Internat. J. Math. & Math. Sci.
Vol. 24, No. 4 (2000) 217–230
S0161171200004737
© Hindawi Publishing Corp.
ELSAYED F. EL SHEHAWEY and WAHED EL SEBAEI
(Received1 March 2000)
Abstract.
The problem of peristaltic transport in a cylindrical tube through a porous medium has been investigated. A perturbation solution is obtained, which satisfies the momentum equation for the case in which the amplitude ratio is small. The results show that the fluidphase mean axial velocity increases with increasing the permeability parameter k . The phenomena of reflux is discussed. Numerical results are reported for various values of the physical parameter.
Keywords and phrases. Peristaltic transport, porous medium.
2000 Mathematics Subject Classification. Primary 76Z05, 76S05.
1. Introduction.
The wordperistaltic stems from the Greek wordperistaltikos, which means clasping andcompressing. It occurs due to the action of a progressive waves which propagates along the length of a distensible tube containing liquid. Peristaltic pumping has been the object of scientific andengineering researchs during the recent past few decades.
The pumping of fluids through muscular tubes by means of peristaltic waves is an important biological mechanism. In particular, peristaltic mechanism may be involved in swallowing food through the oesophagus, urine transport from kidney to bladder through the ureter.
Study of the mechanism of peristalsis from both the mechanical and physiological view points has been the object of scientific research. Since the first investigation of
Latham [5], several theoretical and experimental attempts have been made to under-
standperistaltic action in different situations. A review of much of the early literature
is presentedby Jaffrin andShapiro [4]. A summary of most of the experimentedand
theoretical investigations reported with details of the geometry, fluid, Reynolds number, wavelength parameter, wave amplitude parameter, and wave shape have been
given by Srivastava andSrivastava [13]. Srivastava andSrivastava [14] studiedthe ef-
fects of Poiseuille flow on peristaltic transport of a particulate suspension. Recently
Saxena andSrivastava [11] studiedthe particulate suspension flow inducedby sinu-
soidal peristaltic waves. In another type of studies on peristaltic transport, El Misery,
El Shehawey, andHakkem [1] studiedthe peristaltic motion of an incompressible generalizedNewtonian fluidin a planer channel. Mekheimer, El Shehawey, andElaw [6]
studiedthe peristaltic motion of a particle fluidsuspension in a planer channel. Flow through a porous medium has been of considerable interest in recent years particularly among geophysical fluiddynamicists. Many technical processes involve parallel

218 E. F. EL SHEHAWEY AND W. EL SEBAEI flow of fluids of different viscosity and density through porous media. Such parallel flows exist in packedbedreactors in the chemical industry, in petroleum production engineering, andin many other processes as well. Flows through porous medium occur in filtration of fluids and seepage of water in river beds. Movement of underground, water andoils, beach sandstandstone, limestone, rye bread, wood, the human lung, bile duct, gall bladder with stones, and small blood vessels are some important examples of flow through porous medium. Another example is the seepage under a dam
which is very important [10]. Several works have been publishedby using the generalizedDarcy’s law [12], where the convection acceleration andviscous-stress are taken into account [16].
El Sayed [2] studied the electrohydrodynamic instability of two superposed viscous and streaming fluids through porous medium. Varshney [15] studied the fluctuating
flow of a viscous fluidthrough a porous medium boundedby porous andhorizontal
surface. Raptis et al. [8, 9] studied the steady free convection and mass transfer flow
of a viscous fluid through a porous media bounded by a vertical surface. El Shehawey
et al. [3] studied the peristaltic transport through a porous medium. In the case of high
permeability parameter (as k → ∞
), our result is in agreement with Yih andFung [17].
2. Formulation of the problem.
We shall consider a two-dimensional circular cylindrical tube of radius R , with axisymmetric, moderate-amplitude traveling waves imposed on its wall is considered.
