williamson fluid model for the peristaltic flow of chyme

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2012, Article ID 479087, 18 pagesdoi:10.1155/2012/479087

Research ArticleWilliamson Fluid Model for the Peristaltic Flow ofChyme in Small Intestine

Sohail Nadeem,1 Sadaf Ashiq,1 and Mohamed Ali2

1 Department of Mathematics, Quaid-i-Azam University, Islamabad 45320, Pakistan2 Department of Mechanical Engineering, King Saud University, Riyadh 11451, Saudi Arabia

Correspondence should be addressed to Sohail Nadeem, [email protected]

Received 21 September 2011; Revised 24 December 2011; Accepted 2 January 2012

Academic Editor: Angelo Luongo

Copyright q 2012 Sohail Nadeem et al. This is an open access article distributed under theCreative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

Mathematical model for the peristaltic flow of chyme in small intestine along with insertedendoscope is considered. Here, chyme is treated as Williamson fluid, and the flow is consideredbetween the annular region formed by two concentric tubes (i.e., outer tube as small intestineand inner tube as endoscope). Flow is induced by two sinusoidal peristaltic waves ofdifferent wave lengths, traveling down the intestinal wall with the same speed. The governingequations of Williamson fluid in cylindrical coordinates have been modeled. The resultingnonlinear momentum equations are simplified using long wavelength and low Reynolds numberapproximations. The resulting problem is solved using regular perturbation method in terms of avariant of Weissenberg number We. The numerical solution of the problem is also computed byusing shooting method, and comparison of results of both solutions for velocity field is presented.The expressions for axial velocity, frictional force, pressure rise, stream function, and axial pressuregradient are obtained, and the effects of various emerging parameters on the flow characteristicsare illustrated graphically. Furthermore, the streamlines pattern is plotted, and it is observed thattrapping occurs, and the size of the trapped bolus varies with varying embedded flow parameters.

1. Introduction

The object of this study is to investigate the flow induced by peristaltic action of chyme(treated as Williamson fluid) in small intestine with an inserted endoscope. Williamsonfluid is characterized as a non-Newtonian fluid with shear thinning property (i.e., viscositydecreases with increasing rate of shear stress). Since many physiological fluids behave like anon-Newtonian fluid [1], so chyme in small intestine is assumed to behave like Williamsonfluid. Peristaltic motion is one of the most characteristics fluid transport mechanism inmany biological systems. It pumps the fluids against pressure rise. It involves involuntarymovements of the longitudinal and circular muscles, primarily in the digestive tract

2 Mathematical Problems in Engineering

but occasionally in other hollow tubes of the body, that occur in progressive wavelikecontractions. The waves can be short, local reflexes or long, continuous contractions thattravel the whole length of the organ, depending upon their location and what initiates theiraction.

In human gastrointestinal tract, the peristaltic phenomenon plays a vital rolethroughout the digestion and absorption of food. The small intestine is the largest part ofthe gastrointestinal tract and is composed of the duodenum which is about one foot long,the jejunum (5–8 feet long), and the ileum (16–20 feet long). The rhythmic muscular actionof the stomach wall (peristalsis) moves the chyme (partially digested mass of food) intothe duodenum, the first section of the small intestine, where it stimulates the release ofsecretin, a hormone that increases the flow of pancreatic juice as well as bile and intestinaljuices. Nutrients are absorbed throughout the small intestine. There are blood vessels andvessels contained a fluid called lymph inside the villi. Fat-soluble vitamins and fatty acids areabsorbed into the lymph system. Glucose, amino acids, water-soluble vitamins, and mineralsare absorbed into the blood vessels. The blood and lymph then carry the completely digestedfood throughout the body [2].

The endoscope effect on peristaltic motion occurs in many medical applications.Direct visualization of interior of the hollow gastrointestinal organs is one of the mostpowerful diagnostic and therapeutic modalities in modern medicine. A flexible tube calledan endoscope is used to view different parts of the digestive tract. The tube contains severalchannels along its length. The different channels are used to transmit light to the area beingexamined, to view the area through a camera lens (with a camera at the tip of the tube), topump fluids or air in or out, and to pass biopsy or surgical instruments through [3]. Whenpassed through themouth, an endoscope can be used to examine the esophagus, the stomach,and first part of the small intestine. When passed through the anus, an endoscope can be usedto examine the rectum and the entire large intestine.

