flow under consideration the velocity field is of the

following form

V =[V(X, R, t

), U(X, R, t

), 0]. (2)

The conservation of mass and momentum for the consideredgeometry using the above velocity field yields the followingequations:

∂

∂ R

{(R+ R∗

)V}+ R∗

∂U∂ X= 0, (3)

ρ

[∂ V∂ t+ V

∂ V∂ R+R∗UR∗ + R

∂ V∂ X−

U2

R∗ + R

]= −

∂ p∂ R+

1R∗ + R

∂

∂ R

{(R∗ + R

)τRR}

+

(R∗

R∗ + R

)∂

∂ XτRX −

τX X

R∗ + R, (4)

ρ

[∂U∂ t+ V

∂U∂ R+R∗UR∗ + R

∂U∂ X−

U VR∗ + R

]= −

(R∗

R∗ + R

)∂ p∂ X+

1(R∗ + R

)2 ∂∂ R{(R∗ + R

)2τRX

}+

(R∗

R∗ + R

)∂

∂ XτX X , (5)

where p is the pressure, ρ is the fluid density, t is the time and τRR,τX X , τRX are the components of the extra stress tensor τ. In derivingabove equations, we have used the curvilinear coordinates

(R, X

)with scale factors given by h1 = 1 and h2 =

(R∗ + R

)/R∗.

For the present problem, we choose τ to satisfy the constitutiverelationship proposed by Fosdick and Rajagopal [19] for a thirdgrade fluid and it is given by

τ = µA1 + α1A2 + α2A21 + β1(tr A21

)A1. (6)

In the above equation µ is the coefficient of shear viscosity, α1, α2and β1 are the material constants. The Rivlin–Ericksen tensors Anare defined by

A1 =(grad V

)+(grad V

)T,

An =dAn−1dt+ An−1

(grad V

)+(grad V

)T An−1, n > 1.(7)

The Clausius–Duhem inequality and the requirement of Helmholtzfree energy to be a minimum in equilibrium impose the followingconstraints on the dynamic viscosity µ, the normal stresscoefficients α1 and α2, and the coefficient β1,

µ ≥ 0, α1 ≥ 0, β1 ≥ 0, |α1 + α2| ≤√24µβ1. (8)

It is noted that this constitutive relation is able to predict notonly the normal stress differences, but also the shear thickeningphenomena (since β1 > 0) which indicates the increase in theviscosity by increasing the shear rate.In the fixed frame of reference

(R, X

), the flow in the channel

is unsteady. However, it can be treated as steady in a frame withcoordinate system (r, x) moving with the wave speed c (waveframe). The transformation equations relating the two frames are:

x = X − ct, r = R,

u = U − c, v = V ,(9)

where v and u are the velocity components along r and x-directionsin thewave frame. Eq. (9) helpswriting Eqs. (3)–(5) in the followingform:∂

∂ r

{(r + R∗

)v}+ R∗

∂ u∂ x= 0, (10)

ρ

[−c∂v

∂ x+ v

∂v

∂ r+R∗ (u+ c)R∗ + r

∂v

∂ x−(u+ c)2

R∗ + r

]= −

∂ p∂ r+

1R∗ + r

∂

∂ r

{(R∗ + r

)τr r}

+

(R∗

R∗ + r

)∂

∂ xτr x −

τx x

R∗ + r, (11)

ρ

[−c∂ u∂ x+ v

∂ u∂ r+R∗ (u+ c)R∗ + r

∂ u∂ x+

u vR∗ + r

]= −

(R∗

R∗ + r

)∂ p∂ x+

1

(R∗ + r)2∂

∂ r

{(R∗ + r

)2τr x

}+

(R∗

R∗ + r

)∂

∂ xτx x. (12)

We define the following dimensionless variables

x =2π xλ, η =

ra, u =

uc, v =

v

c,

Re =ρcaµ,

P =2πa2

λµcp, h =

Ha, δ =

2πaλ, k =

R∗

a,

β =β1c2

µa2

(13)

N. Ali et al. / European Journal of Mechanics B/Fluids 29 (2010) 387–394 389

a b

c d

e f

Fig. 2. Variation of u (η) for different values of β at cross-sections x = −π/2 (panels (a), (b)), x = 0 (panels (c), (d)) and x = π/2 (panels (e), (f)) with φ = 0.4 and k = 2.Left panels correspond toΘ = 4 while right panels are forΘ = 1.

and the stream function [17]

u = −∂ψ

∂η, v = δ

kη + k

∂ψ

∂x. (14)

With the help of Eqs. (13) and (14), the Eq. (10) is identicallysatisfied. Eqs. (11) and (12) under long wavelength and lowReynolds number approximations [5,7–10] are reduced to thefollowing dimensionless form

∂p∂η= 0, (15)

−∂p∂x−

1k (η + k)

