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Unsteady Axial Viscoelastic Pipe Flows of an Oldroyd B Fluid

By A. Abu-El Hassan and E. M. El-Maghawry

Submitted: May 24th 2012Reviewed: September 20th 2012Published: February 13th 2013

DOI: 10.5772/53638

Downloaded: 1372

1. Introduction

The unsteady flow of a fluid in cylindrical pipes of uniform circular cross-section has applications in medicine, chemical and petroleum industries [3,4,5]. For viscoelastic fluids, the unsteady axial decay problem for UCM fluid is considered by Rahman et al. [6]; and for Newtonian fluids as a special case. Rajagopal [7] has studied exact solutions for a class of unsteady unidirectional flows of a second-order fluid under four different flow situations. Atalik et al. [8] furnished a strong numerical evidence that non-linear Poiseuille flow is unstable for UCM, Oldroyd-B and Giesekus models. This fact is supported experimentally by Yesilata, [9]. The unsteady flow of a blood, considered as Oldroyd-B fluid, in tubes of rigid walls under specific APGs is concerned by Pontrelli, [10, 11].

Flow of a polymer solution in a circular tube under a pulsatile APG was investigated by Barnes et al. [12, 13].The same problem for a White-Metzner fluid is performed by Davies et al. [14] and Phan-Thien [15]. Recently, periodic APG for a second-order fluid has been studied by Hayat et al. [16]. Numerical simulation based on the role of the pulsatile wall shear stress in blood flow, is investigated by Grigioni et al. [1].

The present paper is concerned with the unsteady flow of a viscoelastic Oldroyd-B fluid along the axis of an infinite tube of circular cross-section. The driving force is assumed to be a time-dependent APG in the following three cases:

  1. APG varies exponentially with time,

  2. Pulsating APG,

  3. A starting flow under a constant APG.

2. Formulation of the problem

The momentum and continuity equations for an incompressible and homogenous fluid are given by

ρdq_dt=P+S__,E1

and

q_=0,E2

where ρ is the material density, q is the velocity field, p is the isotropic pressure and S̿is the Cauchy or extra-stress tensor. The constitutive equation of Oldroyd-B fluid is written as

T__=pI__+S;S__+λ1S__=μ{A1__+λ2A1__}E3

where T̿is the total stress, I̿is the unit tensor, μ is a constant viscosity, λ1 and λ2, (0λ2λ1) are the material time constants, termed as relaxation and retardation times; respectively. The deformation tensor A̿1 is defined by

 A__1=L__+L__T;L__=q_.E4

and “” denotes the upper convected derivative ; i.e. for a symmetric tensor G̿we get,

G__=G__t+q_G__G__L__L__TG__.E5

The symmetry of the problem implies that S̿and q depend only on the radial coordinate r in the cylindrical polar coordinates (r,θ,z) where the z-axis is chosen to coincide with the axis of the cylinder. Moreover, the velocity field is assumed to have only a z-component, i.e.

q_=(0,0,w_),E6

which satisfies the continuity equation (2) identically. The substitution of Eq. (6), into Eqs. (1) and (3) yields the set of equations

Srz+λ1Srzt=μ(wr+λ22wrt),E7
pz=Srzr+1rSrzρwt,E8
pr=pθ=0.E9

Equations (8) and (9) imply that the pressure function takes the form; p=zf(t)+c, so that

pz=f(t).E10

The elimination of Srzfrom (7) and (8) shows that velocity field w(r,t)is governed by:

ρ(λ12wt2+wt)μ(1+λ2t)(2wt2+1rwr)=(1+λ1t)pz.E11

The non-slip condition on the wall and the finiteness of won the axis give

w(r,t)|r=R=0andwr|r=0=0.E12

Introducing the dimensionless quantities

η=rR,τ=μtρR2,φ=μLΔPR2w,λ=λ2λ1andH=λ1μρR2=WeRe,E13

where R is the radius of the pipe, ΔPa characteristic pressure difference, L is a characteristic length, Weand Reare the Weissenberg and Reynolds numbers; respectively, into Eqs. (10),(11) and (12) we get

H2φτ2+φτ[1+λHτ][2φη2+1ηφη]=[1+Hτ]Ψ(τ),E14

with the BCs.

