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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†ebrahim@cam.wits.ac.za
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
September 18, 2008 10:12 WSPC/140-IJMPB 03974
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.
September 18, 2008 10:12 WSPC/140-IJMPB 03974
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 .
September 18, 2008 10:12 WSPC/140-IJMPB 03974
4000 T. Hayat, E. Momoniat & F. M. Mahomed
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)
September 18, 2008 10:12 WSPC/140-IJMPB 03974
Effects of an Endoscope and Fluid on Peristaltic Motion 4001
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)
September 18, 2008 10:12 WSPC/140-IJMPB 03974
4002 T. Hayat, E. Momoniat & F. M. Mahomed
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)
September 18, 2008 10:12 WSPC/140-IJMPB 03974
Effects of an Endoscope and Fluid on Peristaltic Motion 4003
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
September 18, 2008 10:12 WSPC/140-IJMPB 03974
4004 T. Hayat, E. Momoniat & F. M. Mahomed
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)
September 18, 2008 10:12 WSPC/140-IJMPB 03974
Effects of an Endoscope and Fluid on Peristaltic Motion 4005
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)
September 18, 2008 10:12 WSPC/140-IJMPB 03974
4006 T. Hayat, E. Momoniat & F. M. Mahomed
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)
September 18, 2008 10:12 WSPC/140-IJMPB 03974
Effects of an Endoscope and Fluid on Peristaltic Motion 4007
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
September 18, 2008 10:12 WSPC/140-IJMPB 03974
4008 T. Hayat, E. Momoniat & F. M. Mahomed
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.
September 18, 2008 10:12 WSPC/140-IJMPB 03974
Effects of an Endoscope and Fluid on Peristaltic Motion 4009
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.
September 18, 2008 10:12 WSPC/140-IJMPB 03974
4010 T. Hayat, E. Momoniat & F. M. Mahomed
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.
September 18, 2008 10:12 WSPC/140-IJMPB 03974
Effects of an Endoscope and Fluid on Peristaltic Motion 4011
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)
September 18, 2008 10:12 WSPC/140-IJMPB 03974
4012 T. Hayat, E. Momoniat & F. M. Mahomed
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.
September 18, 2008 10:12 WSPC/140-IJMPB 03974
Effects of an Endoscope and Fluid on Peristaltic Motion 4013
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)
September 18, 2008 10:12 WSPC/140-IJMPB 03974
4014 T. Hayat, E. Momoniat & F. M. Mahomed
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.
September 18, 2008 10:12 WSPC/140-IJMPB 03974
Effects of an Endoscope and Fluid on Peristaltic Motion 4015
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.
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