After the pioneering work of Latham [4], a number of analytical, numerical andexperimental studies [5–13] of peristaltic flows of different fluids have been reportedunder different conditions with reference to physiological and mechanical situations. Severalmathematical and experimental models have been developed to understand the chymemovement aspects of peristaltic motion. But less attention has been given to its relevancewith endoscope effect. Lew et al. [14] discussed the physiological significance of carrying,mixing, and compression, accompanied by peristalsis. Srivastava [15] devoted his study toobserve the effects of an inserted endoscope on chyme movement in small intestine. Theimportant studies of recent years include the investigations of Saxena and Srivastava [16, 17],L.M. Srivastava and V.P. Srivastava [18], Srivastava et al. [19], Cotton and Williams [20], andAbd El-Naby and El-Misery [21].

The aim of present investigation is to investigate the peristaltic motion of chyme,by treating it as Williamson fluid, in the small intestine with an inserted endoscope. Formathematical modeling, we consider the flow in the annular space between two concentrictubes (i.e., outer tube as small intestine and inner tube as endoscope). Moreover, the flowis induced by two sinusoidal peristaltic waves of different wave lengths, traveling along thelength of the intestinal wall. The solution of the problem is calculated by two techniques: (i)analytical technique (i.e., regular perturbation method in terms of a variant of Weissenbergnumber We), (ii) numerical technique (i.e., shooting method). The expressions for axialvelocity, frictional force, pressure rise, axial pressure gradient and stream function areobtained and the effects of various emerging parameters on the flow characteristics are

Mathematical Problems in Engineering 3

illustrated graphically. Streamlines are plotted, and trapping is also discussed. Trapping isan important fluid dynamics phenomenon inherent in peristalsis. At high flow rates andocclusions, there is a region of closed stream lines in the wave frame, and thus, some fluid isfound trapped within a wave of propagation. The trapped fluid mass (called bolus) is foundto move with the mean speed equal to that of the wave [9].

2. Mathematical Development

We consider the flow of an incompressible, non-Newtonian fluid, bounded between smallintestine (outer boundary) and inserted cylindrical endoscope (inner boundary). A physicalsketch of the problem is shown in the Figure 1(a). We assume that the peristaltic wave isformed in nonperiodic rush mode composing of two sinusoidal waves of different wavelengths, traveling down the intestinal wall with the same speed c. We consider the cylindricalcoordinate system (R,Z) in the fixed frame, whereZ-axis lies along the centerline of the tube,and R is transverse to it. Also a symmetry condition is used at the centre.

The geometry of the outer wall surface is described as

h(Z, t)= r0 +A1 sin

2πλ1

(Z − ct

)+A2 sin

2πλ2

(Z − ct

), (2.1)

where r0 is the radius of the outer tube (small intestine),A1 and λ1 are the amplitude and thewave length of first wave, A2 and λ2 are the amplitude and the wave length of the secondwave, c is the propagation velocity, t is the time, and Z is the axial coordinate.

The governing equations in the fixed frame for an incompressible Williamson fluidmodel [6] are given as follows:

∂U

∂R+U

R+∂W

∂Z= 0,

ρ

(∂

∂t+U

∂

∂R+W

∂

∂Z

)U = −∂P

∂R+

1

R

∂

∂R

(RτR R

)+

∂

∂Z

(τRZ),

ρ

(∂

∂t+U

∂

∂R+W

∂

∂Z

)W = −∂P

∂Z+

1

R

∂

∂R

(RτR Z

)+

∂

∂Z

(τZ Z

),

(2.2)

where P is the pressure, and U,W are the respective velocity components in the radial andaxial directions in the fixed frame, respectively. Further, the constitutive equation of extrashear stress tensor τ for Williamson fluid [22] is expressed as

τ =[μ∞ +

(μ0 + μ∞

)(1 − Γ

∣∣∣γ̇∣∣∣)−1]

γ̇ , (2.3)


λ1

λ2

A2 A1

r1

r0

r

z

Small intestine

Endoscope

(a)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Numerical solutionPerturbation solution

−1.08

−1.07

−1.06

−1.05

−1.04

−1.03

−1.02

−1.01

−1

w(r,z)

r

(b)

Figure 1: (a) Physical sketch of the problem. (b) Comparison of numerical and perturbation solutions ofaxial velocity w(r, z) forWe = 0.02, dp/dz = 0.7, and δ = 0.1.

where μ∞ is the infinite shear rate viscosity, μ0 is the zero shear rate viscosity, Γ ischaracteristic time and the generalized shear rate γ̇ is expressed in terms of second invariantstrain tensor Π as,

∣∣∣γ̇∣∣∣ =√√√√1

2

∑i

∑j

γ̇ ij γ̇ ji =

√12Π, (2.4)

in which Π = tr (gradV + (gradV)T )2, and V denotes the velocity vector.


