Commun. Theor. Phys. 65 (2016) 66–72 Vol. 65, No. 1, January 1, 2016

Peristaltic Flow of Couple Stress Fluid in a Non-Uniform Rectangular Duct Having

Compliant Walls

R. Ellahi,1,2,∗ M. Mubashir Bhatti,3 C. Fetecau,4 and K. Vafai1

1Department of Mechanical Engineering, University of California Riverside, Riverside, USA

2Department of Mathematics and Statistics FBAS IIUI, Islamabad, Pakistan

3Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai 200444, China4Department of Mathematics, Technical University of Iasi, Romania Academy of Romanian Scientists, 050094 Bucuresti,Romania

(Received March 30, 2015; revised manuscript received June 10, 2015)

Abstract The present study investigates the peristaltic flow of couple stress fluid in a non-uniform rectangular duct

with compliant walls. Mathematical modeling is based upon the laws of mass and linear momentum. Analytic solutions

are carried out by the eigen function expansion method under long-wavelength and low-Reynolds number approximations.

The features of the flow characteristics are analyzed by plotting the graphs of various values of physical parameters of

interest. Trapping bolus scheme is also presented through streamlines.

PACS numbers: 47.63.-bKey words: peristaltic flow, couple stress fluid, compliant walls, eigen function expansion method, analytical

solutions

1 Introduction

Non-Newtonian fluids are of great interest to re-searchers because of their practical importance in bio en-gineering, physiology and industry.[1−11] The couple stressfluid is a special case of non-Newtonian fluid, which is in-tended to take into account the particle size effects. Thestudy of a couple stress fluid is very useful to understandthe various physical problems, because it possesses themechanism to describe rheologically complex fluids suchas liquid crystals, colloidal fluids, liquids containing long-chain molecules as polymeric suspensions, animal and hu-man blood and lubrication etc.

Moreover, peristaltic pumping is a form of fluid trans-port that occurs when a progressive wave of area contrac-tion/expansion propagates along the length of distensibleduct. Peristalsis is an inherent property of many biologi-cal systems having smooth muscle tubes which transportsbiofluids by its propulsive movements, for instance, intra-uterine fluid motion, transport of urine from kidney tothe bladder, vasomotion of the small blood vessels suchas arterioles, venules and capillaries, transport of sperma-tozoa in the ductus efferentes of the male reproductivetract and in many other glandular ducts. The mecha-nism of peristaltic transport has also been exploited forindustrial applications such as blood pumps in heart lungmachine and transport of corrosive fluids where contact offluid with machinery parts is prohibited. A vast amountof literature is now available on peristaltic viscous fluidsunder the assumptions of long wavelength, small wavenumber low Reynolds number and small amplitude ra-tio etc.[12−16] Most of the theoretical investigations have

been carried out by assuming that blood behave like non-Newtonian fluids. This approach provides enough under-standing when peristaltic mechanism is involved in a smallblood vessels, lymphatic vessels, intestine, ductus effer-entes of the male reproductive tract and in transport ofspermatozoa in cervical canal.[17−20]

Furthermore, the constitutive equations in couplestress fluid model involving a number of parameters arevery complicated and higher order in comparison with theNavier–Stokes equations.[21−22] Consequently, such non-linear problems offer interesting challenges to the appliedmathematicians, physicists, modelers and computer sci-entists alike. Some relevant studies on couple stress flu-ids and on compliant walls can be found from the list ofRefs. [23–28].

Literature survey witnesses that the information onthe peristalsis flow of couple stress fluid in a channel withcompliant walls is scant. Therefore the main goal here isto propose the study of peristaltic flow of couple stressfluid in a non-uniform rectangular duct with compliantwalls. The obtained expressions are utilized to discussthe influence of various emerging parameters. The streamlines have also been presented. To the best of authors’knowledge no such study has been investigated before onperistaltic flow of couple stress fluid in a non-uniform rect-angular duct having compliant walls. To the best of ourknowledge the title problem is not studied before. Thispaper runs as follows:

After the introduction in Sec. 1, formulation of theproblem is given in Sec. 2. In Sec. 3, we apply eigen func-tion expansion method under the consideration of long

∗Corresponding author, E-mail: rellahi@engr.ucr.edu; rahmatellahi@yahoo.com

c© 2016 Chinese Physical Society and IOP Publishing Ltd

http://www.iopscience.iop.org/ctp http://ctp.itp.ac.cn

No. 1 Communications in Theoretical Physics 67

wavelength and low Reynolds number in order to obtainthe exact solution. Finally, Sec. 4 is devoted to resultsand discussion.