The fluidis assumedto be Newtonian, viscous, homogeneous, andincompressible through a porous medium occupying a semi-infinite region of the space. The equations governing two-dimensional motion of a viscous incompressible fluid through a porous
medium are (see [7])
∂v r
∂t
+ v r
∂v r
∂r
+ v z
∂v r
∂z
= −
1
ρ
∂P
∂r
+ ν
∂v z
∂t
+ v r
∂v
∂r z
+ v z
∂v
∂z z
= −
1
ρ
∂P
∂z
+ ν
∂ 2
∂r v r
2
+
∂ 2
∂z v
2 r
+
1 r
∂v r
∂r
∂ 2
∂r v
2 z
+
∂ 2
∂z v
2 z
+
1 r
∂v z
∂r
− v r r 2
−
ν k v z
,
−
ν k v r
, (2.1a)
(2.1b) andthe equation of continuity is
∂v r
∂r
+
∂v z
∂z
+ v r r
= 0 , (2.2) where z is the axial coordinate in the direction of wave propagation, r is the radial coordinate, ν is kinematic viscosity, ρ is the density, P is the pressure, t is the time, v r and v z are velocity components in the r - and z -directions, respectively, and k is the permeability parameter. Axisymmetric motion is assumed.
The velocity components can be written in terms of Stoke’s stream function Ψ as v r
=
1 r
∂ Ψ
∂z
, v z
= −
1 r
∂ Ψ
∂r
.
(2.3)

r
PERISTALTIC TRANSPORT IN A CYLINDRICAL TUBE ...
λ
R
Z
219 r = R + a cos ( 2 π/λ)(z − ct) a
Figure 2.1
.
Geometry of the problem.
Eliminating pressure P
between equations (2.1a) and(2.1b) using equation (2.3)
yields the equation for the stream function,
∂
∂t
∇ 2 Ψ +
1 r
∂ Ψ
∂z
∇ 2
∂ Ψ
∂r
−
2 r
∇ 2 Ψ + r
1
2
∂ Ψ
∂r
−
1 r
∂ Ψ
∂r
∇ 2
∂ Ψ
∂z
= ν ∇ 2 ∇ 2 Ψ −
ν k
∇ 2 Ψ ,
(2.4)
∇ 2 is a special operator defined as
∇ 2 =
∂ 2
∂z 2
+
∂ 2
∂r 2
−
1 r
∂
∂r
.
(2.5)
At the boundaries, the fluidis subjectedto the motion of the wall in the form (see
Figure 2.1),
η = a cos
2 π
λ
(z − ct), (2.6) where a is the amplitude, η is the radial displacement from the mean position of the wall, λ is the wavelength, and c is the wave speed. The boundary conditions that must be satisfiedby the fluidon the walls are the no-slip andimpermeability conditions.
The velocity components of the fluidparticles on the wall are thus v z
= 0 , v r
=
∂η
∂t
, (2.7) andin terms of the stream function Ψ ,
∂ Ψ
∂r
= 0 ,
∂ Ψ
∂z
=
2 πacr
λ sin
2 π
λ
(z − ct), we introduce the non-dimensional variables and parameters as follows: r ∗ = r
R
,
Ψ ∗ = z ∗ = z
R
,
Ψ
R 2 c
, t ∗ = v ct
R
,
= v r c
,
P ∗ = v p
ρc 2
,
= v z c
, k ∗ =
η ∗ =
R k
2
,
η
R
,
(2.8)
(2.9a)

220 E. F. EL SHEHAWEY AND W. EL SEBAEI andthe amplitude ratio , the wave number α , andthe Reynolds number Re. These parameters are defined as:
= a
R
, α =
2 πR
λ
, Re = cR
ν
.
