EFFECTS OF AN ENDOSCOPE AND AN ELECTRICALLY CONDUCTING THIRD GRADE FLUID ON PERISTALTIC MOTION

September 18, 2008 10:12 WSPC/140-IJMPB 03974

International Journal of Modern Physics BVol. 22, No. 23 (2008) 3997–4016c© World Scientific Publishing Company

EFFECTS OF AN ENDOSCOPE AND AN ELECTRICALLY

CONDUCTING THIRD GRADE FLUID

ON PERISTALTIC MOTION

T. HAYAT∗, E. MOMONIAT† and F. M. MAHOMED

Centre for Differential Equations, Continuum Mechanics and Applications

School of Computational and Applied Mathematics,

University of the Witwatersrand,

Private Bag 3, Wits 2050, Johannesburg, South Africa†[email protected]

Received 19 October 2006

This investigation deals with the influence of an inserted endoscope and a magnetohy-drodynamic (MHD) third grade fluid on the peristaltic flow under zero Reynolds numberand long wavelength approximations. A uniform magnetic field is applied transverselyto the flow. The mathematical model considers an incompressible fluid through the gapbetween concentric uniform tubes, such that the inner tube is rigid and the outer tubetravels down its wall. The governing non-linear equation has been modeled and thensolved for arbitrary values of Deborah number. The effect of the Hartmann number,Deborah number, volume flow rate, radius ratio and the various parameters enteringinto the problem are discussed on the velocity components, axial pressure gradient,pressure rise and frictional forces on the inner and outer tubes.

Keywords: Magnetohydrodynamic fluid; non-Newtonian fluid; an endoscope.

1. Introduction

Peristaltic flows are generated by the propagation of waves along the flexible walls

of a tube. This type is often involved in the gastro-intestinal tract, esophagus,

glandular ducts, movement of ovum in the fallopian tube and vasomotion in small

blood vessels. The industrial use of this flow mechanism in roller/finger pumps is

well-known. The essential theoretical and experimental fluid mechanics of peristaltic

transport have been reviewed in the references.1–3

The behavior of most of the physiological fluids is known to be non-Newtonian.

Hence in the recent past, quite a good number of investigations4–10 related to

peristaltic transport of non-Newtonian fluids have been carried out with various

∗Permanent Address: Department of Mathematics, Quaid-i-Azam University 45320, Islamabad,Pakistan.†Corresponding author.

3997


3998 T. Hayat, E. Momoniat & F. M. Mahomed

prospectives, in channels or tubes. Most of these analytic studies are asymptotic

expansions with small Reynolds number, wave number, amplitude ratio as the per-

turbation parameter. Siddiqui and Schwarz11 discussed the peristaltic motion of

a third-order fluid with long wavelength and low Reynolds number assumptions.

More recently, Hayat et al.12 examined the peristaltic flow of a third-order fluid

in a tube under the same assumptions as in Ref. 11. In both papers, the fluid

is hydrodynamic and the influence of an endoscope is absent. Numerical methods

have also been used by Takabatake and Ayukawa13 and Takabatake et al.14 to

solve the Newtonian hydrodynamic equations. Wagh and Wagh,15 Sud et al.,16 and

Agrawal and Anwaruddin17 treated blood flow as magnetic fluid flow. Chaturani

and Bharatiya18 discussed the magnetic fluid model for two-phase pulsatile flow of

blood. Also, Mekheimer19 discussed the blood flow as magnetic couple stress fluid

in a non-uniform channel.

Almost all of the above mentioned investigations have been carried out without

an endoscope effect, which occurs in many clinical applications. From the fluid

dynamics point of view, both endoscope and catheter have no difference, but from a

physiological point of view, we cannot use the catheter for the small intestine. More

recently, Misery et al.20 and Hakeem et al.21 analyzed the effect of an endoscope on

hydrodynamic peristaltic flows of the Newtonian and generalized Newtonian fluids,

respectively under long wavelength at low Reynolds number assumption.

However, no attempt has been made to discuss the effects of an endoscope on

the peristaltic flow of a magnetohydrodynamic non-Newtonian fluid. Even no such

attempt is available in the literature for magnetohydrodynamic Newtonian fluid.

The present study is aimed at investigating the effects of an endoscope in a physio-

logical situation. The fluid considered is third grade and electrically conducting. We

consider the magnetohydrodynamic flow through the gap between co-axial uniform

tubes, where the inner tube is rigid and the outer tube has a sinusoidal wave trav-

eling down its wall. The present theoretical study has potential for therapeutic use

in the diseases of heart, blood and blood vessels. Blood is a suspension of cells in

plasmas. Due to the presence of haemoglobin (an iron compound) in red cells, blood

can be regarded as a suspension of magnetic particles (red cells) in non-magnetic

plasma. Assuming blood as a magnetic fluid, it may be possible to control blood

pressure and its flow behavior by using an appropriate magnetic field. Also, for the

flow of blood in arteries with arterial disease like arterial stenosis or arterioscle-

rosis, the influence of magnetic field may be utilized as a blood pump in carrying

out cardiac operations. The paper is organized in the following way. In Sec. 2, the

governing equations for magnetohydrodynamic flow are modeled without any as-

sumption, and the problem is formulated. In Sec. 3, equations for axial velocity

and axial pressure gradient under long wavelength and low Reynolds number are

derived. Further, the expressions for non-dimensional rate of volume flow, pressure

rise and frictional forces on the inner and outer tubes are given. Section 4 reports

the approximate solution for velocity components and the pressure gradient. The

numerical solution is discussed in Sec. 5. In Sec. 6, the results are summarized.


