An analysis of peristaltic motion of compressible convected maxwell fluid

An analysis of peristaltic motion of compressible convected Maxwell fluid An analysis of peristaltic motion of compressible convected Maxwell fluid A Abbasi, , I Ahmad, N Ali, and T Hayat Citation AIP[.]

A Abbasi, I Ahmad, N Ali, and T Hayat

Citation: AIP Advances 6, 015119 (2016); doi: 10.1063/1.4940896

View online: http://dx.doi.org/10.1063/1.4940896

View Table of Contents: http://aip.scitation.org/toc/adv/6/1

Published by the American Institute of Physics

AIP ADVANCES 6, 015119 (2016)

An analysis of peristaltic motion of compressible

convected Maxwell fluid

A Abbasi,1, aI Ahmad,1N Ali,2and T Hayat3,4

1Department of Mathematics, University of Azad Jammu& Kashmir, Muzaffarabad 13100,

Pakistan

2Department of Mathematics and Statistics, International Islamic University, Islamabad

44000, Pakistan

3Department of Mathematics Quaid-I-Azam University, Islamabad 44000, Pakistan

4Nonlinear Analysis and applied Mathematics (NAAM) Research Group, Department of

Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia

(Received 2 December 2015; accepted 11 January 2016; published online 29 January 2016)

This paper presents a theoretical study for peristaltic flow of a non-Newtonian compressible Maxwell fluid through a tube of small radius Constitutive equation

of upper convected Maxwell model is used for the non-Newtonian rheology The governing equations are modeled for axisymmetric flow A regular perturbation method is used for the radial and axial velocity components up to second order

in dimensionless amplitude Exact expressions for the first-order radial and axial velocity components are readily obtained while second-order mean axial velocity component is obtained numerically due to presence of complicated non-homogenous term in the corresponding equation Based on the mean axial velocity component, the net flow rate is calculated through numerical integration Effects of various emerging parameters on the net flow rate are discussed through graphical illustrations It

is observed that the net flow rate is positive for larger values of dimensionless relaxation time λ1 This result is contrary to that of reported by [D Tsiklauri and

I Beresnev, “Non-Newtonian effects in the peristaltic flow of a Maxwell fluid,” Phys Rev E 64 (2001) 036303].” i.e in the extreme non-Newtonian regime, there

is a possibility of reverse flow C 2016 Author(s) All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).[http://dx.doi.org/10.1063/1.4940896]

I INTRODUCTION

Peristaltic motion of fluids has now acquired a special status amongst the recent investigators Such motivations infect stems due to applications of peristaltic transport of fluids in the engineering, industry and physiology In particular such activity is quite prevent in the urine transport from kidney to the bladder, chyme motion in the gastrointestinal tract, locomotion of worms, sperma-tozoa transport, egg moment in the female fallopian tube, roller and figure pumps, dialysis etc Abundant information on the topic is available for viscous materials since the seminal work of Latham,1Shapiro et al.2and Fung and Yih.3Peristaltic transport of viscous materials in the existing studies have been especially examined through different aspects of heat and mass transfer, convec-tive conditions, compliant boundary, various waveforms, symmetric and axisymmetric channels, uniform and non- uniform flow configurations etc All these studies have been presented through the use of one or more assumptions of long wavelength, low Reynolds number, small wave number and small amplitude of wave In continuation reasonable literature also exists at present for the peristaltic transport of non-Newtonian liquids in view of afore stated aspects and assumptions Mention may be made to few studies in this direction through the attempts4 13 and many refs

a Corresponding author: Tel :- +923457347636 (e-mail: aamir_mathematics@yahoo.com )