φ(1,τ)=0andφ(0,τ)η=0,E15

and

Ψ(τ)=LΔppz=LΔpf(t).E16

Equation (14) subject to BCs. (15) is to be solved for different types of APGs; i.e. different forms of the function Ψ(τ).

3. Pressure gradient varying exponentially with time

We consider the two cases of exponentially increasing and decreasing with time APGs separately.

3.1. Pressure gradient increasing exponentially with time

Let,

Ψ(τ)=LΔppz=Keα2τ,E17

and assume that

φ(η,τ)=g(η)eα2τ,E18

where Kand αare constants. The substitution of Eqs. (17) and (18) into Eq. (14) leads to

g+1ηgα2(Hα2+1)λHα2+1g=KHα2+1λHα2+1,E19

while the BCs. (15) reduce to

g(1)=0,g(0)=0E20

A solution of Eq. (19) subject to the BCs. (20) is

g(η)=Kα2[1I0(βη)I0(β)],E21

where I0 (x) is the modified Bessel-functions of zero-order, and

β2=α2(1+Hα2)1+λHα2.E22

Therefore, the velocity field is given by

φ(η,τ)=Kα2[1I0(βη)I0(β)]eα2τ.E23

The solution given by Eq. (23) processes the following properties:

  1. The time dependence is exponentially increasing such that for η ≠ 1 limτϕ(η,τ). It may be recommendable to choose another APG which increases up to a certain finite limit in order to keep ϕ(η,τ)finite.

  2. The present solution depends on the parameter β in the same form as the solution for the UCM [6]. For any value of β the Oldroyd-B fluid exhibits the same form as the UCM- fluid. However, in the present case β depends on λ in addition to H and α2. A close inspection show that limλ0β2=β2for the UCM-fluid while the limλ1β2=α2which coincides with the case of the Newtonian fluid, [8].

  3. The parameter β is inversely proportional to λ where the decay rate increases by increasing the value of H. However, as mentioned above, as λ approaches the value λ = 1 all the curves matches together approaching the value β2= α2 asymptotically. The behavior of β as a function of λ, where H is taken as a parameter is shown in Fig. (1).

Figure 1.

The λβ - relation H= 2, 3, 5, 7, 9, (Bottom to top)

For small values of |β|and by using the asymptotic expansion of I0 (x),

it can be shown that the velocity profiles approaches the parabolic distribution;

Limβ0φ(η,τ)=K(Hα2+1)4(λHα2+1)(1η2)eα2τE24

For the case of large |β|the velocity distribution is given as;

Limβφ(η,τ)=Kα2[11ηeβ(1η)]eα2τE25

This solution is completely different from the parabolic distribution and it depends on ηonly in the neighborhood of the wall. Therefore, such a fluid exhibits boundary effects.

The rising-APG velocity field ϕ(η,τ)is plotted in Figs. (2a) and (2b) as a function of ηat different values of β for α=2and α=5.

Figure 2.

Rising – (a,b) APG velocity filed ;β=5.2,  3.5,  2.5,  2.1 (Bottom to top) Fig. (c) : Decreasing – APG velocity filed ;β=8.7,  3.9,  2.6,  2.1 ( Top to Bottom) Fig. (d) : Decreasing – APG velocity filed ;β=16.4,  9.2,  6.5,  5.3 ( Top to Bottom)

3.2. Pressure gradient decreasing exponentially with time

The solution at present is obtained from the previous case by changing α2by α2. Therefore,

φ(η,τ)=Kα2[1j0(β1η)j0(β1)]eα2τ.E26

where

β12=α2(1Hα2)1λHα2E27

The discussion of this solution is similar to the case of increasing APG except that the velocity decays exponentially with time and the value α2=1/λHis not permissible as it leads to infinite β12; i.e.

limα121/λHβ12E28

The two cases of small and large |β1|produce similar results as the previous solution. Thus

Limβ10ϕ(η,τ)=K4α2β12(1η2)eατ,E29

and

Limβ1φ(η,τ)=Kα2[11ηcos(β1ηπ4)cos(β1π4)]eα2τ.E30

4. Pulsating pressure gradient

The present case requires the solution of Eq. (14) subject to BCs. (15) in the form