Effects of an Endoscope and Fluid on Peristaltic Motion 3999

2. Mathematical Analysis

Let us consider the magnetohydrodynamic flow of a third grade fluid through co-

axial uniform tubes. The outer tube has a sinusoidal wave traveling down its wall,

while the inner tube is rigid. The fluid is electrically conducting and the uniform

magnetic field is applied in the transverse direction. The magnetic Reynolds num-

ber is assumed to be small so that the induced magnetic field is neglected. The

axisymmetric geometry facilitates the choice of the cylindrical polar coordinates R

and Z to study the problem. Here, the Z-axis is along the centerline of the inner

and the outer tubes and R is in the radial direction. The geometry of the two wall

surfaces are given by the equations

r1 = a1 , (1)

r2 = a2 + b sin2π

λ(Z − ct ) . (2)

In the above equations, a1 is the radius of the inner tube, a2 is the radius of the

outer tube at the inlet, c is the wave speed, t is the time, b is the amplitude and λ

is the wavelength. The governing equations for the flow in the laboratory frame are

ρ

[

∂U

∂t+ U

∂U

∂R+ W

∂U

∂Z

]

= − ∂p

∂R+

1

R

∂

∂R(RSRR) +

∂

∂Z(SRZ) − Sθθ

R, (3)

ρ

[

∂W

∂t+ U

∂W

∂R+ W

∂W

Z

]

= − ∂p

∂Z+

1

R

∂

∂R(RSRZ) +

∂

∂Z(SZZ) − σB2

0W , (4)

∂U

∂R+

U

R+

∂W

∂R= 0 , (5)

where U and W are the radial and axial velocities, p is the pressure, B0 is the

applied magnetic field and σ is the electrical conductivity of the fluid.

Using the arguments of modern continuum thermodynamics, the extra stress

tensor for a third grade fluid is given by Fosdick and Rajagopal22 as:

S = µA1 + α1A2 + α2A21 + β(tr A2

1)A1 , (6)

in which µ is the dynamic viscosity, α1, α2 and β are material constants. The Rivlin-

Ericksen tensors A1 and A2 are given through the following equations:

A1 = (grad V) + (grad V)T ,

A2 =

(

∂

∂t+ V · ∇

)

V + A1(grad V) + (grad V)T A1 ,(7)

where grad is the grad operator and V is the velocity. The Clausius-Duhem inequal-

ity and the requirement that the free energy be a minimum in equilibrium imposes

the following restrictions on the material constants:

µ ≥ 0, α1 ≥ 0, β ≥ 0, |α1 + α2| ≤√

24µβ .



A detailed analysis on third grade fluid models has been found in Dunn and

Rajagopal.23

In the laboratory frame (R, Z), the motion is unsteady. In order to carry

out steady state analysis, we move to the wave frame (r, z). The transformations

between the two frames are

z = Z − ct, r = R ,

u = U , w = W − c ,(8)

in which u and w are the velocity components in the wave frame.

The boundary conditions in the wave frame are

w = −c, u = 0 at r = r1 , (9)

w = −c, at r = r2 = a2 + b sin2π

λz . (10)

Using the transformations (8) and defining the non-dimensional variables

r = ra2, z =z

λ, u =

λu

a2c, p =

a22p

cλµ,

r1 =r1

a2=

a1

a2= ε < 1, w =

w

c,

S =a2S

µc, r2 =

r2

a2= 1 + φ sin(2πz) ,

(11)

Eqs. (3) to (5) and boundary conditions (9) and (10) become

Reδ3

[

u∂u

∂r+ w

∂u

∂z

]

= −∂p

∂r+

δ

r

∂

∂r(rSrr) + δ2 ∂

∂z(Srz) − δ

Sθθ

r, (12)

Re δ

[

u∂w

∂r+ w

∂w

∂z

]

= −∂p

∂z+

1

r

∂

∂r[rSrz ] + δ

∂

∂zSzz − M2(w + 1) , (13)

∂u

∂r+

u

r+

∂w

∂z= 0 , (14)

w = −1, u = 0 at r = r1 = ε ,

w = −1, at r = r2 = 1 + φ sin(2πz) ,(15)

where the Reynolds number (Re), wave number (δ), non-dimensional wave ampli-

tude (φ) and Hartmann number (M) are

Re =ρca2

µ, δ =

a2

λ, φ =

b

a2(< 1) ,

M =

√

σ

µB0a2(>

√2) (16)



and

Srr = 2δ∂u

∂r+ 2λ1

[

δ2

(

u∂

∂r+ w

∂

∂z

)

∂u

∂r+ 2δ2

(

∂u

∂r

)2

+∂w

∂r

(

∂w

∂r+ δ2 ∂u

∂z

)

]

+λ2

[

4δ2

(

∂u

∂r

)2

+

(

∂w

∂r+ δ2 ∂u

∂z

)2]

+ 4Γδ

[

2δ2

(

∂u

∂r

)2

+

(

∂w

∂r+ δ2 ∂u

∂z

)2

+2δ2(u

r

)2

+ 2δ2

(

∂w

∂z

)2]

∂u

∂r, (17)

Srθ = 0 = Sθz , (18)

Srz =

(

∂w

∂r+ δ2 ∂u

∂z

)

+ λ1

[(

u∂

∂r+ w

∂

∂z

)(

δ∂w

∂r+ δ3 ∂u

∂z

)

+ δ∂u

∂r

(

∂w

∂r+ δ2 ∂u

∂z

)