2 Formulation of the Problem

Based on the Stokes[29] theory, when the body forcesand body couples are neglected, the continuity equationof an incompressible couple stress fluid is given by

divV = 0 , (1)

and the conservation of linear momentum is reduced to

ρdV

dt= −∇p+µ∇

2V −η∇

4V , (2)

where V is the velocity vector, p is pressure, ρ is density,d/dt is time material derivative, µ is the classical viscosityand η is a new material parameter responsible for couplestress property.

Consider the flow of an incompressible couple stressfluid in a non-uniform duct of rectangular cross sectionhaving width 2d and height 2a+kx. We have chosen Carte-sian coordinate system such that x-axis is taken along theaxial direction, y-axis is taken along the lateral directionand z-axis is taken along the vertical direction of the rect-angular duct (see Fig. 1). The walls of the duct are as-sumed to be flexible and are taken as compliant, on which

waves with small amplitude and long wave length are con-sidered.

Fig. 1 Geometry of the problem.

The peristaltic waves on the walls are represented as

Z = h(x, t) = ±a ± kx ± b sin[2π

λ(x − ct)

]

, (3)

where a is the height of duct, b is the amplitude of thewave, λ is the wavelength, c is the velocity of the propa-gation, t is the time and x is the direction of wave prop-agation. The walls parallel to xz-plane are not distractedand are not subject to any peristaltic wave motion.

In view of velocity filed V [u, 0, w] the projection ofEqs. (1) and (2) on the coordinate system (x, y, z), asshown in Fig. 1, can be reduced to the following form

∂u

∂x+

∂w

∂z= 0 , (4)

ρ(∂u

∂t+ u

∂u

∂x+ w

∂u

∂z

)

= −∂p

∂x+ µ

(∂2u

∂x2+

∂2u

∂y2+

∂2u

∂z2

)

− η

( ∂4u∂x4 + ∂4u

∂y4 + ∂4u∂z4

+2 ∂4u∂x2∂y2 + 2 ∂4u

∂z2∂y2 + 2 ∂4u∂x2∂z2

)

, (5)

0 = −∂p

∂y, (6)

ρ(∂w

∂t+ u

∂w

∂x+ w

∂w

∂z

)

= −∂p

∂z+ µ

(∂2w

∂x2+

∂2w

∂y2+

∂2w

∂z2

)

− η

( ∂4w∂x4 + ∂4w

∂y4 + ∂4w∂z4

+2 ∂4w∂x2∂y2 + 2 ∂4w

∂z2∂y2 + 2 ∂4w∂x2∂z2

)

. (7)

Using the following non-dimensional quantities

x =x

λ, y =

y

d, z =

z

a, u =

u

c, w =

w

cδ, t =

ct

λ, h =

h

a, p =

a2p

µcλ,

Re =ρac

µ, δ =

a

λ, φ =

b

a, β =

a

d, γ =

√

µ

ηa, K∗ =

k

a, η0 =

η0

a, (8)

the resulting Eqs. (4) to (7) after dropping the bars can be written as

∂u

∂x+

∂w

∂z= 0 , (9)

Reδ(∂u

∂t+ u

∂u

∂x+ w

∂u

∂z

)

= −∂p

∂x+ δ2 ∂2u

∂x2+ β2 ∂2u

∂y2+

∂2u

∂z2−

1

γ2

(

δ4 ∂4u

∂x4+ β4 ∂4u

∂y4+

∂4u

∂z4+ 2δ2β2 ∂4u

∂x2∂y2

+ 2δ2 ∂4u

∂z2∂x2+ 2β2 ∂4u

∂y2∂z2

)

, (10)

0 = −∂p

∂y, (11)

Reδ3(∂w

∂t+ u

∂w

∂x+ w

∂w

∂z

)

= −∂p

∂z+ δ2

(

δ∂2w

∂x2+ β2 ∂2w

∂y2+ δ2 ∂2w

∂z2

)

−1

γ2

(

δ6 ∂4w∂x4 + δ2β4 ∂4w

∂y4 + δ2 ∂4w∂z4

+2δ4β2 ∂4w∂x2∂y2 + 2δ4 ∂4w

∂z2∂x2 + 2δ2β2 ∂4w∂y2∂z2

)

. (12)

Under the assumption of long wave length δ ≤ 1 and low Reynolds number Re −→ 0, Eqs. (10) to (12) along the

68 Communications in Theoretical Physics Vol. 65

corresponding boundary conditions are

β2 ∂2u

∂y2+

∂2u

∂z2−

1

γ2

(

β4 ∂4u

∂y4+

∂4u

∂z4+ 2β2 ∂4u

∂y2∂z2

)

−∂p

∂x= 0 , (13)

u(x, y, z, t) = −1 and∂2u

∂y2(x, y, z, t) = 0 at y = ±1 , (14)

u(x, y, z, t) = −1 and∂2u

∂y2(x, y, z, t) = 0 at z = ±h = ±1 ± K∗x ± η0(x, t) , (15)

where η0(x, t) = φ sin 2π(x − t), 0 ≤ φ ≤ 1.