There in It has been noted that from the past studies that much attention to peristaltic transport has been focused to an incompressible viscous and non-Newtonian fluids The relevant studies analyzing peristaltic flows of compressible fluids are very few Such investigations further narrowed down when compressible non-Newtonian fluids are consider To our information, Tsiklauri and Beresnev14discussed the effects of relaxation time in the peristaltic flow of compressible Maxwell fluid The authors especially discussed the net flow rate using linear model of Maxwell fluid Here

it is found that in extreme non-Newtonian regime back flow occurs and the net flow rate is highly oscillatory Hayat et al.15extended the work in Ref.14for linear Jeffrey fluid Their study showed that behavior of net flow rate in linear Jeffrey fluid is less oscillatory than that of linear Maxwell fluid The slip effects on the net flow rate through a tube of small radius were studied theoretically

by EL-Shehawy et al.16They noted that net flow rate is nearly independent of slip parameter for values of tube radius α < 0.001 Moreover, the reversal flow occurs at high values of compress-ibility and slip parameter Mekheimer and Wahab17 discussed the effects of wall compliance on compressible fluid transport induced by surface acoustic wave in a micro channel In all the studies mentioned above linear constitutive equations of Newtonian or non-Newtonian are utilized How-ever, the use of linear constitutive equation makes the applicability of the work to any real flow highly restricted Therefore, there is a need to consider nonlinear constitutive equation such as upper convected Maxwell or Oldroyd equation to bring out more realistic physics In this present paper

we put forward a study where constitutive equation of upper convected Maxwell fluid is used to investigate the interaction of rheology with net flow rate through a tube of small radius To the best

of our information such type of study is not available in the literature The paper is arranged in the following manner The problem is modeled with appropriate assumptions in Section II Solution

of the problem is constructed in Section III SectionIV consists of results and discussion The observations are concluded in SectionV

II MATHEMATICAL FORMULATION

Consider the flow of an upper convected Maxwell fluid through an axisymmetric tube of radius

Rshown in Fig.1 Assume that peristaltic waves of amplitude a, wavelength λ and speed c propa-gate along the tube wall A cylindrical polar coordinate system(r, θ, z) with r in the radial direction and z along the centerline of the tube is chosen for the flow analysis The displacement of tube wall

is expressed as follow:

h(z,t)= R + a cos2πλ (z − ct) (2.1) The equations for the flow of compressible non-Newtonian fluid are given by:

ρ,t+ (ρui

ρai= τi j

FIG 1 Geometry of the problem.

015119-3 Abbasi et al. AIP Advances 6, 015119 (2016)

where ρ is the fluid density,t is the time, uiare the velocity components, τijare the components of Cauchy stress tensor obeying the relation:

and aiare the components of acceleration vector The units of each term in Eq (2.3) are kgm−2s−2

In expression (2.4) p is the pressure, δij is the kronecker delta and the components of extra stress tensor Si jfor upper convected Maxwell fluid19 , 20satisfy

(

1+ λ1D Dt

)

in which

˙

γi j= ui , j+ uj ,i− 2

3u

i

Here λ1is the relaxation time, µ is dynamic viscosity and D/Dt is contravariant convected deriva-tive which for any contravariant vector bisatisfies:

Dbi

Dt =∂b∂t +i vrb,ri −vi

It is further assumed that the following equation of the state holds

1 ρ

dρ

The solution of above equation turns out to be

In above equation k is fluid compressibility and ρ0 is constant density at the reference pressure p0 The relevant no-slip boundary conditions are:

u(h, z,t)= ∂h∂t, w (h, z,t) = 0, (2.10) where it is assumed that u1= uu2= 0,u3= w

The flow rate through the tube is given by:

Q=

 h(z,t) 0

In view of Eq (2.4), Eq (2.3) becomes

ai= −p,i+ Si j

Applying the operator 1+ λ1D

Dt
to the above equation, one has:

(

1+ λ1 D

Dt

)

ai=

(

1+ λ1D Dt

) (−p,i+ Si j

, j)

= −(1+ λ1

D Dt

) (p,i) +

(

1+ λ1D Dt

)

Si j, j

= −(1+ λ1D

Dt

) (p,i) +

 (

1+ λ1 D Dt

)