(2.9b)
The non-dimensional forms of (2.4), (2.6), and (2.8), after dropping the star are:
∂
∂t
∇ 2 Ψ +
1 r
∂ Ψ
∂z
∇ 2
∂ Ψ
∂r
−
2 r
∇ 2 Ψ + r
1
2
∂ Ψ
∂r
−
1 r
∂ Ψ
∂r
∇ 2
∂ Ψ
∂z
=
1
Re
∇ 2 ∇ 2 −
1 k
Ψ , (2.10)
η = cos α(z − t), (2.11)
∂ Ψ
∂r
= 0 ,
∂ Ψ
∂z
= αr sin α(z − t), r = 1 + η.
(2.12)
3. Method of solution.
We expand Ψ and ∂P/∂z in a power series of the small parameter ,
∂P
∂z
=
Ψ = Ψ
0
+ Ψ
1
+ 2 Ψ
2
+··· ,
∂P
∂z
0
+
∂P
∂z
1
+ 2
∂P
∂z
2
+··· .
(3.1)
(3.2)
On substituting (3.1) into (2.10), collecting terms of equal powers of
, andequating the coefficients of like powers on both sides of the equation, we obtain
1
Re
∇ 2 ∇ 2 −
1 k
Ψ
0
∇ 2
∂ Ψ
0
∂r
−
2 r
∇ 2 Ψ
0
+ r
1
2
∂ Ψ
0
∂r
−
1 r
∂ Ψ
0
∂r
∇ 2
∂ Ψ
0
∂z
,
(3.3)
1
Re
∇ 2
=
∂
∂t
∇ 2 Ψ
0
+
1 r
∂ Ψ
0
∂z
∇ 2 −
1 k
Ψ
1
=
1
Re
∇ 2
∂
∂t
+
1 r
∇ 2 Ψ
1
+
∂ Ψ
∂z
0
∇ 2 −
1 k
Ψ
1 r
2
∂ Ψ
∂z
1
∇ 2
∂ Ψ
0
∂r
−
2 r
∇ 2 Ψ
0
+ r
1
2
∂ Ψ
0
∂r
∇ 2
∂ Ψ
1
∂r
−
2 r
∇ 2 Ψ
1
+ r
1
2
∂ Ψ
1
∂r
−
1 r
∂ Ψ
1
∂r
∇ 2
∂ Ψ
0
∂z
−
1 r
∂ Ψ
0
∂r
∇ 2
∂ Ψ
1
∂z
,
(3.4)
=
∂
∂t
∇ 2 Ψ
2
+
1 r
∂ Ψ
∂z
2
∇ 2
∂ Ψ
0
∂r
−
2 r
∇ 2 Ψ
0
+ r
1
2
∂ Ψ
0
∂r
+
1 r
−
1 r
∂ Ψ
∂z
1
∂ Ψ
2
∂r
∇
∇ 2
∂ Ψ
1
∂r
2
∂ Ψ
0
∂z
−
−
1 r
2 r
∇ 2 Ψ
1
+ r
1
2
∂ Ψ
1
∂r
∇ 2
∂ Ψ
1
∂z
−
∂ Ψ
1
∂r
1 r
+
1 r
∂ Ψ
∂z
0
∂ Ψ
0
∂r
∇ 2
∂ Ψ
2
∂z
.
∇ 2
∂ Ψ
2
∂r
−
2 r
∇ 2 Ψ
2
+ r
1
2
∂ Ψ
2
∂r
(3.5)

PERISTALTIC TRANSPORT IN A CYLINDRICAL TUBE ...
221
We expand the boundary conditions (2.12) in power series of
η :
Ψ z
Ψ r
( 1 ) + η Ψ r r
( 1 ) + η Ψ r z
( 1 ) +
( 1 ) +
η 2
2
Ψ r r r
( 1 ) +··· = 0 ,
η 2
2
Ψ zr r
( 1 ) +··· = α sin α(z − t).