The governing equation for the flexible wall may be specified as

L(η0) = p − p0 , (16)

where L is an operator, which is used to represent the motion of stretched membrane with viscosity damping forces

such that[30]

L = m∂2

∂t2+ D

∂

∂t+ B

∂4

∂x4− T

∂2

∂x2+ K . (17)

In the above equation, m is mass per unit area, D is coefficient of the viscosity damping membrane, B is flexural

rigidity of the plate, T is elastic tension in the membrane, K is spring stiffness and p0 is pressure on outside surface ofthe wall due to tension in muscle, which is assumed to be zero here. Using Eqs. (16) and (17) with the continuity ofstress at z = ±1 ± K∗x ± η0, we get

∂p

∂x= E1

∂3η0

∂t2∂x+ E2

∂2η0

∂t∂x+ E3

∂5η0

∂x5− E4

∂3η0

∂x3+ E5

∂η0

∂x, (18)

E1∂3η0

∂t2∂x+ E2

∂2η0

∂t∂x+ E3

∂5η0

∂x5− E4

∂3η0

∂x3+ E5

∂η0

∂x= β2 ∂2u

∂y2+

∂2u

∂z2−

1

γ2

(

β4 ∂4u

∂y4+

∂4u

∂z4+ 2β2 ∂4u

∂y2∂z2

)

(19)

in which E1 = ma3c/λ3µ, E2 = Da3/λ2µ, E3 = Ba3/cλ5µ, E4 = Ta3/cλ3µ, E5 = Ka3/cλµ are the non-dimensionalelasticity parameters.

3 Solution of the Problem

The solution of the non-homogenous partial differential equation given in Eq. (19) can be obtained by using the

following transformation

u(x, y, z, t) = v1(x, y, z, t) + w1(y) . (20)

Using the above transformation in Eqs. (13) to (15), we get

β2 ∂2v1

∂y2+

∂2v1

∂z2−

1

γ2

(

β4 ∂4v1

∂y4+

∂4v1

∂z4+ 2β2 ∂4v1

∂y2∂z2

)

= C0(x, t) , (21)

β4

γ2

d4w1

dy4− β2 d2w1

dy2= 0 , (22)

along with the corresponding boundary conditions

v1(x, y,±h(x), t) = −1 − w1(y) and ∂2v1

∂z2 (x, y,±h(x), t) = 0 ,

v1(±1, y, z, t) = 0 and ∂2v1

∂y2 (±1, y, z, t) = 0 ,

}

(23)

w1(±1) = −1 and∂2w1

∂y2(±1) = 0 . (24)

The exact solutions of Eqs. (21) and (22) along with the corresponding boundary conditions (23) and (24) by meansof eigen function expansion method are obtained as

v1(x, y, z, t) =

∞∑

n=1

( (−1)n4γ2C0(x, t)

(2n − 1)πα20

−(−1)n4γ2C0(x, t)

(2n − 1)πα20(ξ

20 − ξ2

1)

(ξ20 cosh ξ1z

cosh ξ1h−

ξ21 cosh ξ0z

cosh ξ0h

))

cos(2n − 1)π

2y , (25)

w1(y) = −1 . (26)

In view of Eqs. (25) and (26). The solution of Eq. (13) satisfying boundary conditions (14) and (15) can be written as

u(x, y, z, t) = −1 +

∞∑

n=1

[ (−1)n4γ2C0(x, t)

(2n − 1)πα20

−(−1)n4γ2C0(x, t)

(2n − 1)πα20(ξ

20 − ξ2

1)

×(ξ2

0 cosh ξ1z

cosh ξ1h−

ξ21 cosh ξ0z

cosh ξ0h

)]

cos(2n − 1)π

2y , (27)

No. 1 Communications in Theoretical Physics 69

in which

C0(x, t) = φ2π cos 2π(x − t)[(E4 − E1 + 4π2E3)4π2 + E5] + E2[φ4π2 sin 2π(x − t)] , (28)

ξ0 = ±

√

α21 +

√

α41 − 4α2

0

2, ξ1 = ±

√

α21 −

√

α41 − 4α2

0

2, (29)

α0 =

√

γ2β2(

(2n − 1)π

2

)2

+ β4(2n − 1)π

2, α1 =

√

2β2(

(2n − 1)π

2

)2

+ γ2 , (30)

where β 6= 0, 1 and γ 6= 0,∞.

4 Results and Discussion

In this section graphical results are being discussed to see the effects of various emerging parameters and trappingphenomena for different pertinent parameters. For this purpose Figs. 2–4 are sketched to see the variation in thevelocity for both two and three dimensions. In Fig. 2 it is evident that with the increase of E1 the velocity decreasesand the maximum velocity attained in the centre of the channel whereas opposite behavior has been observed in caseof E2.

