Series Solutions for the Peristaltic Flow of a Tangent Hyperbolic Fluidin a Uniform Inclined TubeSohail Nadeem and Noreen Sher Akbar

Department of Mathematics, Quaid-i-Azam University 45320, Islamabad 44000, Pakistan

Reprint requests to S. N.; E-mail: snqau@hotmail.com

Z. Naturforsch. 65a, 887 – 895 (2010); received June 4, 2009 / revised November 6, 2009

In the present investigation we have studied a tangent hyperbolic fluid in a uniform inclined tube.The governing equations are simplified using long wavelength and low Reynold number approxima-tions. The solutions of the problem in simplified form are calculated with two methods namely (i) theperturbation method and (ii) the homotopy analysis method. The comparison of the solutions showa very good agreement between the two results. At the end of the article the expressions of the pres-sure rise and the frictional force are calculated with the help of numerical integration. The graphicalresults are presented to show the physical behaviour of Weissenberg number We, amplitude ratio φ ,and tangent hyperbolic power law index n.

Key words: Series Solutions; Peristaltic Flow; Tangent Hyperbolic Fluid; Uniform Tube.

1. Introduction

The study of non-Newtonians fluids have becomean increasing importance in the last few decades. Thisis mainly due to its large number of applications inmany areas. Such applications are foodmixing andchyme movement in the intestine, flow of plasma, flowof blood, polymer fluids, colloidal suspension, liquidcrystals, animal blood exotic lubricants, and fluid con-taining small additives. Another reason which has at-tracted the attention of various researcher’s to dis-cuss the non-Newtonian fluids is that an universal non-Newton constitutive relation that can be used for allfluids and flows is not available. Therefore, a numberof studies have been presented which deal with differ-ent kinds of non-Newtonian fluid models using differ-ent kinds of geometries [1 – 6]. Peristaltic mechanismis also a rich of research area which has attracted theattention of researchers due to its wide range of appli-cations in physiology and industry. After the first in-vestigation done by Latham [7] a number of studiesdealing with different flow geometries have been dis-cussed in peristaltic flow problem [8 – 18].

In view of the above analysis the purpose of thepresent investigation is multidimensional. The mainaim is to discuss the peristaltic flow of a tangent hy-perbolic fluid model and the secondary purpose is tofind the series solutions of nonlinear equations of thegiven problem with the help of the homotopy analy-

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sis method and the perturbation method. The govern-ing equations of the problem are first simplified andthen the reduced nonlinear equations are solved ana-lytically. The expressions for the pressure rise and fric-tional forces are calculated numerically. The highlightsof the problems are discussed through graphs.

2. Mathematical Model

For an incompressible fluid the balance of mass andmomentum are given by

divV = 0, (1)

ρdVdt

= divS+ρf, (2)

where ρ is the density, V is the velocity vector, S isthe Cauchy stress tensor, f represents the specific bodyforce, and d/dt represents the material time derivative.The constitutive equation for a hyperbolic tangent fluidis given by [5]

S =−PI+τττ, (3)

τττ =[[

η∞ +(η0 +η∞) tanh(Γ γ)n] γ], (4)

in which −PI is the spherical part of the stress dueto constraint of incompressibility, τττ is the extra stresstensor, η∞ is the infinite shear rate viscosity, η0 is the

888 S. Nadeem and N. Sher Akbar · Peristaltic Flow of a Tangent Hyperbolic Fluid

zero shear rate viscosity, Γ is the time constant, n is thepower law index, and γ is defined as

γ =

√12 ∑

i∑

jγ i j γ ji =

√12

Π , (5)

where Π = 12 tr(gradV +(gradV )T )2 is the second in-

variant strain tensor. We consider the constitution (4),the case for which η∞ = 0 and Γ γ < 1. The componentof extra stress tensor, therefore, can be written as

τττ = η0[(Γ γ)n]γ = η0[(1+Γ γ − 1)n]γ= η0[1+ n(Γ γ − 1)]γ.

(6)

3. Mathematical Formulation

Let us consider the peristaltic transport of an in-compressible hyperbolic tangent fluid in a two dimen-sional inclined tube. The flow is generated by sinu-soidal wave trains propagating with constant speed calong the channel walls. The geometry of the wall sur-face is defined as

hhh = a+ bsin2πλ

(Z − ct), (7)

where a is the radius of the tube, b is the wave ampli-tude, λ is the wavelength, c is the propagation velocity,and t is the time. We are considering the cylindrical co-ordinate system (R, Z), in which the Z-axis lies alongthe centerline of the tube and R is transverse to it (seeFig. 1).