Ψ(τ)=LΔPpz=Keinτ;i=1,E31

K and n are constants. Assuming the velocity function has the form

φ(η,τ)=re[f(η)einτ],E32
f+1ηfin(1+inH)(1+inλH)f=K(1+inH)(1+inλH).E33

The solution of this equation satisfying the BCs. (15) is :

f(η)=kin[1I0(βη)I0(β)],β2=in(1+inH)(1+inλH).E34

Hence, the velocity distribution is given by:

ϕ(η,τ)=re{kineinτ[1I0(βη)I0(β)]}.E35

Obviously; for small |β|,

Limβ0f(η)=Kin(β2(1η2)4).E36

and for large |β|

LimβI0(βη)I0(β)=1ηeβ(1η),E37

So that,

φ(η,τ)=re{Kin[11ηeβ(1η)einτ]},E38

where,

β2=in(1+inH)(1+inλH)=11+n2λ2H2[n2H(λ1}+in(1+n2λH2)],E39

or simply,

β=eiθ/2,E40
=n1+n2λ2H2n2H2(1λ)2+(1+n2λH2)2,E41
θ2=-12Tan-1[1+n2λHnH(1-λ)].E42

Substituting from Eqs.(40,41,42) into Eq. (37), we get:

Limβφ(η,τ)=kn{sinnτ1ηe(1η)cos(θ/2)sin[nτ(1ηsinθ2)]}.E43

As λ0, [6], r1=n1+n2H2and θ2θ12=12Tan1(1nH).

Then

φ(η,τ)=kn{sinnτ1ηe(1η)r1(cosθ1/2)sin[nτ(1η)r1(sinθ12)]}E44

The velocity field φ(η,τ) is plotted in Figs. (3a) and (3b); respectively, against η for different values of β. The two limiting cases for small and large |β| are represented in three-dimensional Figs. (4a) and (4b) in order to emphasize the oscillating properties of the solution.

Figure 3.

a) : Pulsating – APG ; n=2, H=5, β=3.7 ,  2.5 ,  1.8 ,  1.5 (b) : Pulsating – APG ; n=5, H=5, β=6.8 ,  4.1 ,  2.9 ,  2.4 [Top to Bottom for all]

Figure 4.

a): Pulsating-APG, n = 2, H = 5, at small |β| ; β=3.7 (b): Pulsating-APG, n = 3, H = 5, at large |β| ; β=6.8

5. Constant pressure gradient

Here we consider the flow to be initially at rest and then set in motion by a constant ABG “-K”. Hence, Ψ(τ); Eq.(14), subject to BCs. (15) reduces to

LΔPPz=K.E45

Therefore, we need to solve the equation

H2Φτ2+Φτ[1+λHτ][1ηΦη+2Φη2]=K,E46

subject to the boundary and initial conditions

ϕ(1,τ) = 0, for τ0 ,

ϕ(η,0)=0,for0η1E47

Equation (46) can be transformed to a homogenous equation by the assumption

Φ(η,τ)=K4(1η2)ψ(η,τ),E48

where Ψ(η,τ) represents the deviation from the steady state solution. Hence,

[τ(1+Hτ)(1+λHτ)(2η2+1ηη)]ψ=0,E49

subject to the boundary and initial conditions

Ψ(1,τ)=0forτ0.E50
ψ(η,0)=K4(1η2)for0η1.E51

Assuming that ψ(η,τ)=F(η)G(τ), Eq.(49) separates to

HG+(1+λHα2)G+α2G=0,E52
F+η1(1+λHα2)F+α2F=0.E53

Equation (52) has the solution,

G(τ)=Aeγ1τ+Beγ2τE54

where γ1 and γ2 are the roots of the Eq.( 52). On the other hand, Eq. (53 ) has the solution

F(η)=J0(αmη).E55

Therefore,

γ1,2=(1+λHαm2)±(1+λHαm2)24αm2H2H.E56

The BCs. (50,51) implies that the constant αm takes all zeros of the Bessel-function J01, α2, ………). Hence,