Si j



, j (2.13)

where we have assume d(D/Dt), j = 0 This assumption was already used by Harris17in deriving the boundary layer equations for upper convected Maxwell fluid Invoking Eq (2.5) into (2.13) we get the following equation

(

1+ λ1 D Dt

)

ai= −

(

1+ λ1D Dt )

p,i+ µ ˙γi j, j (2.14)

Taking u1= u, u2= 0, u3= w we can write Eqs (2.2) and (2.14) in component form as follows:

∂ ρ

∂t +u

∂ ρ

∂r +w

∂ ρ

∂z +ρ

(∂u

∂r +

u

r +∂w∂z

)

ρ

(∂u

∂t +u

∂u

∂r +w

∂u

∂z

) + λ1

 ρ

(∂2u

∂t2 +∂u∂t ∂u∂r +u ∂2u

∂t∂r +

∂w

∂t

∂u

∂z +w

∂2u

∂t∂z )

+

(∂u

∂t +u

∂u

∂r +w

∂u

∂z

) ∂ ρ

∂t +u









ρ* ,

∂2u

∂t∂r +

(∂u

∂r

)2 + u∂∂r2u2 +∂w∂r ∂u∂z +w ∂2u

∂r∂z+ -+ (∂u∂t +u∂u

∂r +w

∂u

∂z

) ∂ ρ

∂r

 + wρ( ∂2u

∂t∂z +

∂u

∂r

∂u

∂z +u

∂2u

∂r∂z +

∂w

∂z

∂u

∂z +w

∂2u

∂z2 )

+ (∂u∂t +u∂u

∂r +w

∂u

∂z

) ∂ ρ

∂z



−∂u

∂r



ρ(∂u

∂t +u

∂u

∂r +w

∂u

∂z

) 

−∂u

∂z



ρ(∂w

∂t + u∂w∂r +w∂w

∂z

)  +∂t∂r +∂2p u∂2p

∂r2 + w∂r∂z∂2p −∂u

∂r

∂p

∂r −

∂u

∂z

∂p

∂z



= −∂p∂r +µ

∂2u

∂r2 +1 r

∂u

∂r −

u

r2+∂∂z2u2 +1

3

∂

∂r

(∂u

∂r −

u

r +∂w∂z

) 

ρ

(∂w

∂t +u

∂w

∂r +w

∂w

∂z

) + λ1

 ρ

(∂2w

∂t2 +∂u∂t ∂w∂r +u∂2w

∂t∂r +

∂w

∂t

∂w

∂z +w

∂2w

∂t∂z )

+

(∂w

∂t +u

∂w

∂r +w

∂w

∂z

) ∂ ρ

∂t +u

 ρ

(∂2w

∂t∂r +

∂u

∂r

∂w

∂r +u

∂2w

∂r2 +∂w∂r ∂w∂z +w ∂2w

∂r∂z )

+

(∂w

∂t +u

∂w

∂r +w

∂w

∂z

) ∂ ρ

∂r



−ρ∂w

∂r

∂u

∂t +u

∂u

∂r +w

∂u

∂z



−ρ∂w

∂z

∂w

∂t +u

∂w

∂r +w

∂w

∂z



+ w







ρ*

,

∂2w

∂t∂z +

∂u

∂r

∂w

∂z +u

∂2w

∂r2 +

(∂w

∂z

)2 + w∂∂z2w2+

-+

(∂w

∂t +u

∂w

∂r +w

∂w

∂z

)∂ ρ

∂z







 +∂t∂z +∂2p u∂2p

∂r2 + w∂∂z2p2 −∂w

∂r

∂p

∂r −

∂w

∂z

∂p

∂z



= −∂p∂z +µ 1

r

∂

∂r

(

r∂w

∂r

) +∂∂z2w2



+µ

3

∂

∂z

(∂u

∂r +

u

Introducing the dimensionless variables ˜h= h/R, ˜ρ = ρ/ρ0,, ˜u = u/c, ˜w = w/c, ˜λ1= λ1R/c, and ˜p= p/ρ0c2, the Eqs (2.15)-(2.17) can be put in the following dimensionless form