(3.6)
(3.7)
On substituting (2.11) and(3.1) into (3.6) and(3.7) andequating coefficients like pow-
ers of on both sides of the equation we obtain
∂ Ψ
0
∂r
∂ Ψ
1
∂r
∂ Ψ
2
∂r
∂ Ψ
0
∂z
∂ Ψ
1
∂z
∂ Ψ
2
∂z
( 1 ) = 0 ,
( 1 ) +
( 1 ) +
( 1 ) = 0 ,
( 1 ) +
( 1 ) +
∂ 2 Ψ
0
∂
∂r
2
2
Ψ
1
∂r 2
( 1 ) cos α(z − t) = 0 ,
( 1 ) cos α(z − t) +
1
2
∂ 3
∂r
Ψ
3
0
( 1 ) cos 2 α(z − t) = 0 ,
∂ 2 Ψ
0
∂r ∂z
∂ 2 Ψ
1
∂r ∂z
( 1 ) cos α(z − t) = α sin α(z − t),
( 1 ) cos α(z − t) +
1
2
∂
∂r
3 Ψ
2
0
∂z
( 1 ) cos 2 α(z − t) = 0 .
(3.8)
(3.9)
(3.10)
(3.11)
(3.12)
(3.13)
Equations (3.3), (3.8), and(3.11), together with a condition of uniform pressure gra-
dient (∂P/∂z)
0
= constant, are satisfiedto yieldthe classical Poiseuille flow for a fluid through a porous medium
Ψ
0 r
= ¯ I
0
√
1 k r − I
0
√
1 k r r , (3.14) where
¯ =
I
0
(
Re
1 / k k)
∂P
∂z
0
, (3.15) is the Poiseuille flow parameter for a fluidthrough a porous medium, I
0 modified Bessel functions of the first kind.
and I
1 are the
Equation (3.4) with the corresponding boundary conditions (3.9) and (3.12) shows
that a solution can be chosen in the form
Ψ
1
= φ
1
(r )e iα(z − t) + φ ∗
1
(r )e − iα(z − t) , (3.16) where the asterisk denote a complex conjugate.
On substituting (3.16) into (3.4), (3.9), and(3.12), we obtain the Orr-Sommerfeld
equation with the corresponding boundary conditions as follows: d 2 dr 2
−
1 r d dr
− α 2 +
× d 2 dr 2
1 k
+ iα Re 1 +
−
1 r d dr
− α 2
¯ I
0
√
1 k
φ
1
+ iα Re ¯ r
− I
0
1 k
I
0
√ r k
√ r k r − √
2 k
I
1
√ r k
φ
1
= 0 ,
(3.17)

222
φ
E. F. EL SHEHAWEY AND W. EL SEBAEI
1
( 1 ) =
2 k
I
1
√
1 k
, φ
1
( 1 ) = −
1
2
, (3.18) where the primes denote differentiation with respect to r .
Equation (3.5) with the corresponding boundary conditions (3.10) and (3.13)
shows that a solution can be chosen in the form
Ψ
2
= φ
20
(r ) + φ
22
(r )e 2 iα(z − t) + φ ∗
22
(r )e − 2 iα(z − t) .
(3.19)
φ
On substituting (3.19) into (3.5), (3.10), and(3.13), we obtain the two equations for
20 and φ
22 with the corresponding boundary conditions as follows:
∇
2
=
∇
2
− iα Re
1 k r
φ
−
20
3 r
φ
1
φ ∗
1
+ φ
1
φ ∗
1
+ r
3
2
φ
1
− φ ∗
1
φ
1
−
∇
2
∇
2
−
1 k
= − 2 iα Re
φ
22
1 − i ¯ I
0 r
3
2
φ
√
1 k
1
φ
−
∗
1
I
+
0
φ
1
√ r
φ k
∗
1
φ
−
∗
1
φ
∇
2
+
∗
1
φ
3 r
φ
1
φ
1
22
,
φ ∗
1
+
+
2 iα iα r r
Re ¯
Re
−
φ
1
φ
1
1 k
I
0
√ r k
− φ
1
φ
1 r
−
+ √
2 k
I
1
3 r
φ
1
φ
1
√ r
+ k
1 r
φ
(φ
1
)
22
2 + r
3
2
φ
1
φ
1
+
2 r
α 2 (φ
1
) 2 ,
(3.20)
(3.21) with
∇
2
φ
20
φ
22
φ
22
= d 2 dr
( 1 ) +
( 1 ) +
( 1 ) +
2
2
1
2
1
1
−
φ
1 r
φ
1
1 d dr
, ∇
(
( 1 ) + φ
1
1 ) +
1
8
¯
2
(
−
=
1 ) dr
√
1 k d 2
+
I
1
2
1
4
−
¯
1 r
√
1 k d
− dr
√
1
−
− 4 α 2 , k
1 k
I
1
I
0
√
1
√
1 k k
4
φ
1
( 1 ) = 0 .