We consider the case μ∞ = 0 and Γ|γ̇ | < 1 for constitutive equation (2.3). Hence, theextra stress tensor can be written as

τ = μ0

[(1 − Γ

∣∣γ̇∣∣)−1]γ̇ ,

= μ0

[(1 + Γ

∣∣∣γ̇∣∣∣)]γ̇ ,

(2.5)

such that

τR R = 2μ0

(1 + Γ

∣∣∣γ̇∣∣∣)∂U∂R

,

τR Z = μ0

(1 + Γ

∣∣∣γ̇∣∣∣)(∂U

∂Z+∂W

∂R

),

τZ Z = 2μ0

(1 + Γ

∣∣∣γ̇∣∣∣)∂W∂Z

,

∣∣∣γ̇∣∣∣ =⎡⎣2(∂U

∂R

)2

+

(∂U

∂Z+∂W

∂R

)2

+ 2

(∂W

∂Z

)2⎤⎦

1/2

.

(2.6)

In the fixed coordinates (R,Z), the flow is unsteady. It becomes steady in a wave frame(r, z)moving with the same speed as the wave moves in theZ-direction. The transformationsbetween the two frames are

r = R, z = Z − ct,u = U, w =W − c,

(2.7)

in which u and w are the velocities in the wave frame. The corresponding boundary con-ditions in the wave frame are

w = −c, at r = r1,

w = −c, at r = h = r0 +A1 sin2πλ1

(z) +A2 sin2πλ2

(z),(2.8)

where r1 is the radius of the inner tube (endoscope). In order to reduce the number ofvariables, we introduce the following nondimensional variables:

R =R

r0, r =

r

r0, Z =

Z

λ1, z =

z

λ1, W =

W

c, w =

w

c, τ =

r0τ

cμ0,

U =λ1U

r0c, u =

λ1u

r0c, p =

r0P

cλ1μ0, t =

ct

λ1, Re =

ρcr0μ0

, We =Γcr0,

h =h

r0= 1 + φ1 sin 2πz + φ2 sin 2πγz, r1 =

r1r0, δ =

r1r0, φ1 =

A1

r0,

φ2 =A2

r0, γ =

λ1λ2, ε =

r0λ1.

(2.9)


Here Re,We, and ε represent the Reynolds number, Weissenberg number, and wave number,respectively. Moreover, φ1 and φ2 are nondimensional amplitudes of the waves, δ is theannulus aspect ratio, and γ represents the wave length ratio between two waves. By using(2.7) and (2.9), we get

∂u

∂r+u

r+∂w

∂z= 0,

Reε3(u∂

∂r+w

∂

∂z

)u = −∂p

∂r+ε

r

∂

∂r(rτ rr) + ε2

∂

∂z(τ rz),

Reε

(u∂

∂r+w

∂

∂z

)w = −∂p

∂z+1r

∂

∂r(rτ rz) + ε

∂

∂z(rτzz),

(2.10)

and components of the extra stress tensor take the following form:

τ rr = 2ε[1 +We

∣∣γ̇∣∣]∂u∂r,

τ rz =[1 +We

∣∣γ̇∣∣](∂u

∂zε2 +

∂w

∂r

),

τzz = 2ε[1 +We

∣∣γ̇∣∣]∂w∂z

,

∣∣γ̇∣∣ =[2ε2(∂u

∂r

)2

+(∂w

∂r+∂u

∂zε2)2

+ 2ε2(∂w

∂r

)2]1/2

.

(2.11)

Under the assumption of long wavelength ε � 1 and low Reynolds number approxi-mations, the above equations are further reduced to

∂p

∂r= 0, (2.12)

−∂p∂z

+1r

∂

∂r(rτ rz) = 0, (2.13)

τ rz =∂w

∂r+We

(∂w

∂r

)2

. (2.14)

Equation (2.12) shows that p = p(z). Now putting expression of τ rz in (2.13) we get

−∂p∂z

+1r

[r

(∂w

∂r+We

(∂w

∂r

)2)]

= 0. (2.15)

The corresponding nondimensional boundary conditions thus obtained are

w = −1 at r = δ,

w = −1 at r = h = h(z) = 1 + φ1 sin 2πz + φ2 sin 2πγz.(2.16)


3. Solution of the Problem

3.1. Perturbation Solution

Since (2.15) is a nonlinear equation and the exact solution may not be possible, therefore, inorder to find the solution, we employ the regular perturbation method in terms of a variantof Weissenberg numberWe. For perturbation solution, we expand w and p as

w = w0 +Wew1 +O(W2

e

),

p = p0 +Wep1 +O(W2

e

).