∂ ρ

∂t +u

∂ ρ

∂r +w

∂ ρ

∂z +ρ

(∂u

∂r +

u

r +∂w∂z

)

ρ

(∂u

∂t +u

∂u

∂r +w

∂u

∂z

) + λ1

 ρ

(∂2u

∂t2 +∂u∂t ∂u∂r +u ∂2u

∂t∂r +

∂w

∂t

∂u

∂z +w

∂2u

∂t∂z )

+

(∂u

∂t +u

∂u

∂r +w

∂u

∂z

) ∂ ρ

∂t +u









ρ* ,

∂2u

∂t∂r +

(∂u

∂r

)2 + u∂∂r2u2 +∂w∂r ∂u∂z +w ∂2u

∂r∂z+ -+ (∂u∂t +u∂u

∂r +w

∂u

∂z

) ∂ ρ

∂r

 + wρ( ∂2u

∂t∂z +

∂u

∂r

∂u

∂z +u

∂2u

∂r∂z +

∂w

∂z

∂u

∂z +w

∂2u

∂z2 )

+ (∂u∂t +u∂u

∂r +w

∂u

∂z

) ∂ ρ

∂z



−∂u

∂r



ρ(∂u

∂t +u

∂u

∂r +w

∂u

∂z

) 

−∂u

∂z



ρ(∂w

∂t + u∂w∂r +w∂w

∂z

)  +∂t∂r +∂2p u∂2u

∂r2 + w∂r∂z∂2p −∂u

∂r

∂p

∂r −

∂u

∂z

∂p

∂z



= −∂p∂r + 1

Re

∂2u

∂r2+1 r

∂u

∂r −

u

r2+∂∂z2u2+1

3

∂

∂r

(∂u

∂r −

u

r +∂w∂z ) 

015119-5 Abbasi et al. AIP Advances 6, 015119 (2016)

ρ

(∂w

∂t +u

∂w

∂r +w

∂w

∂z

) + λ1

 ρ

(∂2w

∂t2 +∂u∂t ∂w∂r +u∂2w

∂t∂r +

∂w

∂t

∂w

∂z +w

∂2w

∂t∂z )

+

(∂w

∂t +u

∂w

∂r +w

∂w

∂z

) ∂ ρ

∂t +u

 ρ

(∂2w

∂t∂r +

∂u

∂r

∂w

∂r +u

∂2w

∂r2 +∂w∂r ∂w∂z +w ∂2w

∂r∂z )

+

(∂w

∂t +u

∂w

∂r +w

∂w

∂z

) ∂ ρ

∂r



−ρ∂w

∂r

∂u

∂t +u

∂u

∂r +w

∂u

∂z



−ρ∂w

∂z

∂w

∂t +u

∂w

∂r +w

∂w

∂z



+ w







ρ*

,

∂2w

∂t∂z +

∂u

∂r

∂w

∂z +u

∂2w

∂r2 +

(∂w

∂z

)2 + w∂∂z2w2+

-+

(∂w

∂t +u

∂w

∂r +w

∂w

∂z

)∂ ρ

∂z







 +∂t∂z +∂2p u∂2p

∂r2 + w∂∂z2p2 −∂w

∂r

∂p

∂r −

∂w

∂z

∂p

∂z



= −∂p∂z + 1

Re

 1 r

∂

∂r

(

r∂w

∂r

) +∂∂z2w2



+ 1

3Re

∂

∂z

(∂u

∂r +

u

where the tilde sign is dropped for simplicity

Equation (2.9) in dimensionless form reads as

In Eqs (2.15)-(2.21) Re= (ρ0cR/µ), is the Reynolds number λ1 is the dimensionless relaxation time and χ= (k ρρ0c2