+ 2δ∂w

∂r

∂w

∂z+ 2δ3 ∂u

∂r

∂u

∂z

+ δ∂w

∂z

(

∂w

∂r+ δ2 ∂u

∂z

)]

+ 2λ2δ

[

∂u

∂r

(

∂w

∂r+ δ2 ∂u

∂z

)

+∂w

∂z

(

∂w

∂r+ δ2 ∂u

∂z

)]

+ 2Γ

[

2δ2

(

∂u

∂r

)2

+

(

∂w

∂r+ δ2 ∂u

∂z

)2

+ 2δ2(u

r

)2

+ 2δ2

(

∂w

∂z

)2]

(

∂w

∂r+ δ2 ∂u

∂z

)

, (19)

Sθθ = 2δu

r+ 2λ1δ

2

[(

u∂

∂r+ w

∂

∂z

)

u

r+ 2

(u

r

)2]

+ 4λ2δ2(u

r

)2

+ 4δΓ

[

2δ2

(

∂u

∂r

)2

+

(

∂w

∂r+ δ2 ∂u

∂z

)2

+ 2δ2(u

r

)2

+ 2δ2

(

∂w

∂z

)2]

u

r,

(20)

Szz = 2δ∂w

∂z+ 2δ2λ1

[

(

u∂

∂r+ w

∂

∂z

)

∂w

∂z+

∂u

∂z

(

∂w

∂r+ δ2 ∂u

∂z

)

+ 2

(

∂w

∂z

)2]

+ λ2

[

(

∂w

∂r+ δ2 ∂u

∂z

)2

+ 4δ2

(

∂w

∂z

)2]

+ 4δΓ

[

2δ2

(

∂u

∂r

)2

+

(

∂w

∂r+ δ2 ∂u

∂z

)2

+ 2δ2(u

r

)2

+ 2δ2

(

∂w

∂z

)2]

∂w

∂z,

(21)



where

λ1 =α1c

µa2, λ2 =

α2c

µa2, Γ =

βc2

µa22

.

3. Analysis for Large Wavelength at Low Reynolds Number

We note that solution of Eqs. (12) to (14) in general is impossible. Thus, similar

to Shapiro et al.,24 Siddiqui and Schwarz,11 Rao and Usha25 and Hayat et al.12 we

assume that the wavelength is long and thus from Eqs. (12) to (14) and Eqs. (17)

to (21), we have

0 = −∂p

∂r, (22)

0 = −∂p

∂z+

1

r

∂

∂r[rSrz ] − M2(w + 1) , (23)

where

Srz =∂w

∂r+ 2Γ

(

∂w

∂r

)3

. (24)

Elimination of pressure from Eqs. (22) and (23) yields

∂

∂r

[

1

r

∂

∂r

{

r

(

∂w

∂r+ 2Γ

(

∂w

∂r

)3)}]

− M2 ∂w

∂r= 0 . (25)

Moreover, Eq. (22) indicates that p is not a function of r. Hence from Eqs. (23) and

(24), we can write for axial pressure gradient as:

dp

dz=

1

r

∂

∂r

{

r

(

∂w

∂r+ 2Γ

(

∂w

∂r

)3)}

− M2(w + 1) . (26)

4. Rate of Volume Flow

The instantaneous volume flow rate in the laboratory frame is defined by

Q = 2π

∫ r2

r1

W RdR , (27)

in which r1 is a constant and r2 depends upon Z and t. Making use of transforma-

tions (8) and integrating, we obtain from Eq. (27) as

Q = q + πc(r22 − r2

1) , (28)



where the volume flow rate in the wave frame is

q = 2π

∫ r2

r1

wrdr . (29)

The mean time flow over a period T1(= λ/c) at a fixed Z-position is defined by

Q =1

T1

∫ T1

0

Qdt . (30)

Substituting Eqs. (28) and (29) into Eq. (30) and then integrating, we obtain

Q = q + πc

(

a22 − a2

1 +b2

2

)

. (31)

Now, Eqs. (29) and (31) in non-dimensional variables can be written as

F =

∫ r2

r1

rwdr , (32)

Θ = F +1

2

(

1 − ε2 +φ2

2

)

, (33)

where

F =q

2πa22c

, Θ =Q

2πca22

. (34)

The pressure rise (∆Pλ) and frictional forces on the inner (F(i)λ

)and outer tubes

(F(o)λ

) in non-dimensional form are

∆Pλ =

∫ 1

0

(

dp

dz

)

dz , (35)

F(i)λ

=

∫ 1

0

r21

(

−dp

dz

)

dz , (36)

F(o)λ

=

∫ 1

0

r22

(

−dp

dz

)

dz . (37)

5. Approximate Solution

By inspection, we find that Eq. (26) admits a solution of the form

w(r, z) =Mr2

8√

Γ− λ1(z)

M2+

1

2M√

Γ− 1 , (38)

where dp/dz = G(z). We cannot impose both the boundary conditions (15) on (38)

as it has only one unknown function λ1(z) to be determined. Instead, we proceed



to solve Eq. (26) as an ordinary differential equation in w where the dependence of

the solution on z comes through by imposing the boundary condition at r = r2 =

1 + φ sin 2πz. We investigate approximate solutions admitted by Eq. (26) of the

form

w(r) = w0(r) + M2w1(r) + Γw2(r), Γ, M2 << 1 . (39)

Substituting (39) into Eq. (26) and separating to first-order in terms of the small

parameters Γ and M2, we find that

d

dr

[

rdw0

dr

]

− rλ(z) = 0 , (40)

d

dr

[

rdw1

dr

]