(3.1)

To first order, the expressions for axial velocity and axial pressure gradient satisfyingboundary conditions (2.16) directly yield

w(r, z) = −1 + 0.25dp

dz(a45) +We

dp

dz(a46 − a47), (3.2)

dp

dz= a48(a49 + a50 − a51 + a52), (3.3)

where the involved quantities are defined in Appendix.The expressions of pressure rise Δp and the frictional forces F0 and F1 at the outer and

inner boundaries, respectively, in their nondimensional forms, are given as

Δp =∫1

0

dp

dzdz, (3.4)

F0 =∫1

0δ2(−dpdz

)dz, (3.5)

F1 =∫1

0h2(−dpdz

)dz. (3.6)

where dp/dz is defined through (3.3), and flow rate Q in dimensionless form is defined as

Q = q − π(⟨h2⟩− δ2), (3.7)

where q is flow rate in the fixed frame of reference, and 〈h2〉 is square of displacement of thewalls over the length of an annulus, defined as

⟨h2⟩=∫1

0

(h2)dz. (3.8)


Table 1: Numerical and perturbation solutions for axial velocity w(r, z).

rNumerical Sol. Perturbation Sol. Error

w(r, z) w(r, z)0.10 −1.0000 −1.0000 0.00000.15 −1.0284 −1.0286 0.00020.20 −1.0470 −1.0473 0.00030.25 −1.0599 −1.0601 0.00020.30 −1.0687 −1.0689 0.00020.35 −1.0745 −1.0748 0.00030.40 −1.078 −1.0781 0.00010.45 −1.0794 −1.0795 0.00010.50 −1.0790 −1.0790 0.00000.55 −1.0770 −1.0770 0.00000.60 −1.0734 −1.0734 0.00000.65 −1.0685 −1.0684 0.00010.70 −1.0622 −1.0622 0.00000.75 −1.0546 −1.0547 0.00010.80 −1.0461 −1.0460 0.00010.85 −1.0362 −1.0361 0.00010.90 −1.0251 −1.0252 0.00010.95 −1.0128 −1.0131 0.00031.0 −1.0000 −1.0000 0.0000

Also in order to establish stream lines, we obtain stream function by using the followingrelation:

u = −1r

∂ψ

∂z, w =

1r

∂ψ

∂r, (3.9)

which yields

ψ =r

480(a30) + a31

[(a32 − a33 − a34 + 5a26a212 − a35

)

×a36 − a37 − a38 + a39 + a40 + a41 − a42 + a43 + a44],

(3.10)

where the involved quantities are defined in Appendix.

3.2. Numerical Solution

Equation (2.15) is solved numerically by using shooting method [23]. The numerical resultfor axial velocity w(r, z) is compared with the perturbation result, and both results reveal avery good agreement with each other, as demonstrated in Table 1 and Figure 1(b).


0 0.5 1 1.5 2 2.5

0

100

200

300

400

500

600

700

800

−200

−100

−0.5

∆p

Q

We = 0.01We = 0.03

We = 0.05We = 0.07

(a)

0 0.5 1 1.5 2

0

100

200

300

400

500

600

700

−100−0.5

γ = 0.6γ = 0.7

γ = 0.8γ = 0.9

∆p

Q

(b)

0 0.5 1 1.5 2 2.5

0

500

1000

1500

−0.5

−500

φ2 = 0.03φ2 = 0.05

φ2 = 0.07φ2 = 0.09

∆p

Q

(c)

0 0.5 1 1.5 2 2.50

10

20

30

40

50

60

70

80

90

100

−0.5

δ = 0.06δ = 0.07

δ = 0.08δ = 0.09

∆p

Q

(d)

Figure 2: Variation of pressure rise per wavelength (Δp) for different values of (a) We with φ1 = 0.03, φ2 =0.05, Q = 0.2, δ = 0.7 and γ = 0.9, (b) γ with φ1 = 0.03, φ2 = 0.05, Q = 2, δ = 0.7, andWe = 0.01, (c) φ2 withφ1 = 0.01, Q = 0.2, γ = 0.9, δ = 0.7, and We = 0.01, and (d) δ with φ1 = 0.03, φ2 = 0.05, Q = 0.2, γ = 0.9,andWe = 0.01.

4. Graphical Results and Discussion

In this section, the graphical representations of the obtained solutions are demonstratedalong with their respective explanation. The expressions for pressure rise and frictionalforces are not found analytically; therefore, MATHEMATICA software is used to performthe integration in order to analyze their graphical behavior. It is also pertinent to mentionthat the values of all embedded flow parameters are considered to be less than 1.