) is a compressibility parameter The boundary conditions in dimensionless form can be expressed as follows:

u(1+ η (z,t) , z,t) =∂h∂t, w (1 + η (z,t) , z,t) = 0, (2.22) where η(z,t)= ε cos α (z − t), ε = (a/R) is the amplitude ratio and α = (2πR/λ) denotes the dimensionless pore radius

III SOLUTION PROCEDURE

For the solutions of Eqs (2.18)-(2.21) subject to boundary conditions (2.22), we expand pres-sure, velocity components and density as16 – 18

p= p0+ ε p1(r, z,t)+ ε2p2(r, z,t)+ ,

u= ε u1(r, z,t)+ ε2

u2(r, z,t)+ ,

w = ε w1(r, z,t)+ ε2w2(r, z,t)+ ,

ρ = 1 + ε ρ1(r, z,t)+ ε2ρ2(r, z,t)+ ,

(3.1)

Substituting Eq (3.1) into Eqs (2.18)-(2.21) and comparing the coefficient of various powers of ε,

we get a closed set of equations at first (ε) and second (ε2) orders For the first order system, we seek the solution in the following form

u1(r, z,t)= U1(r) eiα(z−t)+ U1(r) e−iα(z−t), w1(r, z,t)= W1(r) eiα(z−t)+ W1(r) e−iα(z−t), p1(r, z,t)= P1(r) eiα(z−t)+ P1(r) e−iα(z−t),

ρ1(r, z,t)= χP1(r) eiα(z−t)+ χP1(r) e−iα(z−t),

(3.2)

where over bars denotes the complex conjugates Now adopting the same procedure as given in Ref.10we arrive at the following equations and boundary condition in terms of U1,W1,P1

(1 − iαλ1)P1′+ 1

Re

(

U′′1+U′1

r −

U1

r2 −α2U1

) + 1 3Re

d dr

(

U′1+U1

r − iαw1

)

= iα(1 − iαλ1) U1, (3.3)

−(1 − iαλ1)P1+ 1

Re

(

w′′

1+w′1

r −α2w1

) + iα 3Re

d dr

(

U′1+U1

r − iαw1

)

= iα(1 − iαλ1)w1, (3.4)

U1(1)= −iα

Omitting the detail of calculations, we get the equation for U1as follows:

B( d2

dr2+1 r

d

dr −

1

r2−ν2) ( d2

dr2+1 r

d

dr −

1

r2−β2

)

where B= 1 −iα χ

γ , ν2= β2−αB 2, γ = (1 − iαλ1)Re −iα χ

3 , β2= α2− iαRe(1 − iαλ1) The solution of (3.7) is given by

U1(r)= c1I1(νr)+ c2I1( βr) (3.8) where c1and c2are the complex constants and I1is the modified Bessel function of the first kind of order 1 The expression for p1is:

p1(r)= c3+ c1ν2νγ−β2I0(νr) , (3.9)

in which c3is complex constant and I0is the modified Bessel function of the first kind of order zero Similarly

w1(r)=iαc1ν I0(νr)+iαc2β I0( βr)+ χc3 (3.10)

In view of Eq (3.6) c3 turns out to be zero and c1 and c2 are obtained by using the boundary conditions given in (3.7) Now we proceed to find the second order solution To this end we assume

u2(r, z,t)= U20(r)+U2(r) e2iα(z−t)+ ¯U2(r) e−2iα(z−t), w2(r, z,t)= W20(r)+W2(r) e2iα(z−t)+ ¯W2(r) e−2iα(z−t), p2(r, z,t)= P20(r)+P2(r) e2iα(z−t)+ ¯P2(r) e−2iα(z−t), ρ2(r, z,t)= D20+D2(r) e2iα(z−t)+ ¯D2(r) e−2iα(z−t)

(3.11)