− r(w0 + 1) = 0 , (41)

d

dr

[

rdw2

dr

]

+ 2d

dr

[

r

(

dw0

dr

)3]

= 0 . (42)

Solving Eqs. (40)–(42) we find that

w0(r) = k2(z) +r2G(z)

4+ k1(z) log(r) , (43)

w1(r) =r4G(z)

64

+r2 (1 − k1(z) + k2(z)) + 4k3(z) +

(

r2k1(z) + 4k4(z))

log(r)

4, (44)

w2(r) =k1(z)

3

r2+ k5(z) − 3 r2k1(z)G(z)

2

4− r4G(z)

3

16+ k6(z) log(r) , (45)

where ki(z), i = 1, . . . , 6 are arbitrary functions to be determined from the boundary

conditions (15). We can use the results obtained for w(r) to determine u(r) from

Eq. (14). We look for solutions of the form

u(r) = u0(r) + M2u1(r) + Γu2(r), Γ, M2 << 1 . (46)

We substitute (46) into Eq. (14) and separate to first-order in terms of the small

parameters Γ and M2. We find that

u0(r)

r+ log(r)k1

′(z) + k2′(z) +

r2G′(z)

4+ u0

′(r) = 0 , (47)

u1(r)

r− r2k1

′(z)

4+

r2 log(r)k1′(z)

4

+r2k2

′(z)

4+ k3

′(z) + log(r)k4′(z) +

r4G′(z)

64+ u1′(r) = 0 , (48)



u2(r)

r+

3k1(z)2k1

′(z)

r2− 3r2G(z)

2k1

′(z)

4+ k5

′(z) + log(r)k6′(z)

− 3r2k1(z)G(z)G′(z)

2− 3r4G(z)

2G′(z)

16+ u2′(r) = 0 (49)

where ′ indicates differentiation with respect to z. Solving the first-order ordinary

differential equations (47)–(49), we find that

u0(r) =16b1(z) + 4r2(1 − 2 log(r))k1

′(z) − 8r2k2′(z) − r4G′(z)

16r, (50)

u1(r) =b2(z)

r− (1/384)[r(6r2(−5 + 4 log(r))k1

′(z) + 24r2k2′(z)

+ 192k3′(z) − 96k4

′(z) + 192 log(r)k4′(z) + r4G′(z))] , (51)

u2(r) =1

32r[8(4b3(z) − 12k1(z)

2log(r)k1

′(z) − 2r2k5′(z) + r2k6

′(z)

− 2r2 log(r)k6′(z)) + 12r4k1(z)G(z)G′(z)

+ G(z)2(6r4k1′(z) + r6G′(z))] , (52)

where bi(z), i = 1, 2, 3 are arbitrary functions of z to be determined from the

boundary conditions (15).

Imposing the boundary conditions (15) and separating the resulting equations

to first-order in terms of the small parameters Γ and M 2, we obtain

b1(z) =1

16[ε2((−4 + 8 log(ε))k1

′(z) + 8k2′(z) + ε2G′(z))] , (53)

b2(z) =1

384[6ε4(−5 + 4 log(ε))k1

′(z)

+ 24ε2(ε2k2′(z) + 8k3

′(z) + 4(−1 + 2 log(ε))k4′(z)) + ε6G′(z)] , (54)

b3(z) =1

32[96k1(z)2 log(ε)k1

′(z) + 8ε2(2k5′(z) + (−1 + 2 log(ε))k6

′(z))

− 12ε4k1(z) G(z)G′(z) − G(z)2(6ε4k1

′(z) + ε6G′(z)−)] , (55)

k1(z) =G(z)(ε − r2(z))(ε + r2(z))

4(− log(ε) + log(r2(z))), (56)

k2(z) =(4 + ε2G(z)) log(r2(z)) + log(ε)(−4 − G(z)r2(z)2)

4(log(ε) − log(r2(z))), (57)



k3(z) = {64[log(ε) − log(r2(z))]2}−1

×{G(z)(3 log(ε)2r2(z)

4+ ε2 log(r2(z))(3ε2 log(r2(z))

+ 4(ε− r2(z))(ε + r2(z)))

+ log(ε)(−4(ε − r2(z))r2(z)2(ε + r2(z)) − 3 log(r2(z))(ε4 + r2(z)4)))} ,

(58)

k4(z) = {64[log(ε) − log(r2(z))]2}−1

×{G(z)(ε− r2(z))(ε + r2(z))(−4(ε − r2(z))(ε + r2(z))

+ 3(log(ε) − log(r2(z)))(ε2 + r2(z)2))} , (59)

k5(z) = {64ε2[log(ε) − log(r2(z))]4r2(z)

2}−1

×{G(z)3(4ε2log(ε)

4r2(z)

6+ log(r2(z))r2(z)

2(4ε6log(r2(z))

3

+ 12ε4log(r2(z))2(ε − r2(z))(ε + r2(z))

+ (−ε2 + r2(z)2)3) + 12ε2log(ε)

2log(r2(z))r2(z)

2(ε4 + ε2r2(z)

2 − 2r2(z)4

+ log(r2(z))(ε4 + r2(z)4)) − 4ε2log(ε)3r2(z)2(3(ε − r2(z))r2(z)2(ε + r2(z))

+ log(r2(z))(ε4 + 3r2(z)4)) + ε2 log(ε)((ε − r2(z))3(ε + r2(z))3

− 4log(r2(z))2r2(z)

2(3(ε − r2(z))(ε + r2(z))(2ε2 + r2(z)

2)

+ log(r2(z))(3ε4 + r2(z)4))))} , (60)

k6(z) = {64ε2[log(ε) − log(r2(z))]4r2(z)

2}−1

×{G(z)3(−((ε − r2(z))4(ε + r2(z))4)

+ 4ε2[log(ε) − log(r2(z))]2(ε − r2(z))r2(z)

2(ε + r2(z))

× (−3(ε − r2(z))(ε + r2(z)) + (log(ε) − log(r2(z)))(ε2 + r2(z)2)))} .