Fig. 2 Velocity profile for different values of E1 & E2 for fixed φ = 0.6, K∗ = 0.5, γ = 8, E3 = 0.01, E4 = 0.2, E5 = 0.1(a) for 2-dimensional, (b) for 3-dimensional.

Fig. 3 Velocity profile for different values of E3 & E4 for fixed φ = 0.6, K∗ = 0.5, E1 = 0.1, E2 = 0.3, E5 = 0.1,

β = 0.1 (a) for 2-dimensional, (b) for 3-dimensional.

Figure 3 depicts the velocity profile for various values of E3 and E4. It can be seen that velocity increases withthe increase of E3 while his attitude remains same when E4 increases. It can be noticed from Fig. 4 that velocityincreases with the increment of E5. It is also observed that by increasing the couple stress parameter γ, the velocityfield increases. This due fact that the elasticity of the walls there is less resistance to the flow and so that the velocityfield increases.

The phenomenon of trapping is another interesting topic in peristaltic transport. The formation of an internallycirculating bolus of the fluid by closed stream lines is called trapping and this trapped bolus pushed ahead long theperistaltic wave. Generally, in the wave frame stream lines have contours that are identical to immobile walls. It isoriginated due to internally circulating bolus in a fluid enclosed by stream lines. Some stream lines are dividing undercertain conditions due to the sustenance of stagnation point. The formulation of this internal circulating bolus of thefluid encloses by stream lines is known as trapping and this trapped bolus pushed ahead along the peristaltic wave.

70 Communications in Theoretical Physics Vol. 65

The size of bolus is measured by the volume of the fluid which is surrounded by closed stream lines. The variation

in heat transfer coefficient z(x) for various values of emerging parameters γ, E1, E2, E3, E4, and E5 are analyzed in

Figs. 5 to 8.

Fig. 4 Velocity profile for different values of E5 & γ for fixed φ = 0.6, K∗ = 0.5, E2 = 0.3, E3 = 0.01, E4 = 0.2,

β = 0.1 (a) for 2-dimensional, (b) for 3-dimensional.

Fig. 5 Stream lines for different values of γ, (a) for γ = 4, (b) for γ = 5, (c) for γ = 6, (d) for γ = 7. The otherparameters are φ = 0.6, K∗ = 0.5, E1 = 0.1, E2 = 0.3, E3 = 0.01, E4 = 0.2, E5 = 0.1, β = 0.1.

The stream lines for various values of couple stress pa-rameter γ are shown in Fig. 5. It can be noticed thattrapped bolus decreases by increasing the couple stressparameter γ while the bolus reduces in numbers. FromFig. 6, it reveals that with the increase of wall tension E1

the size of bolus and trapped bolus increases. With therise of mass characterizing parameter E2 the size of bolusgradually increases in Fig. 7. Figure 8 depicts that whendamping nature of the wall E3 increases then the trap-

ping bolus decreases. The size of bolus increases in the

region x ∈ [0, 0.5] while opposite behavior is observed in

the region x ∈ [0.5, 1]. It is found that the nature of heat

transfer coefficient is oscillatory. This is expected due to

propagation of sinusoidal waves. The key learning of the

presented analysis is that the peristaltic pumping is ap-

plicable in colloidal fluids, crystals liquids and biological

fluids.

No. 1 Communications in Theoretical Physics 71

Fig. 6 Stream lines for different values of E1, (a) for E1 = 2, (b) for E1 = 3, (c) for E1 = 4, (d) for E1 = 5. The otherparameters are φ = 0.6, K∗ = 0.5, γ = 8, E2 = 0.3, E3 = 0.01, E4 = 0.2, E5 = 0.1, β = 0.1.

Fig. 7 Stream lines for different values of E2, (a) for E2 = 0.2, (b) for E2 = 0.4, (c) for E2 = 0.7, (d) for E2 = 1. Theother parameters are φ = 0.6, K∗ = 0.5, γ = 8, E1 = 0.1, E3 = 0.01, E4 = 0.2, E5 = 0.1, β = 0.1.

The motion of differential Newtonian and non-Newtonian biological fluids in physiological system can easily be

examined thoroughly to solve the different diagnostic problems that generated in living body. It is also useful for

transport of slurries, sensitive or corrosive fluids, sanitary fluid, and noxious fluids in the nuclear industry.

72 Communications in Theoretical Physics Vol. 65

Fig. 8 Stream lines for different values of E3, (a) for E3 = 0.01, (b) for E3 = 0.05, (c) for E3 = 0.08, (d) for E3 = 0.15.

The other parameters are φ = 0.6, K∗ = 0.5, γ = 8, E1 = 0.1, E2 = 0.3, E4 = 0.2, E5 = 0.1, β = 0.1.

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