Figures 2(a) to 2(d) are graphs of pressure rise Δp versus flow rate Q to show theeffects of different parameters on pumping rate. For peristaltic pumping, we divide the whole


0 0.5 1 1.5 2

0

2000

4000

−8000

−6000

−4000

−2000

F1

Q

−0.5

We = 0.01We = 0.03

We = 0.05We = 0.07

(a)

0

200

400

0 0.5 1 1.5 2Q

−0.5−1400

−1200

−1000

−800

−600

−400

−200

F1

γ = 0.6γ = 0.7

γ = 0.8γ = 0.9

(b)

01000200030004000

0 0.5 1 1.5 2Q

−0.5−7000−6000−5000−4000−3000−2000−1000

F1

φ2 = 0.03φ2 = 0.05

φ2 = 0.07φ2 = 0.09

(c)

0 0.5 1 1.5 2Q

−0.5

−90

−80

−70

−60

−50

−40

−30

−20F

1

δ = 0.03δ = 0.05

δ = 0.07δ = 0.09

(d)

Figure 3: Variation of frictional force at the inner wall (F1) for different values of (a) We with φ1 = 0.03,φ2 = 0.05, Q = 2, δ = 0.7, and γ = 0.9, (b) γ with φ1 = 0.03, φ2 = 0.05, Q = 0.2, δ = 0.7, andWe = 0.01, (c) φ2with φ1 = 0.01, Q = 0.2, γ = 0.9, δ = 0.7 andWe = 0.01, and (d) δ with φ1 = 0.03, φ2 = 0.05, Q = 0.2, γ = 0.9andWe = 0.01.

region into three parts. The region corresponding to Δp > 0 and Q > 0 is known as theperistaltic pumping region. At Δp = 0 is the free pumping region. And the region at Δp < 0and Q > 0 is called augmented pumping. Figure 2(a) shows that the pumping rate decreasesby increasing the values of Weissenberg number We, and this behavior remains the same inall three pumping regions. Figure 2(b) indicates the effect of the wavelength ratio γ on Δp.Here, pressure rise decreases with an increase in γ peristaltic pumping region, and after acritical value of Q = 1.4, it increases in the free and augmented pumping regions. Figure 2(c)explains the effect of the amplitude ratio φ2 on Δp. Here, pressure rise decreases with an


0

200F

0

0 0.5 1 1.5 2Q

−0.5−1000

−800

−600

−400

−200

We = 0.01We = 0.03

We = 0.05We = 0.07

(a)

0

100

200

0 0.5 1 1.5 2Q

−0.5−700

−600

−500

−400

−300

−200

−100

F0

γ = 0.6γ = 0.7

γ = 0.8γ = 0.9

(b)

0.6 0.8 1 1.2 1.4 1.6 1.8 2

0

100

200

300

Q

−400

−300

−200

−100

F0

φ2 = 0.03φ2 = 0.05

φ2 = 0.07φ2 = 0.09

(c)

0

0 0.5 1 1.5 2Q

−0.5−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

F0

δ = 0.03δ = 0.05

δ = 0.07δ = 0.09

(d)

Figure 4: Variation of frictional force at the outer wall (F0) for different values of (a) We with φ1 = 0.03,φ2 = 0.05, Q = 0.2, δ = 0.7, and γ = 0.9, (b) γ with φ1 = 0.03, φ2 = 0.05, Q = 2, δ = 0.7, andWe = 0.01, (c) φ2with φ1 = 0.01,Q = 0.2, γ = 0.9, δ = 0.7, andWe = 0.01, and (d) δ with φ1 = 0.03, φ2 = 0.05, Q = 0.2, γ = 0.9andWe = 0.01.

increase in value of φ2 in the free and peristaltic pumping region, and after a critical value ofQ = 1.5, it increases in the augmented pumping region. Figure 2(d) shows that pressure risedecreases when annulus aspect ratio δ increases.

Similarly the effects of We, γ , φ2, and δ on frictional forces are plotted in Figures 3-4.Figures 3(a) to 3(d) represent the variation of the frictional force at the outer wall (F0) andFigures 4(a) to 4(d) indicate the variation of the frictional force at the inner wall (F1) with


0 0.5 1 1.50

200

400

600

800

1000

1200

z

dp/dz

φ1 = 0.03φ1 = 0.05

φ1 = 0.07φ1 = 0.09

(a)

150

200

250

300

350

400

450

500

0 0.5 1 1.5z

dp/dz

φ2 = 0.02φ2 = 0.05

φ2 = 0.07φ2 = 0.09

(b)

100

200

300

400

500

600

700

800

900

1000

0 0.5 1 1.5z

dp/dz

We = 0.1We = 0.3

We = 0.5We = 0.7

(c)

100

120

140

160

180

200

220

240

260

280

300

0 0.5 1 1.5z

dp/dz

γ = 0.8γ = 0.85

γ = 0.9γ = 0.95

(d)

0 0.5 1 1.50

100

200

300

400

500

600

700

δ = 0.2δ = 0.4

δ = 0.6δ = 0.8

z

dp/dz

(e)

Figure 5: Pressure gradient dP/dz verses z for different values of (a) φ1 with φ2 = 0.06, Q = 0.2, γ = 0.9,δ = 0.5, and We = 0.05, (b) φ2 with φ1 = 0.06, Q = 0.2, γ = 0.9, δ = 0.5, and We = 0.05, (c) We. withφ1 = 0.04, φ2 = 0.06, Q = 0.2, δ = 0.5, and γ = 0.9, (d) δ with φ1 = 0.04, φ2 = 0.06, Q = 0.2, γ = 0.9 andWe = 0.05, and (e) γ with φ1 = 0.04, φ2 = 0.06, Q = 0.2, δ = 0.5 andWe = 0.05.