−
1 k
I
0
= 0 ,
√
1 k
= 0 ,
(3.22)
(3.23)
(3.24)
(3.25)
Thus, we obtained a set of differential equations together with the corresponding boundary conditions which are sufficient to determine the solution of the problem up to the secondorder in .
Now, our main purpose is to findout solutions of differential equations for φ
1
, al-
though equation (3.17) for
φ
1 is the fourth-order ordinary differential equation with variable coefficients. However, we can restrict our investigation to the case of pumping of an initially stagnant fluid, which is not subject to imposed pressure gradient.
Thus, in this case (∂P/∂z)
0
= 0, which means constant ¯ vanishes andwe wouldbe able to obtain a simple closedform analytical solution of this interesting case of free pumping.

PERISTALTIC TRANSPORT IN A CYLINDRICAL TUBE ...
φ
1
(r ) = c
1 r I
1
(βr ) + c
3 r I
1
(αr ),
223
4. Free pumping (originally stationary fluid).
Let us consider the case in which the pressure gradient (∂P/∂z)
0 vanishes. In this case, there will be no flow if the wall motion stops. Hence ¯ =
0, then (3.17) becomes
d 2 dr 2
−
1 r d dr
− β 2 d 2 dr 2
−
1 r d dr
− α 2 φ
1
= 0 , (4.1) in which β 2 = α 2 − iα Re − 1 /k
and the boundary conditions (3.18) with ¯
= 0 becomes
φ
1
( 1 ) = 0 , φ
1
( 1 ) = −
1
2
.
(4.2)
The boundary condition (4.2), together with the condition that the velocity must
remain finite at r = 0, then leadto the solution
(4.3) where c
1
= c
3
=
2 (αI
0
(α)I
1
− αI
0
(β) −
(α)
βI
0
β)I
1
(α)
2 αI
0
− βI
0
(β)
(α)I
1
(β) − βI
0
(β)I
1
(α)
,
.
(4.4a)
(4.4b)
On substituting (4.3) andits conjugate into (3.20) and(3.23), we obtain
∇
2
∇
2
−
= − α 2
1 k
Re 2
φ
20
2 c
1 c ∗
1
+ c
1 c ∗
3 r β ∗ I
1
(βr )I
0 r βI
0
(βr )I
1
β ∗ r + r βI
0
(βr )I
1
β ∗ r − 2 I
1
(βr )I
1
β
(αr ) + r αI
1
(βr )I
0
(αr ) − 2 I
1
(βr )I
1
(αr )
∗ r
+ c c
3 r β ∗ I
1
(αr )I
0
β ∗ r + r αI
0
(αr )I
1
β ∗ r − 2 I
1
β ∗ r I
1
(αr ) ,
(4.5)
φ
20
( 1 ) = −
1
2 c
1
β 2 I
1
(β) + c
3
+ c ∗
3
+ c
1
βI
0
(β) + c ∗
1
β ∗ I
0
α 2 I
β ∗
1
(α) + c ∗
1
β
+ c
3
+ c ∗
3
∗ 2
I
1
β ∗
αI
0
(α) = D.