∂

∂η

{(η + k)2 τηx

}= 0. (16)

Eliminating the pressure we get the following compatibilityequation

∂

∂η

[1

k (η + k)∂

∂η

{(η + k)2 τηx

}]= 0, (17)

where

τηx = −∂2ψ

∂η2−

1η + k

(1−

∂ψ

∂η

)

− 2β{∂2ψ

∂η2+

1η + k

(1−

∂ψ

∂η

)}3. (18)

It should be emphasized here that under long wavelengthapproximation the terms with normal stress coefficients α1 andα2 vanish from the Eq. (17). However the Eq. (17) is still capableof predicting shear thickening effects due to the presence ofparameter β . It is also noted that in the limit when k → ∞ the

390 N. Ali et al. / European Journal of Mechanics B/Fluids 29 (2010) 387–394

Fig. 3. Variation of u (η) for different values of k (β = 2, x = π/2) with φ = 0.4andΘ = 1.64.

Eq. (17) reduces to the governing equation of peristaltic flow of athird grade fluid in a planar channel [5].The dimensional volume flow rate in the laboratory frame is

defined as

Q =∫ H

−HUdR (19)

in which H is a function of X and t . The above expression in thewave frame becomes

F =∫ H

−Hu dr (20)

where H is a function of x alone. From Eqs. (9), (19) and (20) wecan writeQ = F + 2cH. (21)The time-averaged flux over a period T at a fixed position X is

Q =1T

∫ T

0Q dt. (22)

Invoking Eq. (21) into Eq. (22) and then integrating one has

Q = q+ 2ac. (23)If we define the dimensionless mean flows Θ in the laboratoryframe, and q in the wave frame, as:

Θ =Qac, q =

Fac, (24)

the Eq. (23) reduces toΘ = q+ 2, (25)

where

q = −∫ h

−h

∂ψ

∂ηdη = −(ψ(h)− ψ(−h)). (26)

Selecting ψ (h) = −q/2, we have ψ (−h) = q/2. Therefore, theappropriate boundary conditions in the wave frame are

ψ = −q2,

∂ψ

∂η= 1 at η = h = 1+ φ sin x,

ψ =q2,

∂ψ

∂η= 1 at η = −h = −1− φ sin x,

(27)

where φ = b/a is the amplitude ratio.The dimensionless pressure rise over onewavelength is defined

by

1p =∫ 2π

0

dpdxdx. (28)

3. Method of solution

The boundary value problem consisting of the Eq. (17) and theboundary conditions (27) is solved for the stream functionψ (η, x)by employing the built in routine for solving nonlinear ordinarydifferential equation via shootingmethod using the computationalsoftwareMathematica. Our results obtained for the limiting casesβ = 0 and k → ∞ are in excellent agreement with the previousstudies in the literature.

4. Results and discussion

In this section, we will discuss some of the important featuresof the peristaltic motion such as flow characteristics, pumpingcharacteristics and trapping phenomenon. Our focus will be onthe effects of curvature parameter (k) and Deborah number (β)on the above-mentioned features of peristaltic motion. First, wewill discuss how flow features are altered with rheology of thefluid and the curvature of the channel by examining Figs. 2 and 3.Then, we will analyze the pumping characteristics with the help ofFig. 4. Finally, the trapping phenomenon is studied with the aid ofFigs. 5–8.

4.1. Flow characteristics

In Fig. 2 the axial velocity u is plotted as a function of η at threedifferent cross-sections x = −π/2 (narrow part of the channel),x = 0 (mean width of the channel) and x = π/2 (wider part of thechannel) and for two different values of the flow rate Θ = 4 andΘ = 1. Fig. 3 shows the variation of u with η for different values

a b

Fig. 4. Variation of1pwithΘ for different values of β (k = 5) (panel (a)) and k (β = 0.2). Here φ = 0.4.

N. Ali et al. / European Journal of Mechanics B/Fluids 29 (2010) 387–394 391

a b

c d

e f

Fig. 5. Streamlines for different values of k. Panels (a)-(f) correspond to values of k = (3, 4, 10, 50, 100, 500) respectively. The other parameters are β = 0.2,Θ = 1.5 andφ = 0.8.

of k at the cross-section x = −π/2 and Θ = 1.64. The followingobservations have been inferred from these figures.

• The profiles of axial velocity are not symmetric about the centreline of the channel (η = 0) and themaxima in the profiles shiftstowards the lowerwall for the small values of k i.e. for a channelwith large curvature. However, as k→∞ the profiles becomesymmetric about the centre line. It is also noted that themaximain the above profiles is an increasing function of k.• The variation of u (η) for different values of β reveals that themaximum in the velocity curve for a Newtonian fluid (β = 0)lies below the maxima in velocity curves for a non-Newtonianfluid (non-zero values of β). Further, when the prescribed flowrate is large (Θ = 4), the effects of β on the velocity profilesare significant for its small values (up to β = 0.1). But for a

relatively small prescribed flow rate (Θ = 1), the effects of βon the velocity are visible for its large values (up to β = 0.5).These results are quite in accordance with the previous studiesof the peristaltic flow of a third grade fluid in a planar channel.• The magnitude of the velocity at the cross-section x = −π/2 isgreater in comparison with its magnitude at the cross-sectionsx = 0 and x = π/2.