0.2 0.4 0.6 0.8 1 1.2 1.40

0.2

0.4

0.6

0.8

z

r

(a)

0.2 0.4 0.6 0.8 1 1.2 1.40

0.2

0.4

0.6

0.8

z

r

(b)

Figure 6: Streamlines pattern for (a) φ2 = 0.08 (b) φ2 = 0.09 with φ1 = 0.04, Q = 0.2, δ = 0.4, γ = 0.9 andWe = 0.03.

0.2 0.4 0.6 0.8 1 1.2 1.40

0.2

0.4

0.6

0.8

z

r

(a)

0.2 0.4 0.6 0.8 1 1.2 1.40

0.2

0.4

0.6

0.8

z

r

(b)

Figure 7: Streamlines pattern for (a)We = 0.01 (b)We = 0.02 with φ1 = 0.04, φ2 = 0.06, Q = 0.2, γ = 0.9 andδ = 0.4.

flow rate Q. It can be noted that the phenomena presented in these figures possess oppositecharacter to the pressure rise for any given set of parameters. Also the graphs of both F0 andF1 show similar behavior when compared to their respective parameters. It is significant tomention that inner friction force F1 attains higher magnitude than outer friction force F0 withincreasing values of any given set of parameters.

In order to discuss the effects of variation of various parameters on the axialpressure gradient dp/dz, MATHEMATICA has been used for the numerical evaluation ofthe analytical results, and the results are graphically presented in Figures 5(a) to (5(e). Inthese figures, the pressure gradient distribution for various values of φ1, φ2, We, γ , andδ is depicted. It is observed that pressure gradient increases with increasing the values ofpreviously mentioned parameters.


0.2 0.4 0.6 0.8 1 1.2 1.40

0.2

0.4

0.6

0.8

z

r

(a)

0.2 0.4 0.6 0.8 1 1.2 1.40

0.2

0.4

0.6

0.8

z

r

(b)

Figure 8: Streamlines pattern for (a) γ = 0.8 (b) γ = 0.9 with φ1 = 0.04, φ2 = 0.06, Q = 0.2, δ = 0.4 andWe = 0.03.

0.2 0.4 0.6 0.8 1 1.2 1.40

0.2

0.4

0.6

0.8

z

r

(a)

0.2 0.4 0.6 0.8 1 1.2 1.40

0.2

0.4

0.6

0.8

z

r

(b)

Figure 9: Streamlines pattern for (a) δ = 0.4 (b) δ = 0.5 with φ1 = 0.04, φ2 = 0.06, Q = 0.2, γ = 0.9 andWe = 0.03.

The influence of various parameters on streamlines pattern is depicted in Figures 6,7, 8, 9, and 10. It is noted that trapping is observed in all these cases. Figures 6, 7, 8, and 9show that the size of trapped bolus increases for higher values of Weissenberg number (We),wave length ratio (γ), and amplitude (φ2). Figure 9 exhibits the effects of radius ratio (δ) onstreamlines pattern. It can be seen that the size and number of the trapped bolus increase withan increase in δ. However, the size and number of trapped bolus decrease with increasingvalues of flow rate as shown in Figure 10.


0.2 0.4 0.6 0.8 1 1.2 1.40

0.2

0.4

0.6

0.8

z

r

(a)

0.2 0.4 0.6 0.8 1 1.2 1.40

0.2

0.4

0.6

0.8

z

r

(b)

Figure 10: Streamlines pattern for (a) Q = 0.3 (b) Q = 0.5 with φ1 = 0.04, φ2 = 0.06, γ = 0.9, δ = 0.4 andWe = 0.03.

5. Conclusion

The peristaltic flow of chyme (treated as Williamson fluid) in small intestine with an insertedendoscope is investigated. The flow is considered between annular space of small intestineand inserted endoscope and is induced by two sinusoidal peristaltic waves of different wavelengths, traveling along the length of the intestinal wall. Long wavelength and low Reynoldsnumber approximations are used to simplify the resulting equations. The solution of theproblem is calculated using analytical technique (i.e., regular perturbation method) andnumerical technique (i.e., shooting method). Also results of axial velocity for both solutionsare compared and found a very good agreement between them.