Fig. 1. Geometry of the problem.

The governing equations in the fixed frame for anincompressible flow are

∂U∂ R

+UR+

∂W∂ Z

= 0, (8)

ρ(

∂∂ t

+U∂

∂ R+W

∂∂ Z

)U =

−∂ P∂ R

+1R

∂∂ R

(Rτττ RR)+∂

∂ Z(τττ RZ)+

τττ θ θR

,

(9)

ρ(

∂∂ t

+U∂

∂ R+W

∂∂ Z

)W =

−∂ P∂ Z

+1R

∂∂ R

(Rτττ RZ)+∂

∂ Z(τττ ZZ)−ρgsinα.

(10)

Introducing a wave frame (r, z) moving with velocity caway from the fixed frame (R, Z) by the transforma-tions

z = Z − ct, r = R, (11)

w = W − c, u = U , (12)

where U ,W and u, w are the velocity components inthe radial and axial directions in the fixed and movingcoordinates, respectively.

The corresponding boundary conditions are

∂ w∂ r

= 0, u = 0 at r = 0, (12a)

w =−c, u =−cdhdz

at r = h = a+ bsin2πλ

(z).(12b)

Defining

R =Ra, r =

ra, Z =

Zλ, z =

zλ,

W =Wc, w =

wc, τττ =

aτττcµ

, U =λUac

,

u =λ uac

, P =a2Pcλ µ

, t =ctλ, δ =

aλ,

Re =ρcaµ

, h =ha= 1+φ sin2πz,

γ =ac

γ, E =µc

ρga2 ,

(13)

where µ is the viscosity of the fluid.

S. Nadeem and N. Sher Akbar · Peristaltic Flow of a Tangent Hyperbolic Fluid 889

Using the above non-dimensional quantities and theresulting equations can be written as

∂u∂ r

+ur+

∂w∂ z

= 0, (14)

Reδ 3(

u∂∂ r

+w∂∂ z

)u =

−∂P∂ r

− δr

∂∂ r

(rτττ rr)− δ 2 ∂∂ z

(τττrz),

(15)

Reδ(

u∂∂ r

+w∂∂ z

)w =

−∂P∂ z

− 1r

∂∂ r

(rτττ rz)− δ∂∂ z

(τττzz)− sinαE

,

(16)

where

τττrr =−2[1+ n(Weγ− 1)]∂u∂ r

,

τττrz =−[1+ n(Weγ− 1)](

∂u∂ z

δ 2 +∂w∂ r

),

τττzz =−2δ [1+ n(Weγ− 1)]∂w∂ z

,

γ =

[2δ 2

(∂u∂ r

)2

+

(∂u∂ z

δ 2 +∂w∂ r

)2

+ 2δ 2(

∂w∂ z

)2]1/2

,

(17)

in which δ , Re, We represent the wave, Reynolds, andWeissenberg numbers, respectively. Under the assump-tions of long wavelength δ � 1 and low Reynoldsnumber, neglecting the terms of order δ and higher,(12), (15), and (16) take the form

∂P∂ z

+sinα

E=

1r

∂∂ r

[r(

1+ n(

We∂w∂ r

− 1))

∂w∂ r

],

(18)

∂P∂ r

= 0, (19)

∂w∂ r

= 0, at r = 0,

w =−1, at r = h = 1+φ sin2πz.(20)

4. Solution of the Problem

4.1. Perturbation Solution

For perturbation solution, we expand w, F , and P as

w = w0 +Wew1 +O(We2), (21)

F = F0 +WeF1+O(We2), (22)

P = P0 +WeP1+O(We2). (23)

Expressions for velocity profile and pressure gradientsatisfying the boundary conditions (20) can be writtenas

w =−1+(

r2 − h2

4(1− n)

)∂P∂ z

+sinα

Er2 − h2

4(1− n)

+ We(a11(r3 − h3)),

(24)

dPdz

=−8E(1− n)(2F+ h2)− h2 sinα

Eh4

+ We(−12

5a11h

).