The differences between the previous studies and the oneconducted here is that in a curved channel the fluid accumulatestowards the lower wall and transported with the velocity that isnot symmetric about the centre line of the channel. Moreover,at different cross-sections the fluid attains a maximum velocitywhich decreases in magnitude in the curved channel.

392 N. Ali et al. / European Journal of Mechanics B/Fluids 29 (2010) 387–394

a b

c d

e f

Fig. 6. Streamlines for different values of β . Panels (a)–(f) correspond to values of β = (0, 0.2, 0.5, 1, 10, 20) respectively. The other parameters are k = 5, Θ = 1.5 andφ = 0.8.

4.2. Pumping characteristics

The variation of pressure rise over one wavelength (1p) withthe average flow rate in the fixed frame (Θ) for different values ofβ(panel (a)) and k (panel (b)) is presented in Fig. 4. Fig. 4(a) illustratesthat 1p in the pumping region (1p > 0,Θ > 0) increases byincreasing β for a fixed flow rate Θ . This means that in pumpingregion, peristalsis has to work against a greater pressure rise fora third grade fluid in comparison with the viscous fluid. In freepumping case (when 1p = 0) the average flow rate Θ increasesby increasing β . The case when 1p < 0,Θ > 0 is known ascopumping. Here it is seen that for an appropriately chosen1p < 0the flow rate decreases by increasing β .Fig. 4(b) shows that with a decrease in k i.e., an increase in

the curvature of the channel,1p decreases in the pumping region

for a fixed flow rate. Thus for a third grade fluid the presence ofcurvature in the channel causes a decrease in1p. Interestingly, fora Newtonian fluid the presence of curvature increases 1p in thepumping region [17,18]. Our study also reveals the similar resultfor a Newtonian fluid but shows a deviation for a third grade fluiddue to the presence of the parameter β .

4.3. Trapping

Figs. 5–8 show the effects of k and β on the trappingphenomenon. In Fig. 5, the effects of k on trapping are analyzedfor a third grade fluid (β = 0.2). We can see that for small valuesof k i.e., for a curved channel, the bolus/circulating region is notsymmetric about the centre line. It actually splits into two bolusesappearing in the upper and lower halves of the channel. The size

N. Ali et al. / European Journal of Mechanics B/Fluids 29 (2010) 387–394 393

a b

Fig. 7. Streamlines for different values of β . Panels (a), (b) correspond to values of β = (0, 0.2) respectively. The other parameters are k = 2,Θ = 1.5 and φ = 0.8.

a b

Fig. 8. Streamlines for different values of k. Panels (a), (b) correspond to values of k = (3,∞) respectively. The other parameters are β = 0.2,Θ = 1 and φ = 0.8.

and shape of boluses in two halves are different from each other.However as k increases the bolus attains its symmetric shape.The effects of β on the size and circulation of boluses for a

curved channel (k = 5) are shown in Fig. 6. We can see that thesize and circulation of two boluses increases by increasing β . It isfairly interesting to note that the increase in size and circulation oftwo boluses stops for large values of β . A comparison of boluses forthe Newtonian and third grade fluids is shown in Fig. 7. It is notedthat the size of boluses in third grade fluid is large in comparisonwith their sizes for a Newtonian fluid.It is also observed that the presence of curvature changes the

lower trapping limit ofΘ . To elucidate this fact we have made theFig. 8. This figure shows that for Θ = 1, the central streamlineη = 0 splits to enclose a bolus of a fluid in a straight channel.However, for a curved channel bolus appears only for values of Θgreater than one. Thus one can conclude that the lower trappinglimit of Θ increases in a curved channel when compared to astraight channel.

5. Concluding remarks

A study is conducted for peristaltic flow of a third grade fluidin a curved channel under long wavelength and low Reynoldsnumber assumptions. The resulting nonlinear ordinary differentialequation is solved for the stream function by the shooting methodusing Runge–Kutta algorithm. Some basic features of peristalticmotion such as trapping and pumping are discussed with the helpof the obtained solutions. Special emphasis is given to the effectsof curvature parameter (k) and Deborah number (β) on these basic

features. It is worth mentioning that such a mathematical studyof peristaltic motion of a third grade fluid in a curved channelis never done before. We hope that this study may be helpful inunderstanding the peristaltic flows in more realistic situations.

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