(61)

We can calculate G(z) = dp/dz in terms of the flow rate, F , by substituting

Eq. (39) into Eq. (32). We find that

G(z) = −

( 23 )

1

3 d1(z)(

− 9d0(z)d2(z)2+√

3

√

4d1(z)3d2(z)

3+ 27d0(z)

2d2(z)

4)

1

3

+

(

− 9d0(z)d2(z)2

+√

3

√

4d1(z)3d2(z)

3+ 27d0(z)

2d2(z)

4)

1

3

21

3 32

3 d2(z), (62)



d0(z) =ε2 − 2F − r2(z)

2

2, (63)

d1(z) = {384[log(ε) − log(r2(z))]2}−1

×{−((ε2 − r2(z)2)(ε2(6ε2M2 + 4(−6 + ε2M2)log(ε)

2

+ log(ε)(24 − 9ε2M2 − 8(−6 + ε2M2) log(r2(z)))

+ log(r2(z))(−24 + 9ε2M2 + 4(−6 + ε2M2) log(r2(z))))

+ 4(−3ε2M2 + (−6 + ε2M2)log(ε)2

− 2 log(ε)(3 + (−6 + ε2M2) log(r2(z)))

+ log(r2(z))(6 + (−6 + ε2M2) log(r2(z))))r2(z)2

+ M2(6 + 4 log(ε)2

+ log(ε)(9 − 8 log(r2(z)))

+ log(r2(z))(−9 + 4 log(r2(z))))r2(z)4))} , (64)

d2(z) = {768ε2[log(ε) − log(r2(z))]4r2(z)

2}−1{γ (−3ε10 − ε8(−15 + 4[log(ε)

− log(r2(z))]2[3 − 2 log(ε) + 2 log(r2(z))]2)r2(z)2

− 6ε6(5 + 2(−9 + 4 log(ε) − 4 log(r2(z)))[log(ε) − log(r2(z))]2)r2(z)

4

− 6ε4(−5 + 2(9 + 4 log(ε) − 4 log(r2(z)))[log(ε) − log(r2(z))]2)r2(z)

6

+ ε2(−15 + 4[3 + 2 log(ε) − 2 log(r2(z))]2[log(ε) − log(r2(z))]

2)r2(z)

8

+ 3r2(z)10)}. (65)

It is easy to show that for M = 0 and Γ = 0, we have that

G(z) =dp

dz= [16(−1.13943− log(1 + 0.2 sin(2πz)))

× (−6.1648− 0.04 cos(4πz) + 0.8 sin(2πz))]

× [(−1.8352 + 0.04 cos(4πz) − 0.8 sin(2πz))

× (−0.722602− 2.2448 log(1 + 0.2 sin(2πz))

× 0.04 cos(4πz)(0.139434 + log(1 + 0.2 sin(2πz)))

+ 0.8(−0.139434− log(1 + 0.2 sin(2πz))) sin(2πz))]−1 . (66)

6. Discussion

When solving Eqs. (40) to (42), we do not impose the boundary conditions (15)

immediately. Instead, we impose the boundary conditions on Eq. (39) once we have

solved for w0, w1 and w2. The pressure gradient G(z) = dp/dz is determined in

terms of the flow rate using Eq. (32). Equation (32) is solved exactly, so there

is no linear dependance on Γ and M . This is because we have not performed a



A

0

0.2

0.4

0.6

0.8

1

z0.32

0.34

0.36

0.38

0.4

0.42

Ε

0

200

400

600dp��

dz

0

0.2

0.4

0.6

0.8

1

z

B

0

0.2

0.4

0.6

0.8

1

z-4

-3.5

-3

-2.5

-2

F

0

200

400

600dp��

dz

0

0.2

0.4

0.6

0.8

1

z

C

0

0.2

0.4

0.6

0.8

1

z0.2

0.25

0.3

0.35

0.4

Φ

0

500

1000

1500

2000dp��dz

0

0.2

0.4

0.6

0.8

1

z

Fig. 1. Plot of (66) showing the effects of changing ε, flow rate and wave amplitude. (A) F = −2,φ = 0.2, (B) ε = 0.32, φ = 0.2 and (C) F = −2, ε = 0.32.

perturbation analysis on G(z). The results are plotted in Figs. 1 to 10. The pressure

rise ∆Pλ and the frictional forces on the inner F(i)λ

and outer tubes F(o)λ

respectively

are determined from Eqs. (35)–(37) by numerical integration using NIntegrate in

MATHEMATICA.

In Figs. 1 to 4, we plot the effects of changing ε, flow rate F and wave amplitude

φ on the velocity components, the pressure gradient dp/dz, pressure rise ∆Pλ and

the frictional forces on the inner F(i)λ

and outer tubes F(o)λ

respectively for zero

Hartmann number M and zero Deborah number Γ. From Fig. 1 we note that the

change in pressure gradient dp/dz has a wave-like motion with increasing z. As ε,

flow rate F and wave amplitude φ increase, the amplitude of dp/dz also increases.