(4.6)
The right-handside of (4.5) is a complicatedfunction of
r . We evaluate the righthandside numerically andrepresent the result approximately by a polynomial of the following form:
∇
2
∇
2
−
1 k
φ
20
= − α 2 Re 2
S i = 1
B i r 2 i , (4.7) for different values of α andRe different B i
must be used. Solving (4.7), we obtain
φ
20
= L
1
+ √ i k
L
2
I
0
√ r k
− α 2 Re 2
S i = 1
B i r 2 i + 1
2 i( 2 i + 2 ) 2
, where L
1 and L
2 are constants.
(4.8)

224 E. F. EL SHEHAWEY AND W. EL SEBAEI
Defining the following function:
G(r ) =
S i = 1
B i r 2 i + 1
2 i( 2 i + 2 ) 2
,
we can write the boundary condition (4.6) in the form
φ
20
( 1 ) = D = L
1
+ √ i k
L
2
I
0
√
1 k
on substituting (4.9) and(4.10) into (4.8), we get
− α 2 Re
2
G( 1 ),
(4.9)
(4.10)
φ
20
(r ) = α 2 Re 2 G( 1 ) + D r + √ i k
L
2
I
0
√ r k
− I
0
√
1 k r − α 2 Re 2 G(r ), (4.11)
if each term of (3.1), (3.2), (3.16), (3.19), and(4.3) is time-averagedover one period, we
obtain the mean pressure gradient
∂z
2
=
1 r Re
+ iα r 2
−
φ
1
φ
20
φ ∗
1
+
1 r
φ
20
−
− φ ∗
1
φ
1
− r
1
2
1 r
−
φ
1 k
1
φ ∗
1
φ
20
− φ ∗
1
φ
1
+ O 3
substitution of (4.11) into (4.12) yields:
∂z
2
= − α 2 Re −
G r
(r )
+ r
1
2
G (r ) − r
1
3
G(r ) +
1 kr
G(r )
− 2 c
1 c ∗
1
I
1
(βr )I
1
− c ∗
1 c
3
I
1
(αr )I
1
β
β ∗ r − c
1 c ∗
3
I
1
(βr )I
1
(αr )
∗ r +
H
Re
= Z +
H
Re
.
,
(4.12)
(4.13)
The time-averagedpressure gradient is not constant across the tube, but has a perturbation which varies with the radius. This perturbation function, which is denoted by Z
, is the expression enclosedin the brackets in (4.13).
The solution for the mean axial velocity (averagedover time) is v z
= −
= 2
1 r
2 φ
20
− α 2 Re 2 G( 1 ) −
G(r ) r
− D + k Re
∂P
∂z
2
I
0
1 /
√
2 I
0 k
1 /
− I
0 r / k − 1
√ k
.
(4.14) v z r =
0 (i.e., at the centre of the tube) using (4.13), this condition is
= 0 at
∂z
2 c.r
=
H
Re
, (4.15) where
H =
[ 2 I
0 k[I
0
( 1 /
( 1 /
√
√ k) − 1 ] k) − 1 ]
α 2 Re 2 G( 1 ) + D .
(4.16)

PERISTALTIC TRANSPORT IN A CYLINDRICAL TUBE ...
225
5. Stokes approximation.
As Re → 0 (the Reynolds number is considered so small that the inertia terms in the governing equations are negligible).
Equation (4.8) becomes
φ
20
(r ) = L
1
+ √ i k
L
2
I
0
√ r k
,
we can write the boundary condition equation (4.6) in the form
(5.1)
φ
20
( 1 ) = D = L
1
+ √ i k
L
2
I
0
on substituting (5.2) into (5.1) we get
√
1 k
, (5.2)
φ
20
(r ) = D + √ i k
L
2
I
0
√ r k
− I
0
√
1 k the solution for the mean-axial velocity is r , v z
= −
1 r
2 φ
20
(r ) = − 2 D + √ i k
L
2
I
0
√ r k
− I
0
√
1 k
,
(5.3)
(5.4) where
L
2
=
I
0
(
− i
1 /
√
√ kD k) − 1
.