Ψ(η,τ)=m=1J0(αmη)G(τ),E57
Ψ(η,τ)=m=1J0(αmη)(Ameγ1mτ+Bmeγ2mτ).E58

Τhe initial condition (50) and BCs. (51) will not be sufficient to evaluate the constants Am and Bm. Hence, it is required to employ another condition. We assume that G(τ) is smooth about the value τ = 0 and can be expanded in a power series about τ = 0. Assuming G(τ) to be linear function of τ in the domain about τ = 0, then G=0in Eq. (52). Hence

(1+λHαm2)Gm`(0)+αm2Gm(0)=0,E59
Gm(τ)=Ameγ1mτ+Bmeγ2mτ,E60

From which we obtain

Am[(1+λHαm2)γ1m+αm2]+Bm[(1+λHαm2)γ2m+αm2]=0.E61

To determine the constants Am and Bm we firstly satisfy the remaining condition (51). Owing to Eq. (58) and the initial condition, Eq. (51), we notice that,

Ψ(η,0)=m=1(Am+Bm)J0(αmη)=K4(1η2).E62

Via the Fourier–Bessel series, Eq. (62) leads to,

Am+Bm=K2J12(αm)01η(1η2)J0(αmη)dη.E63

Performing this integration we get

Am+Bm=2Kαm3J1(αm)Kαm2J0(αm)J12(αm).E64

From Eqs. (61) and (64) we obtain :

Am=[(1+λHαm2)γ2m+αm2](1+λHαm2)(γ2mγ1m)[2Kαm3J1(αm)Kαm2J0(αm)J12(αm)],E65
Bm=[(1+λHαm2)γ1m+αm2](1+λHαm2)(γ1mγ2m)[2Kαm3J1(αm)Kαm2J0(αm)J12(αm)].E66

Finally, the velocity field has the series representation

ϕ(η,τ)=K4(1η2)m=1J0(αmη)(1+λHαm2)(γ2mγ1m){[(1+λHαm2)γ2m+αm2]eγ1mτ

[(1+λHαm2)γ1m+αm2]eγ2mτ}[2Kαm3J1(αm)Kαm2J0(αm)J12(αm)].E67

The constant-APG velocity field φ(η,τ) as a function of η shown in Fig. (5).

Figure 5.

The velocity distribution for constant – APG taking H=0.2, τ=0.1 where the summation is taken for α1=2.4, α2=5.8, α3=8.4

6. Results and discussion

The behavior of |β| as a function of λ where H is taken as a parameter is shown in Fig. (1). The behavior of β is inversely proportional to λ while it is fast-decreasing for higher H-values. For any β-value, the Oldroyd-B fluid exhibits the same form as the UCM-fluid. A close inspection of β2 = α2(1+α2H)/(1+λα2H) shows that UCM-fluid is obtained by limλ0β2=β2while limλ0β2=α2leads to the case of Newtonian fluid. For small values of |β| as well as |βη| and by using the asymptotic expansion of I0(x), it can be shown that the velocity profiles approaches the parabolic distribution.

For decay-APGs, Figs. (2a) and (2b) show that the velocity profiles of Oldroyd-B and UCM fluids are parabolic for small values of |βη| while for large |βη| they are completely different from this situation. The solutions depend on η only in the neighboring of the wall. Therefore, such fluids exhibit boundary layer effects [17]].

For pulsating-APG, the velocity distribution is represented in Figs. ( 3a ) and ( 3b). The smallest value of β in both curves is almost parabolic as shown by Eq. (36) while the largest value exhibits boundary effect as reviled by Eq.( 43 ). To emphasize the oscillating nature of the solution a three-dimensional diagrams (4a) and (4b) for the smallest and largest values of |β| are respectively sketched.

Grigioni, et al [1], studided the behavior of blood as a viscoelastic fluid using the Oldroyd-B model. The results obtained for the velocity distribution stands in agreement with the obtained results in the present work.

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A. Abu-El Hassan and E. M. El-Maghawry (February 13th 2013). Unsteady Axial Viscoelastic Pipe Flows of an Oldroyd B Fluid, Rheology - New Concepts, Applications and Methods, Rajkumar Durairaj, IntechOpen, DOI: 10.5772/53638. Available from:

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