The performed analysis can be concluded as follows:

(1) the peristaltic pumping rate decreases with increasing the values of φ2,We, γ , and δ.This shows that the effects of these parameters on the pressure rise are qualitativelysimilar;

(2) frictional forces show an opposite behavior to that of pressure rise in peristaltictransport;

(3) the inner friction force F1 attains higher magnitude than outer friction force F0 withincreasing values of any given set of parameters;

(4) Pressure gradient increases with increasing the values of all embedded parametersthat is, φ1, φ2,We, γ , and δ;

(5) an increase in radius ratio (δ) results in the increase of the size and numberof trapped bolus. Also the size of trapped bolus increases for higher values ofamplitude rate (φ2), Weissenberg number (We), and wave length ratio (γ);

(6) moreover, it is observed that the size and number of trapping bolus decrease withincreasing values of flow rate (Q).


Appendix

The values of quantities appearing in the expressions (3.2), (3.3), and (3.4) are given as

a11 = logh + log r, a12 = logh + log δ, a13 = h2 − r2,a14 = h2 − δ2, a15 = h − r, a16 = h − δ, a17 = h + δ,

a18 = h2 + δ2, a19 = h2 + r2, a20 = h4 − δ4, a21 = h6 − δ6,a22 = h3 + δ3, a23 = h5 − δ5, a24 = h2 + δ2, a25 = h3 − δ3,a26 = h7 − δ7, a27 = h4 + δ4, a28 = log δ3 − logh3,

a29 = logh log δ, a30 =240ra28 + 720ra29a12

a312,

a31 =r(2Q + a14)(

300hδa216a log δ317(−a14 + a24a12)

)3 ,

a32 = −10Wea316a

317a12(2Q + a14),

a33 = hδa12(5a216a

217

)(−24QWe + a14a12(−12We + a17)),

a34 = 10a14a12(a23 − 8QWe(a24 + hδ) − 4hδ(We(a14 + a20) + a25)),

a35 = 32Qa212We

(a27 + hδa14 + h2δ2

)+ 16We(a21 − hδa27 + 5hδa23)a312,

a36 = 30r3 − 8r4We −15rWea

316a17

δa312

(a17h

+ logh),

a37 =1a312

10r(h3We + δ2(3 − 7Weδ) + h2(−3 + 6Weδ) logh2

),

a38 = 20rδ2(3 −Weδ) logh3 + 5a16r log r

(3a12

a216a217

hδ− 12a14a212

),

a39 = 4W2e a12(a24 + hδ − a17),

a40 =ra16

a312

(15We

ha14 log δ + 80h2We + 20δ(−3 + 4Weδ)

),

a41 = 20h(−3 + 7Weδ)a29 + a29 logh(60h − 20h3We + 120δ2 − 40δ3We

),

a42 = 70h2We + 10δ(−3 +Weδ) + 10h(−3 + 7Weδ) log δ2,

a43 =ra29 log δ

a312

(−120h2 + 40h3We − 60δ2 + 20δ3We

),

a44 =20rh2 log δ3

a312(3 − hWe), a45 = −a13 + a11a16

a12a17,

a46 =a1512

(a19 + hr) +a215a16a

217

16a212+a216a

217a11

16a212,


a47 =a15a144a12

+a11a1612a12

(a18 + hδ) +a316a

217a11

16hδ(a12)3,

a48 =1

5hδa216a317(−a14 + a18a12)3

,

a49 = 8((2Q + a14)

(−5hδa316a17a12

)+ 10hδa14(a23 − hδa22)a212 + a212

),

a50 = 8We

(−10a316a317(2Q + a14)2 + 60hδa216a

217a12(2Q + a14)

),

a51 = 40hδa14a212(2Qa18 + a20 + a14)(2Qa18 + a20 + a14),

a52 = 16hδa3122Q(a20 + h2δa17 + hδ3

)+ a21 + hδa20.

(A.1)

Acknowledgments

The first and second author is thankful to higher education of Pakistan for the financialsupport and the third author extends his appreciation to the deanship of Scientific Researchat king Saud University for funding this work through the research group Project no. RGP-VPP-080.

References

[1] L. M. Srivastava and V. P. Srivastava, “Peristaltic transport of a non-Newtonian fluid: applicationsto the vas deferens and small intestine,” Annals of Biomedical Engineering, vol. 13, no. 2, pp. 137–153,1985.

[2] S. Keshav, The Gastrointestinal System at a Glance, Wiley-Blackwell, Malden, Mass, USA, 2003.[3] H. G. Beger, A. Schwarz, and U. Bergmann, “Progress in gastrointestinal tract surgery: the impact of

gastrointestinal endoscopy,” Surgical Endoscopy and Other Interventional Techniques, vol. 17, no. 2, pp.342–350, 2003.

