INTRODUCTION
Peristalsis is a mechanism of pumping fluids in a tube by means
of a contractile ring around the tube which pushes the fluid forward.
By peristaltic pumping we mean transporting the fluid by a wave of contraction
or expansion from a region of lower pressure to higher pressure.
The problem of peristaltic flow in uniform and non uniform channel has
been studied by many authors for both Newtonian and non Newtonian fluids.
Some of these studies have been done by Abd El Naby et al. (2004),
Elshehawey and Husseny (2002), Mekheimer (2005), Mishra and Ramachandra
(2003), Misra and Pandey (2001) and Elshehawey and Sobh (2001).
Recently, some studies have been done to understand the influence of
an inserted endoscope on peristaltic motion of some types of fluids. Some
of these studies were made by Abd El Naby and El Misery (2002) and El
Misery et al. (2003).
The effect of the porous medium on peristaltic transport of a Generalized
Newtonian fluid in a uniform planar channel has been studied by Elshehawey
et al. (2000). Also, Sobh (2004) studied the peristaltic transport
of a magnetoNewtonian fluid through a porous medium in uniform channel.
It has been shown that the pressure rise increases as the permeability
decreases.
Since peristalsis is now well known to physiologists to be one of the
major mechanisms for fluid transport in many biological systems and most
biological organs are generally observed to be nonuniform, we purpose
to study the effect of porous medium on peristaltic motion of a Newtonian
fluid in a nonuniform channel.
Because of the complexity of the governing equations, we shall consider
the case of creeping flow. The problem is formulated and a perturbation
solution with wave number as a parameter is obtained to second order.
The velocity field and pressure gradient are obtained in explicit forms.
Also, the pressure rise per unit wavelength is computed numerically and
is plotted with the variation of the time. Moreover, the average pressure
rise is graphed versus flow rate for various values of permeability parameter
k.
FORMULATION AND ANALYSIS
Consider the two dimensional flow of an incompressible Newtonian
fluid in an infinite channel of nonuniform thickness with a sinusoidal
wave traveling down its wall. The geometry of the wall surface is defined
as:
With
where,
is the half width of the channel at any axial distance from the inlet,
a is the half width of the channel at the inlet, K_{1}<<1
is constant whose magnitude depends on the length of the channel and exit
and inlet dimensions, b the is the amplitude of the wave, λ is the
wavelength, c is propagation velocity of the wave, t is the time.
In the moving coordinates
which travel in the direction
with the same speed as the wave, the flow is steady but in the fixed coordinates,
the flow in the channel can be treated as unsteady. The coordinate frames
are related through (Shapiro et al., 1969).
where, and
are the velocity components in the fixed and the moving frames, respectively.
Using the following nondimensional parameters
and
where, δ is the wave number, Re is the Reynolds number,and φ
is the amplitude ratio, φ = b/a<1, the nondimensional equations
of motion are:
Considering the creeping flow (Re = 0), taking into account that δ
is small and eliminating the pressure from Eq. 7 and
8, we obtain the following system of differential equations:
with the continuity Eq. 6.
The nomdimensional boundary conditions are:
RATE OF VOLUME FLOW
The instantaneous volume flow rate in the stationary frame is given
by:
where,
is a function of
and .
On substituting Eq. 3 and 4 into
Eq. 12 and then integrating, one finds:
Where:
is the rate of volume flow in the moving frame (wave frame) and independent
of time. Here
is a function of
only.
The timemean flow over a period T = λ/c at a fixed position is
defined as:
Substituting from Eq. 13 into Eq. 15
and using Eq. 1 and integrating, we obtain
Defining the dimensionless timemean flows Θ and F as follows:
Equation 16 can be rewritten as:
Where:
METHOD OF SOLUTION
We expand the following quantities in a power series of the small
parameter δ as follows:
The use of expansions (20) with Eq. 6, 9,
10 and 11 gives the following systems:
System of order zero
With the dimensionless boundary conditions
The solution of this system for U_{0} subject to the boundary
conditions is:
The instantaneous volume flow rate F_{0} in the moving coordinates
is given by:
which implies that
Using Eq. 21, 25 and 26,
we obtain the alternative form of U_{0} and V_{0} as:
Where:
and
System of order one: Equating the coefficients of δ on both
sides in Eq. 6, 9, 10
and 11, we get
with the boundary conditions
Solving this system for U_{1}, after using Eq.
27, we obtain
Again, the alternative form for U_{1}, in which ∂p_{1}/∂X
is replaced by an equivalent expression in term of F_{1}, is given
by:
Where:
and
Using the continuity equation, one finds
System of order two: Equating the coefficients of δ^{2}
on both sides in Eq. 6, 9, 10
and 11, we get
with the boundary conditions
Solving this system for U_{2}, after using Eq.
27, we obtain
The instantaneous volume flow rate F_{2} is given by:
Solving the above equation for ∂p_{2}/∂X, we get
Where:
Substituting Eq. 45 into Eq. 44 we
obtain the alternative form of U_{2} as:
Where:
Now, the axial velocity component U and the pressure gradient can be
expressed, to second order where F_{0} = Fδ F_{1}δ^{2}
F_{2}, as:
The pressure rise per wave length Δp_{λ}(t) is given
by:
RESULTS AND DISCUSSION
It is clear that we have obtained analytical form, to second order
of δ, for the velocity field and the pressure gradient, Eq.
49 and 50. As k tends to infinity, we obtainthe
velocity field and the pressure gradient of peristaltic flow of Newtonian
fluid in nonuniform channel.

Fig. 1: 
The pressure rise versus the time
at δ = 0.02, φ = 0.8 and Θ = 0 

Fig. 2: 
The pressure rise versus the time at δ = 0.02,
φ = 0.8 and Θ = 0.1 
We shall now compute the dimensionless pressure rise Δp_{λ}(t)
over the channel length for different values of the dimensionless flow
rate Θ and permeability parameter k.
The average pressure rise is then evaluated by averaging Δp_{λ}(t)
over one period of the wave. As the integral in Eq. 51
is not integrable in the closed form, it is evaluated numerically using
the MATHEMATICA package. Following Srivastava and Srivastava (1984), we
take K_{1} = 3a/λ and then plot Eq. 51.
Figure 13 represent the variation of dimensionless
pressure rise with the dimensionless time for δ = 0.02, φ = 0.8, (k
= 0.05, 0.1, 1) and Θ = 0, 0.1 and 0.2, respectively. We have observed
that the pressure rise decreases with increasing flow rate and becomes maximal
at zero flow rate. Also, it is clear from these figures that the pressure rise
decreases with increasing permeability parameter k. Also, the behavior of the
pressure rise is the same for various values of flow rate Θ.

Fig. 3: 
The pressure rise versus the time at δ = 0.02,
φ = 0.8 and Θ = 0.2 

Fig. 4: 
The average pressure rise versus flow rate at δ
= 0.02 and φ = 0.8 
The average pressure rise versus the flow rate is plotted in Fig.
4 for δ = 0.02, φ = 0.8, (k = 0.05, 0.1, 1). As shown, the
average pressure rise decreases as the permeability parameter increases.
This is because of the resistance caused by the porous medium.