(25)

4.2. Homotopy Analysis Method (HAM) Solution

In this section, we have found the HAM solutionof (18) to (20). For that we choose

w0 =−1+(

r2 − h2

4(1− n)

)∂P∂ z

+sinα

Er2 − h2

4(1− n), (26)

as the initial guess. Further, the auxiliary linear opera-tor for the problem is taken as

Lwr(w) =1r

∂∂ r

(r

∂w0

∂ r

), (27)

which satisfies

Lwr(w0) = 0. (28)

From (18) to (20) we can define the following zeroth-order deformation problems:

(1− q)Lwr[ w(r,q)−w0(r)] = qhwNwr[w(r,q)], (29)

∂ w(r,q)∂ r

= 0, at r = 0,

w(r,q) =−1, at r = h.(30)

In (29) and (30), hw denote the non-zero auxiliary pa-rameter, q ∈ [0,1] is the embedding parameter and

Nwr[w(r,q)] = (1− n)∂ 2w∂ r2 +

(1− n)r

∂w∂ r

+nWe

r

(∂w∂ r

)2

+ 2nWe∂ 2w∂ r2

∂w∂ r

− dPdz

.

(31)

890 S. Nadeem and N. Sher Akbar · Peristaltic Flow of a Tangent Hyperbolic Fluid

Obviously,

∂ w(r,0)∂ r

= w0, w(r,1) = w(r), (32)

when q varies from 0 to 1, then w(r,q) varies from ini-tial guess to the solution w(r). Expanding w(r,q) inTaylor series with respect to an embedding parame-ter q, we have

w(r,q) = w0(r)+∞

∑n=1

wm(r)qm, (33)

wm =1

m!∂ mw(r,q)

∂qm

∣∣∣∣q=0

. (34)

Differentiating the zeroth-order deformation m-timeswith respect to q and then dividing by m! and finallysetting q = 0, we get the following mth-order deforma-tion problem:

Lw[wm(r)− χmwm−1(r)] = hwRwr(r), (35)

where

Rwr = (1− n)w′′m−1 +

1r(1− n)w′

m−1

+nWe

r

m−1

∑i=0

w′m−1−kw′

k

+ 2nWem−1

∑k=0

w′m−1−kw′′

k −dPdz

(1− χm).

(36)

χm =

{0, m ≤ 1,

1, m > 1.(37)

The solution of the above equation with the help ofMathematica can be calculated and presented as fol-lows:

wm(r) = limM→∞

[ M

∑m=0

a0m,0

+2M+1

∑n=1

( 2M

∑m=n−1

2m+1−n

∑k=1

akm,nrn+2

)],

(38)

where a0m,0 and ak

m,n are constants.

5. Graphical Results and Discussion

In this section the pressure rise, frictional forces,and axial pressure gradient are discussed carefully and

Fig. 2. h-curve for the velocity field.

Fig. 3. Comparison of velocity field for We = 0.1, α = 0.1,φ = 0.3, z = 0.1, dP/dz = 0.8, n = 0.4, E = 15, and h =−1.

shown graphically (see Figs. 4 to 15). The pressurerise is calculated numerically by using Mathematica.Figures 4 to 7 show the pressure rise ∆P against thevolume flow rate Θ for different values of angle ofinclination α , Weissenberg number We, amplitude ra-tio φ , and tangent hyperbolic power law index n. Thesefigures indicate that the relation between pressure riseand volume flow rate are inversely proportional to eachother. It is observed from Figure 4 that with the in-crease in α , pressure rise increases. It is analyzedthat through Figures 5 and 6 that with an increasein n and We, pressure rise increases for 0 ≤ Θ ≤ 0.3,while decreases for 0.31 ≤ Θ ≤ 1. Figure 7 show thatwith an increase in φ the pressure rise decreases for0 ≤ Θ ≤ 0.5 while increases for 0.51 ≤ Θ ≤ 1. Peri-staltic pumping occurs in the region 0 ≤ Θ ≤ 0.3 forFigures 5 and 6, 0 ≤Θ ≤ 0.5 for Figure 7, otherwise

S. Nadeem and N. Sher Akbar · Peristaltic Flow of a Tangent Hyperbolic Fluid 891

Fig. 4. Pressure rise versus flow rate for We = 0.1, n = 0.02,φ = 0.01, and E = 0.2.