The inclusion of non-oscillatory terms in W2, U2, P2 and ρ2 results from the fact that peristaltic flow is essentially nonlinear second effect and adding such terms in first order gives a trivial solu-tion Without going into detail of algebra involved, we get the following equations and boundary conditions for W20(r):

W20′′(r)+1

rW20(r)= (1 + iαλ1)1

r

d

dr r W1(r) ¯U1(r)+ ¯W1(r) U1(r)


+Reλ1iα(U1(r) ¯W1(r) − 3 ¯U1(r) W1(r) − U1′ (r) ¯P′(r) − ¯U1(r) P1(r))′

−2α2(iα(W1(r) ¯W1(r) − W1(r) ¯P1− P1W1) − W1¯ (r) ¯P′(r) − ¯W1′(r) P1(r)′

, (3.12) W20(1)+ 0.5W′

1(1)+ 0.5 ¯W1(1)= 0 (3.13) The dimensionless flow rate induced by the traveling wave is

Q(z,t)= 2π

 1 0

In view of (3.1), we can write

Q(z,t)= 2π

 ϵ

 1 0 rW1(r, z,t) dr+ ϵ2

 1 0 rW2(r, z,t) dr



The dimensionless flow rate averaged over one period 2π/α of time is defined as net dimensionless flow rate i.e

⟨Q⟩ =2π/α1

 2π/α

015119-7 Abbasi et al. AIP Advances 6, 015119 (2016)

Invoking (3.15) into (3.16), we get

⟨Q⟩ = 2πε2

 1 0

As evident from (3.17), the evaluation of⟨Q⟩ depends on W20(r) which can be obtained by Eqs (3.12) and (3.13) However, Eq (3.12) cannot be solved with the well-known analytical methods due to presence of complicated non- homogenous term and thus numerical solution is obtained using Matlab built-in routine bvp4c The Matlab built-in routine bvp4c is a finite difference code that implements the 3-stage Lobatto IIIa formula This is a collocation formula and the collocation polynomial provides a C1-continuous solution that is fourth order accurate uniformly in [a, b] For multipoint boundary value problems, the solution is C1-continuous within each region, but conti-nuity is not automatically imposed at the interfaces Mesh selection and error control are based on the residual of the continuous solution Analytical condensation is used when the system of alge-braic equations is formed For further details readers are referred to.21After obtaining the solution, numerical integration is utilized to get the net flow rate⟨Q⟩

IV RESULTS AND DISCUSSION

This section displays the graphical results illustrating the effects of various emerging parame-ters on the net flow rate

In Fig 2, the net flow rate ⟨Q⟩ is plotted against the dimensionless pore radius α for both linear (panel (a)) and convected Maxwell models (panel (b)) for different values of λ1i.e λ1= 100 and λ1= 1000 For the purpose of comparison we have included the net flow rate curve for New-tonian fluid (λ1= 0) in panel (a) Panel (a) comprises of three distinct curves The solid curve

is for Newtonian fluid It is noted that for a Newtonian fluid ⟨Q⟩ shows increasing behavior for

0 ≤ α ≤ 0.003 and then becomes independent of α The dashed curve in panel (a) indicates the behavior of flow rate for linear Maxwell fluid when λ1= 100 Here we can see that ⟨Q⟩ increases with α up to a certain value and then started to decrease with α The behavior of flow rate with

α for linear Maxwell model is shown through the dotted curve in panel (a) when λ1= 1000 This curve predicts oscillatory behavior of⟨Q⟩ with α Moreover, the negative values attained by ⟨Q⟩ confirm the possibility of back flow The above results are not new and already reported in the literature by Tsiklauri and Beresnev.14However, for the convected Maxwell model the results are quite different It is observed from (dash-dotted curve) in panel (b) that in contrast to linear Maxwell model with λ1= 100, the flow rate is oscillatory and attains negative values for convected Maxwell model with λ1= 100 With further increase in λ1i.e λ1= 1000 the flow rate for convected Maxwell model (solid line in panel (b)) becomes oscillatory and negative similar to the flow rate for linear Maxwell model when λ1= 1000 A close look at the flow rate curve of convected Maxwell model for λ1= 1000 reveals very interesting results We see that flow rate for linear Maxwell fluid for

FIG 2 Plot of ⟨Q⟩ verses α when χ = 0.6, ϵ = 0.001 and Re = 10000.