Figure 2 shows an increase in pressure rising with increasing ε, flow rate F and

wave amplitude φ. From Figs. 3 and 4, we note a decrease of the frictional force on

the inner and outer tubes, F(i)λ

and F(o)λ

respectively, with increasing ε, flow rate

F and wave amplitude φ. We note that the magnitude of the flow rate is increasing

in the negative direction.



0.32 0.34 0.36 0.38 0.4 0.42

Ε

140

160

180

200

220

DPΛ

A

F=-2, Φ=0.2

-3.75-3.5-3.25 -3 -2.75-2.5-2.25 -2

F

125

150

175

200

225

250

275

DPΛ

B

Ε=0.32, Φ=0.2

0.2 0.25 0.3 0.35 0.4

Φ

100

200

300

400

500

DPΛ

C

F=-2, Ε=0.32

Fig. 2. Plot of the pressure rise ∆Pλ showing the effects of changing ε, flow rate and waveamplitude for M = 0 and Γ = 0. (A) F = −2, φ = 0.2, (B) ε = 0.32, φ = 0.2 and (C) F = −2,ε = 0.32.

0.32 0.34 0.36 0.38 0.4 0.42

Ε

-40

-35

-30

-25

-20

-15

-10

FΛHiL

A

F=-2, Φ=0.2

-4 -3.5 -3 -2.5 -2

F

-30

-25

-20

-15

-10

FΛHiL

B

Ε=0.32, Φ=0.2

0.2 0.25 0.3 0.35 0.4

Φ

-45

-40

-35

-30

-25

-20

-15

-10

FΛHiL

C

F=-2, Ε=0.32

Fig. 3. Plot of the frictional force on the inner tube F(i)λ

showing the effects of changing ε, flowrate and wave amplitude for M = 0 and Γ = 0. (A) F = −2, φ = 0.2, (B) ε = 0.32, φ = 0.2 and(C) F = −2, ε = 0.32.



0.32 0.34 0.36 0.38 0.4 0.42

Ε

-180

-160

-140

-120

-100

FΛHoL

A

F=-2, Φ=0.2

-4 -3.5 -3 -2.5 -2

F

-240

-220

-200

-180

-160

-140

-120

-100

FΛHoL

B

Ε=0.32, Φ=0.2

0.2 0.25 0.3 0.35 0.4

Φ

-200

-180

-160

-140

-120

-100

FΛHoL

C

F=-2, Ε=0.32

Fig. 4. Plot of the frictional force on the outer tube F(o)λ

showing the effects of changing ε, flowrate and wave amplitude for M = 0 and Γ = 0. (A) F = −2, φ = 0.2, (B) ε = 0.32, φ = 0.2 and(C) F = −2, ε = 0.32.

In Figs. 5 to 8, we examine the effects of changing the Hartmann number M

and Deborah number Γ on the results indicated in Figs. 1 to 4. From Fig. 5, we

note a decrease in the maximum wave amplitude for increasing values of ε, flow

rate F and wave amplitude φ. There is a slight change in the wave profile due to

the decrease in maximum amplitude of dp/dz. From Fig. 6A, we note that when

M = 0.05 and Γ = 0.05, there is a decrease in pressure rise with increasing ε rather

than an increase when compared to Fig. 2A. Figures 6B and 6C show an increase in

pressure rise ∆Pλ with increasing flow rate F and wave amplitude φ. The values of

the pressure rise ∆Pλ in Fig. 6 are much less than those indicated in Fig. 2. When

comparing Fig. 3 to Fig. 7, we note that both indicate a decrease of the frictional

force on the inner tube F(i)λ

with increasing ε, flow rate F and wave amplitude φ.

Once again, the values of the frictional force on the inner tube F(i)λ

are much less

in Fig. 7 than in Fig. 3. From Figs. 8A and 8C, we note that for M = 0.05 and

Γ = 0.05, the frictional force on the outer tube increases with increasing ε and

increasing wave amplitude φ when compared to Figs. 4A and 4C for M = 0 and

Γ = 0. Figure 8B indicates a decrease in the frictional force on the outer tube F(o)λ

with increasing flow rate, as indicated in Fig. 4B. Once again, the values of the

frictional force on the outer tube F(o)λ

as indicated in Fig. 8 are much less than

those indicated in Fig. 7.



A

0

0.2

0.4

0.6

0.8

1

z0.32

0.34

0.36

0.38

0.4

0.42

Ε

6

8

10dp��

dz

0

0.2

0.4

0.6

0.8

1

z

B

0

0.2

0.4

0.6

0.8

1

z

-4

-3.5

-3

-2.5

-2

F

6

8

10

12

14dp��

dz

0

0.2

0.4

0.6

0.8

1

z

C

0

0.2

0.4

0.6

0.8

1

z

0.2

0.25

0.3

0.35

0.4

Φ

5

10

15dp��dz

0

0.2

0.4

0.6

0.8

1

z

Fig. 5. Plot of (62) showing the effects of changing ε, flow rate and wave amplitude when M =0.05 and Γ = 0.05. (A) F = −2, φ = 0.2, (B) ε = 0.32, φ = 0.2 and (C) F = −2, ε = 0.32.

The velocity components w and u are plotted in Fig. 9 for different values of

the Hartmann number M and the Deborah number Γ. We note that the Hartmann

number M decreases both the velocity components w and u, while Γ affects the

shape of the velocity profile. Thus, we can conclude that the effect of Γ on the

velocity components w and u is to increase the nonlinearity of the profiles, while

the Hartmann number M affects the amplitudes.