[4] T. W. Latham, Fluid Motion in a Peristaltic Pump, M.S. thesis, Massachusetts Institute of Technology,Cambridge, Mass, USA, 1966.

[5] S. Srinivas and R. Gayathri, “Peristaltic transport of a Newtonian fluid in a vertical asymmetricchannel with heat transfer and porous medium,” Applied Mathematics and Computation, vol. 215, no. 1,pp. 185–196, 2009.

[6] S. Srinivas, R. Gayathri, and M. Kothandapani, “The influence of slip conditions, wall properties andheat transfer on MHD peristaltic transport,” Computer Physics Communications, vol. 180, no. 11, pp.2115–2122, 2009.

[7] M. Kothandapani and S. Srinivas, “Peristaltic transport of a Jeffrey fluid under the effect of magneticfield in an asymmetric channel,” International Journal of Non-Linear Mechanics, vol. 43, no. 9, pp. 915–924, 2008.

[8] S. Srinivas and M. Kothandapani, “Peristaltic transport in an asymmetric channel with heat transfer.A note,” International Communications in Heat and Mass Transfer, vol. 35, no. 4, pp. 514–522, 2008.

[9] A. H. Sharpio, M. Y. Jaffrin, B. R. P. Rao, and S. I. Weinberg, “Peristaltic pumping with longwavelength and low Reynolds number,” Journal of Fluid Mechanics, vol. 37, no. 4, pp. 799–825, 1969.

[10] S. Nadeem and N. S. Akbar, “Numerical solutions of peristaltic flow of Williamson fluid with radiallyvarying MHD in an endoscope,” International Journal for Numerical Methods in Fluids, vol. 66, no. 2, pp.212–220, 2011.

[11] S. Nadeem, N. S. Akbar, and M. Y. Malik, “Numerical solutions of peristaltic flow of a Newtonianfluid under the effects of magnetic field and heat transfer in a porous concentric tubes,” Zeitschrift furNaturforschung A, vol. 65, no. 5, pp. 369–389, 2010.


[12] S. Nadeem and N. S. Akbar, “Effects of heat transfer on the peristaltic transport of MHD Newtonianfluid with variable viscosity: application of a domain decomposition method,” Communications inNonlinear Science and Numerical Simulation, vol. 14, pp. 3844–3855, 2009.

[13] S. Nadeem, T. Hayat, N. S. Akbar, and M. Y. Malik, “On the influence of heat transfer in peristalsiswith variable viscosity,” International Journal of Heat andMass Transfer, vol. 52, no. 21-22, pp. 4722–4730,2009.

[14] H. S. Lew, Y. C. Fung, and C. B. Lowenstein, “Peristaltic carrying and mixing of chyme in the smallintestine,” Journal of Biomechanics, vol. 4, no. 4, pp. 297–315, 1971.

[15] V. P. Srivastava, “Effects of an inserted endoscope on chyme movement in small intestine—atheoretical model,” Applications and Applied Mathematics, vol. 2, no. 2, pp. 79–91, 2007.

[16] M. Saxena and V. P. Srivastava, “Particulate suspension flow induced by sinusoidal peristaltic waves,”Japanese Journal of Applied Physics, vol. 36, no. 1 A, pp. 385–390, 1997.

[17] V. P. Srivastava and M. Saxena, “A two-fluid model of non-Newtonian blood flow induced byperistaltic waves,” Rheologica Acta, vol. 34, no. 4, pp. 406–414, 1995.

[18] L. M. Srivastava and V. P. Srivastava, “Peristaltic transport of blood: casson model. II,” Journal ofBiomechanics, vol. 17, no. 11, pp. 821–829, 1984.

[19] L. M. Srivastava, V. P. Srivastava, and S. N. Sinha, “Peristaltic transport of a physiological fluid: partI. Flow in non-uniform geometry,” Biorheology, vol. 20, no. 2, pp. 153–166, 1983.

[20] P. B. Cotton and C. B. Williams, Practical Gastrointestinal Endoscopy, Oxford University Press, London,UK, 3rd edition, 1990.

[21] A. E. H. Abd El Naby and A. E. M. El Misiery, “Effects of an endoscope and generalized Newtonianfluid on peristaltic motion,” Applied Mathematics and Computation, vol. 128, no. 1, pp. 19–35, 2002.

[22] I. Dapra and G. Scarpi, “Perturbation solution for pulsatile flow of a non-Newtonian Williamsonfluid in a rock fracture,” International Journal of Rock Mechanics and Mining Sciences, vol. 44, no. 2, pp.271–278, 2007.

[23] R. L. Burden and J. D. Faires, Numerical Analysis, PWS Publishing Company, Boston, Mass, USA, 5thedition, 1985.

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