(5.5)
6. Numerical results and discussion.
In order to study the behavior of the solution, numerical calculations for several values of α,k , andRe were carriedout. Equa-
tion (4.14) shows that the mean axial velocity is dominated by the
D term andthe parabolic term k Re
∂P
∂z
2
I
0
( 1 /
√
2 I
0 k)
( 1 /
− I
0 k)
(r /
− 1
√ k)
.
(6.1)
The constant D , which initially a rose from the non-slip conditions of the axial velocity on the wall andis relatedto the mean-velocity at the boundary of the tube (at r = 1) by ¯ z
= 2 D . The parabolic term is due to the time-averagedsecondorder pressure gradient set up by the peristaltic motion. In addition to the two terms mentioned above, the velocity has a perturbation term which is a function of r , given by
G( 1 ) −
G(r ) r
.
(6.2)
The variation of D with α for various values of Re and k
is depicted in Figures 6.1,
6.2, and6.3. The numerical results show that
D decreases with increasing Re and increases with increasing k .
The effects of k and (∂P/∂z)
2 on the mean-velocity distribution and reversal flow are
displayed in Figures 6.4–6.11. The results reveal that the mean-velocity distribution
increases with increasing the permeability parameter k and (∂P/∂z)
2
. We notice that

226 E. F. EL SHEHAWEY AND W. EL SEBAEI
3 .
7
3 .
8
3 .
9
D − 4
4
4
.
.
1
2
4 .
3
0 k = 0 .
5 k = 0 .
7 k = 1 .
0 k = 3 .
0
0 .
2 0 .
4 0 .
6 0 .
8 1
α
Figure 6.1
.
Variation of D with wave number α andpermeability parameter k at Re = 1 .
0.
3 .
8
− 4
D 4
4
.
.
2
4 k = 0 .
5 k = 0 .
7 k = 1 .
0 k = 3 .
0
4 .
6
0 0 .
2 0 .
4 0 .
6 0 .
8 1
α
Figure 6.2
.
Variation of D with wave number α andpermeability parameter k at Re = 10 .
0.
− 3 .
7
− 3 .
8
− 3 .
9
D
− 4
− 4 .
1
− 4 .
2
− 4 .
3
Re = 1 .
00
Re = 3 .
00
Re = 5 .
00
Re = 10 .
0
0 0 .
2 0 .
4 0 .
6 0 .
8 1
α
Figure 6.3
.
Variation of D with wave number α andReynolds number Re at k = 0 .
5.

r
0 .
1
PERISTALTIC TRANSPORT IN A CYLINDRICAL TUBE ...
5
0 k = 1 .
0 k = 1 .
5 k = 2 .
0 k = 3 .
0 v z
/ 2
− 0 .
5
− 1
− 1 0 1 2 3 4
Figure 6.4
.
The effect of the permeability parameter k on the mean-velocity distribution and reversal flow at Re = 5 .
0, (∂P/∂z)
2
= − 5, and α = 0 .
1.
1 r
0 .
5
0 k = 1 .
0 k = 1 .
5 k = 2 .
0 k = 3 .
0 v z
/ 2
227
− 0 .
5
− 1
− 6 − 4 − 2 0 2 4
Figure 6.5
.
The effect of the permeability parameter k on the mean-velocity distribution and reversal flow at Re = 10 .
0, (∂P/∂z)
2
= − 5, and α = 0 .
1.
1 r
0 .
5
0
− 0 .
5
− 1
4 k = 1 .
0 k = 1 .
5 k = 2 .
0 k = 3 .
0
4 .
5 v z
/ 2
5 5 .
5 6
Figure 6.6
.
The effect of the permeability parameter k on the mean-velocity distribution and reversal flow at Re = 5 .
0, (∂P/∂z)
2
= 2 .
0, and α = 0 .
1.