(a) (b)

FIG 3 Plot of ⟨Q⟩ verses α when χ = 0.6, ϵ = 0.001 and Re = 10000.

λ1= 1000 oscillates about the mean value zero (roughly) Howeve, no such mean value exists for the case of convected Maxwell model Further⟨Q⟩ attains lower negative values for linear Maxwell fluid in comparison with convected Maxwell fluid This indicates strong back flow induced by peristaltic motion for linear Maxwell fluid in contrast to convected Maxwell fluid where the back flow induced by peristaltic waves is weak

The behavior of the flow rate ⟨Q⟩ of linear and convected Maxwell models for λ1= 10000

is shown in Fig 3 Fig 3(a) indicates that for large values of λ1, flow rate in case of linear Maxwell model is highly oscillatory The negative values attained by⟨Q⟩ in Fig.3(a)are indicator

of back flow Fig.3(b)illustrates the behavior of⟨Q⟩ in case of convected Maxwell models when λ1= 10000 Here, the behavior of ⟨Q⟩ is clearly oscillatory but no back flow occurs A quantitative comparison of Fig 3(a)and Fig.3(b)depicts higher values of⟨Q⟩ for convected Maxwell model when compared to linear Maxwell model For example, if we take α= 0.001 then the value of flow rate for linear Maxwell model is 99.8% less than the corresponding value of convected Maxwell model

The plots of flow rate for Newtonian(λ1= 0), linear and convected Maxwell models are shown

in Fig.4when λ1= 100 This Fig shows that the qualitative behavior of ⟨Q⟩ is similar for all three fluids However, quantitatively ⟨Q⟩ attains lower values for convected Maxwell fluid in compar-ison with linear Maxwell and Newtonian fluids For instance, the maximum value of flow rate for

FIG 4 Plot of ⟨Q⟩ verses χ when α = 0.001, ϵ = 0.001 and Re = 10000.

015119-9 Abbasi et al. AIP Advances 6, 015119 (2016)

FIG 5 Plot of ⟨Q⟩ verses χ when α = 0.001, ϵ = 0.001 and Re = 10000.

convected Maxwell fluid is 30% less than corresponding values of flow rate for linear Maxwell model

The plots for large value of λ1i.e.λ1= 10000 against compressibility parameter χ for both linear and convected models are shown in the Figs.5(a)and5(b) Again the behavior of⟨Q⟩ for both models is similar qualitatively However, a quantitative difference arises which increases by increas-ing χ Here, for χ= 1,⟨Q⟩ for linear Maxwell model is almost 18% less than its corresponding values for convected Maxwell model

V CONCLUDING REMARKS

Constitutive equation of Maxwell model is utilized to study compressible flow through a cylin-drical tube of small radius Perturbation expansion in dimensionless amplitude ratio is employed and net flow rate up to second-order is obtained through numerical integration A comparison of net flow rate for linear and convected Maxwell models is presented The following interesting observations are made

• Maxwell model predicts higher negative values of ⟨Q⟩ than linear Maxwell model Thus reverse flow induced by peristaltic waves is weak for convected Maxwell fluid model in when compared with linear Maxwell model

• For large values of λ1 i.e λ1= 10000, convected Maxwell model predicts positive values of

⟨Q⟩ for all values of α and thus rules out the possibility of reverse flow in contrast to linear Maxwell model

• For λ1= 10000 and α = 0.001, the net flow rate for convected Maxwell model is almost 99% higher than its corresponding value for linear Maxwell model

• The qualitative behavior of ⟨Q⟩ with χ is similar for both the models However, a qualitative

difference arises which increases by increasing χ

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