We next solve Eqs. (26) and (14) numerically subject to conditions (15) where

we choose G(z) = dp/dz = e−z2

. In Eq. (14), we approximate the derivative ∂w/∂z

using a forward difference [w(r, z + h) − w(r, z)]/h where the functions w(r, z) are

interpolated functions determined by using NDSolve in MATHEMATICA. These

results are indicated in Fig. 10. As noted earlier, the Hartmann number reduces

the amplitude of the velocity components while Γ increases the non-linearity of the

velocity profiles. For G(z) = dp/dz = e−z2

, we find that

∆Pλ = 0.746824 , (67)



0.32 0.34 0.36 0.38 0.4 0.42

Ε

7.3

7.4

7.5

7.6

7.7

7.8

DPΛ

A

F=-2, Φ=0.2

-3.75-3.5-3.25 -3 -2.75-2.5-2.25 -2

F

8

8.5

9

9.5

10

DPΛ

B

Ε=0.32, Φ=0.2

0.2 0.25 0.3 0.35 0.4

Φ

7.75

8

8.25

8.5

8.75

9

9.25

DPΛ

C

F=-2, Ε=0.32

Fig. 6. Plot of the pressure rise ∆Pλ showing the effects of changing ε, flow rate and waveamplitude for M = 0.05 and Γ = 0.05. (A) F = −2, φ = 0.2, (B) ε = 0.32, φ = 0.2 and(C) F = −2, ε = 0.32.

0.32 0.34 0.36 0.38 0.4 0.42

Ε

-1.3

-1.2

-1.1

-1

-0.9

-0.8

FΛHiL

A

F=-2, Φ=0.2

-4 -3.5 -3 -2.5 -2

F

-1

-0.95

-0.9

-0.85

-0.8

-0.75

FΛHiL

B

Ε=0.32, Φ=0.2

0.2 0.25 0.3 0.35 0.4

Φ

-0.925

-0.9

-0.875

-0.85

-0.825

-0.8

FΛHiL

C

F=-2, Ε=0.32

Fig. 7. Plot of the frictional force on the inner tube F(i)λ

showing the effects of changing ε, flowrate and wave amplitude for M = 0.05 and Γ = 0.05. (A) F = −2, φ = 0.2, (B) ε = 0.32, φ = 0.2and (C) F = −2, ε = 0.32.



0.32 0.34 0.36 0.38 0.4 0.42

Ε

-7.3

-7.2

-7.1

-7

-6.9

FΛHoL

A

F=-2, Φ=0.2

-4 -3.5 -3 -2.5 -2

F

-9

-8.5

-8

-7.5

-7

FΛHoL

B

Ε=0.32, Φ=0.2

0.2 0.25 0.3 0.35 0.4

Φ

-7.3

-7.28

-7.26

-7.24

-7.22

-7.2

FΛHoL

C

F=-2, Ε=0.32

Fig. 8. Plot of the frictional force on the outer tube F(o)λ

showing the effects of changing ε, flowrate and wave amplitude for M = 0.05 and Γ = 0.05. (A) F = −2, φ = 0.2, (B) ε = 0.32, φ = 0.2and (C) F = −2, ε = 0.32.

0.4 0.6 0.8 1 1.2

r

-10

-8

-6

-4

-2

0

w

z=0 z=0.4z=0.8

(a)

0.4 0.6 0.8 1 1.2

r

-4

-2

0

2

4

u

z=00.4

z=0.8

(b)

Fig. 9. Plot of the velocity w from (39) and u from (46) where F = −2, φ = 0.2 and ε = 0.32for different values of the Hartmann number M and the Deborah number Γ at z-values indicated.(− − −), M = 0 and Γ = 0; ( ), M = 0.1 and Γ = 0.001; (• • •) M = 0 and Γ = 0.01.

F(i)λ

= −0.746824ε , (68)

F(o)λ

= −0.746824− 0.228696φ− 0.375771φ2 . (69)



0.4 0.6 0.8 1

r

-1.06

-1.04

-1.02

-1

w

z=0 z=0.4z=0.8

(a)

0.4 0.6 0.8 1

r

-0.04

-0.02

0

0.02

0.04

u

z=0

0.4

z=0.8

(b)

Fig. 10. Plot of the numerical solution of (26) for w and (14) for u where we have taken dp/dz =

e−z2

for different values of the Hartmann number M and the Deborah number Γ at the values ofz indicated. (− − −), M = 0 and Γ = 0; ( ), M = 5 and Γ = 0; (• • •) M = 0 and Γ = 5;(. . .), M = 5 and Γ = 15.

7. Concluding Remarks

In the present work, a theoretical model for peristaltic flow of MHD third grade fluid

between two tubes has been developed. The expressions for velocity components,

axial pressure gradient, pressure rise and frictional forces on the inner and outer

tubes have been found and employed to explore the response of Deborah number,

Hartmann number, an endoscope, wave amplitude and flow rate. As the radius of the

small intestine, a = 1.25 cm, is small compared with the wavelength λ = 8.01 cm,

so the wave number a/λ = 0.156. Moreover, the chyme in the intestine is a non-

Newtonian fluid. Thus, the present work serves as a model for the study of flow

of chyme through small intestines and discloses some underlying physics of various

parameters of interest. Based on the above facts, the following conclusions can be

drawn:

1. With the increase of Deborah number, the non-linearity in the velocity profiles

increases.

2. The increase in Hartmann number reduces the velocity components.

3. The magnitude of axial pressure gradient increases for large values of ε, F and

φ.

4. An increase in Hartmann number increases the pressure rise.

5. The pressure rise is found to increase with the increase in ε, F and φ.

6. The effect of Hartmann number is more obvious as the Deborah number

increases.