228 E. F. EL SHEHAWEY AND W. EL SEBAEI
1 r
0 .
5
0 k = 1 .
0 k = 1 .
5 k = 2 .
0 k = 3 .
0 v z
/ 2
− 0 .
5
− 1
4 5 6 7 8
Figure 6.7
.
The effect of the permeability parameter k on the mean-velocity distribution and reversal flow at Re = 10 .
0, (∂P/∂z)
2
= 2 .
0, and α = 0 .
1.
1
∂P
∂z 2
∂P
∂z 2
= − 2 .
0
= 2 .
0
0 .
5 r
0 v z
/ 2
− 0 .
5
− 1
3 .
4 3 .
6 3 .
8 4 4 .
2 4 .
4
∂P
∂z 2
∂P
∂z 2
= 4 .
0
= 8 .
0
4 .
6
Figure 6.8
.
The effect of the mean second-order pressure gradient (∂P/∂z)
2 on the mean-velocity distribution and reversal flow at Re = 1 .
0, k = 0 .
5, and
α = 0 .
1.
1
∂P
∂z 2
∂P
∂z 2
= 5 .
00
= 10 .
0
0 .
5 r
0 v z
/ 2
− 0 .
5
− 1
∂P
∂z 2
∂P
∂z 2
= 15 .
0
= 20 .
0
6 4 4 .
5 5 5 .
5
Figure 6.9
.
The effect of the mean second-order pressure gradient (∂P/∂z)
2 on the mean-velocity distribution and reversal flow at Re = 1 .
0, k = 0 .
5, and
α = 0 .
1.

1
0 .
5 r 0
− 0 .
5
− 1
0
PERISTALTIC TRANSPORT IN A CYLINDRICAL TUBE ...
1
0 .
5 v z
/ 2
0
− 0 .
5
− 1
1 2
(a) k = 0 .
5 .
3 0 1 2
(b) k = 0 .
7 .
1
0 .
5
3
1
0 .
5
0
− 0 .
5
0
− 0 .
5 r
− 1 − 1
− 1 0 1 2 3 − 2 − 1 0 1 2 3
(c) k = 1 .
0 .
(d) k = 3 .
0 .
Figure 6.10
.
The effect of the permeability parameter k on the meanvelocity distribution and reversal flow at Re
α = 0 .
1.
= 1 .
0, (∂P/∂z)
2
= − 10 .
0, and
1
0 .
5
0
0 .
1
5 v z
/ 2 0
− 0 .
5 − 0 .
5
− 1
0 .
1
5
3 .
8 4 4
(a)
.
2 4 k = 0
.
.
4 4
5 .
.
6 4 .
8 5
4
− 1
3 .
75 4 4 .
25 4 .
5 4 .
75 5 5 .
25
(b) k = 0 .
7 .
1
0 .
5
229
0
− 0 .
5
− 1
0
− 0 .
5
4 4 .
25 4 .
5 4 .
75 5 5 .
25 5 .
5
− 1
4 4 .
5 5 5 .
5
(c) k = 1 .
0 .
(d) k = 3 .
0 .
Figure 6.11
.
The effect of the permeability parameter k on the meanvelocity distribution and reversal flow at Re
α = 0 .
1.
= 1 .
0, (∂P/∂z)
2
= 10 .
0, and
6

230 E. F. EL SHEHAWEY AND W. EL SEBAEI as k increases the velocity increases forwardfor (∂P/∂z)
2 the velocity increases backward(increasing the flux flow).
> 0, while for (∂P/∂z)
2
< 0
For (∂P/∂z)
2
< (∂P/∂z)
2 c.r
there is no reflux andif (∂P/∂z)
2
> (∂P/∂z)
2 c.r
will be reflux andbackwardflow in the neighborhoodof the centre line occurs.
, there
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Elsayed F. El Shehawey and Wahed El Sebaei: Department of Mathematics, Faculty of Education, Ain Shams University, Heliopolis, Cairo, Egypt