Fig. 5. Pressure rise versus flow rate for We = 0.2, α = π/2,φ = 0.05, and E = 1.

Fig. 6. Pressure rise versus flow rate for n = 0.05, α = π/2,φ = 0.05, and E = 1.

Fig. 7. Pressure rise versus flow rate for n = 0.05, α = π/2,We = 0.2, and E = 1.

Fig. 8. Frictional force versus flow rate for We = 0.1, n =0.02, φ = 0.01, and E = 0.2.

Fig. 9. Frictional forces versus flow rate for We = 0.2, α =π/2, φ = 0.05, and E = 1.

892 S. Nadeem and N. Sher Akbar · Peristaltic Flow of a Tangent Hyperbolic Fluid

Fig. 10. Frictional forces versus flow rate for n = 0.05, α =π/2, φ = 0.05, and E = 1.

Fig. 11. Frictional forces versus flow rate for n = 0.05, α =π/2, We = 0.2, and E = 1.

Fig. 12. Pressure gradient versus z for φ = 0.01, n = 0.04,We = 0.1, E = 0.3, and Θ =−4.

Fig. 13. Pressure gradient versus z for φ = 0.05, α = π/2,We = 0.2, E = 0.3, and Θ =−4.

Fig. 14. Pressure gradient versus z for φ = 0.05, α = π/2,n = 0.2, E = 0.3, and Θ =−4.

Fig. 15. Pressure gradient versus z for n = 0.05, α = π/2,We = 0.2, E = 1, and Θ =−4.

S. Nadeem and N. Sher Akbar · Peristaltic Flow of a Tangent Hyperbolic Fluid 893

Fig. 16. Shear stress versus z for n= 0.05, α = π/2, φ = 0.2,E = 1, and Θ =−4.

Fig. 17. Shear stress versus z for n= 0.05, We= 0.2, φ = 0.2,E = 1, Θ =−4, and α = π/2.

Fig. 18. Streamlines for different values of Θ ; (a) Θ = 0.1; (b) Θ = 0.2. The other parameters are We = 0.1, φ = 0.1, andn = 0.02.

Fig. 19. Streamlines for different values of n; (c) n = 0.1; (d) n = 0.2. The other parameters are We = 0.1, φ = 0.1, andΘ = 0.5.

894 S. Nadeem and N. Sher Akbar · Peristaltic Flow of a Tangent Hyperbolic Fluid

Fig. 20. Streamlines for different values of φ ; (e) φ = 0.1; (f) φ = 0.2. The other parameters are We = 0.1, n = 0.1, andΘ = 0.5.

Fig. 21. Streamlines for different values of We; (g) We = 0.1; (h) We = 0.2. The other parameters are φ = 0.1, n = 0.1, andΘ = 0.5.

augmented pumping occurs. Figures 8 to 11 describethe variation of frictional forces. It is seen that fric-tional forces have opposite behaviour as comparedwith the pressure rise. Figures 12 to 15 are preparedto see the behaviour of the pressure gradient for dif-ferent parameters. It is observed that for z ∈ [0,0.5]and [1.1,1.5] the pressure gradient is small, while thepressure gradient is large in the interval z ∈ [0.51,1].Moreover, it is seen that an increase in α , n, and Wecauses the decrease in the pressure gradient, while thepressure gradient increases with the increase in φ . Fig-ures 16 and 17 are prepared to see the variation ofthe shear stress for different values of α and We. Itis observed that with an increase in α and We, theshear stress increases. Figure 18 illustrates the stream-

line graphs for different values of the time mean flowrate Θ . It is observed that when we increase Θ the sizeof trapped bolus increases. Figure 19 shows the stream-lines for different values of (power law index) n. It isanalyzed that with the increase in n size and numberof trapping bolus decreases. Figure 20 illustrates thestreamline graphs for different values of φ . It is ob-served that with the increase in φ the trapped bolus in-creases. The streamlines for different values of We areshown in Figure 21. It is evident from the figure that thesize of the trapped bolus increases by increasing We.

Acknowledgement

The authors thank the Higher Education Commis-sion for financial support of this work.

S. Nadeem and N. Sher Akbar · Peristaltic Flow of a Tangent Hyperbolic Fluid 895

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