7. With the increase of Deborah number, the pressure rise increases. Thus, the

magnitude of the pressure rise is greater for the third grade fluid than for a

Newtonian fluid.

8. The pressure rise is larger for an MHD fluid than for a hydrodynamic fluid.

9. Frictional forces have the opposite behavior when compared to the pressure

rise.



10. Friction force on the outer tube is greater than the friction force on the inner

tube.

11. Setting Γ = 0, we obtain the analysis for the MHD Newtonian fluid. In addition,

for Γ = 0 and M = 0, we have the results for hydrodynamic Newtonian fluid.

Acknowledgments

TH thanks the DECMA research centre and the School of Computational and

Applied Mathematics at the University of the Witwatersrand, Johannesburg, for

financial support and hospitality. Dr. Hayat is further grateful to the Higher Ed-

ucation Commission (HEC) for their financial support. EM acknowledges support

received from the National Research Foundation under grant number 2053745.

References

1. M. Y. Jaffrin and A. H. Shapiro, Peristaltic pumping, Ann. Rev. Fluid Mech. 128,13–35 (1971).

2. H. J. Rath, Peristaltic Stromungen (Springer-Verlag, Berlin, 1980).3. L. M. Srivastava and V. P. Srivastava, Peristaltic transport of blood: Casson model –

II, J. Biomech. 17, 821–829 (1984).4. K. K. Raju and R. Devnathan, Peristaltic motion of a non-Newtonian fluid, Rheol

Acta. 11, 170–178 (1972).5. K. K. Raju and R. Devanathan, Peristaltic motion of a non-Newtonian fluid; Part-II,

Rheol. Acta. 13, 944–948 (1974).6. G. Radhakrishnamacharya, Long wavelength approximation to peristaltic motion of

power law fluid, Rheol. Acta. 21, 30–35 (1982).7. G. Bohme and R. Friedrich, Peristaltic flow of viscoelastic liquids, J. Fluid Mech.

128, 109–122 (1983).8. S. Usha and R. A. Ramachandra, Peristaltic transport of two layered power law fluid,

Trans. ASME J. Biomech. Eng. 119, 483–488 (1997).9. J. B. Shukla, R. S. Parihar and B. R. P. Rao, Effects of stenosis on non-Newtonian

flow of blood in an artery, Bull. Math. Biol. 42, 283–294 (1980a).10. L. M. Srivastava and V. P. Srivastava, Peristaltic transport of power law fluid; Appli-

cation to the ductus of efferentus of the reproductive tract, Rheol. Acta. 27, 428–433(1988).

11. A. M. Siddiqui and W. H. Schwarz, Peristaltic motion of a third order fluid in a planarchannel, Rheol. Acta 32, 47–56 (1993).

12. T. Hayat, Y. Wang, A. M. Siddiqui, K. Hutter and S. Asghar, Peristaltic transport ofa third order fluid in a circular cylindrical tube, Math. Models and Methods in Appl.

Sci. 12, 1691–1706 (2002).13. S. Takabatake and K. Ayukawa, Numerical study of two-dimensional peristaltic flows,

J. Fluid Mech. 122, 439–365 (1982).14. S. Takabatake, K. Ayukawa and A. Mori, Peristaltic pumping in circular cylindrical

tubes; A numerical study of fluid transport and its efficiency, J. Fluid Mech. 193,267–283 (1988).

15. D. K. Wagh and S. D. Wagh, Blood flow considered as magnetic fluid flow, Proc. Phil.

Fluid Dyn. III, 311–315 (1992).16. V. K. Sud, G. S. Sephon and R. K. Mishra, Pumping action on blood flow by a

magnetic field, Bull. Math. Biol. 39, 385–390 (1977).



17. H. L. Agrawal and B. Anwaruddin, Peristaltic flow of blood in a branch, Ranchi Univ.

Math. J. 15, 111–118 (1984).18. P. Chaturani and S. S. Bharatiya, Magnetic fluid model for two–phase pulsatile flow

of blood, Acta Mech. 149, 97–114 (2001).19. Kh. S. Mekheimer, Peristaltic flow of blood under effect of a magnetic field in a

non-uniform channels, Appl. Math. Computation 153(3), 763 (2004).20. A. M. E. Misery, A. E. Hakeem, A. E. Naby and A. H. E. Nagar, Effects of a fluid

with variable viscosity and an endoscope on peristaltic motion, J. Phy. Soc. Japan

72, 89–93 (2003).21. A. E. Hakeem, A. E. Naby and A. E. M. E. Misiery, Effects of an endoscope and

generalized Newtonian fluid on peristaltic motion, Appl. Math. Computation. 128,19–35 (2002).

22. R. L. Fosdick and K. R. Rajagopal, Thermodynamics and stability of fluids of thirdgrade, Proc. R. Soc. London A 339, 351–377 (1980).

23. J. E. Dunn and K. R. Rajagopal, Fluids of differential type — critical review andthermodynamic analysis, Int. J. Eng. Sci. 33(5), 689–729 (1985).

24. A. H. Shapiro, M. Y. Jaffrin and S. L. Weinberg, Peristaltic pumping with longwavelengths at low Reynolds number, J. Fluid Mech. 37, 799–825 (1969).

25. A. R. Rao and S. Usha, Peristaltic transport of two immiscible viscous fluids in acircular tube, J. Fluid Mech. 298, 271–285 (1995).

EFFECTS OF AN ENDOSCOPE AND AN ELECTRICALLY CONDUCTING THIRD GRADE FLUID ON PERISTALTIC MOTION

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