الفهرس 
IntroductionOnly 14 pages are availabe for public view
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Abstract
The results obtained from studying the two parts can be summarized as: The first
part in chapters (2, 3) the problem study analytically by using a perturbation technique
which satisfies the momentum equations for the amplitude ratio is small:
In Chapter (2), we studied the peristaltic transport of a particle-fluid suspension of
an incompressible fluid for small amplitude wave in a two-dimensional channel with
slip at the walls, which achieve a good simulation of motion of liquid through a
channel. We obtained the case of zero imposed pressure gradient (free pumping). The
critical value for the pressure gradient is obtained by equating the mean velocity with
zero at the center line of the channel. Pumping against a positive pressure gradient
greater than that critical value, there will be a backward (reflux flow) in the
neighborhood of the center line occurs. There will be no reflux flow if the pressure
gradient smaller than the critical value.
We found that, the perturbation function (y) increases with increasing particle
concentration C. The perturbation function (y) decreases with increasing α and R, but
the increment in (y) is large at lower values of wave number α and Reynolds number
R. Also we noticed that the presence of Knudsen number Kn results a decrease in (y)
for α ≤ 0.3, but the decrement in (y) is small, for α ≥ 0.4 the effect of Knudsen number
Kn vanishes. Also we noticed that at R < 3 the presence of Knudsen number Kn results
increase in (y), but for R ≥ 3 the effect of Kn vanishes.
The critical value for the pressure gradient ̅̅̅̅ decreases with
increasing particle concentration C, However, C, have significant influence over
only at higher values of α. We observed that the critical reflux
pressure is lower for particle-fluid suspension that for particlefree
fluid. That means, presence of particle in the fluid favors reversal flow. The effect
of presence of Knudsen number Kn on the critical value for the pressure gradient, for α
≤ 0.4, decreases with increasing Knudsen number Kn, but for α
> 0.4, increases with increasing Knudsen number Kn.
The mean-velocity distribution decreases with increasing the value of and
R. Also we notice that the value ofdecreases by increasing the
Knudsen number Kn. Further, we notice that at R < 3, the presence of Knudsen number
Kn results increases in the mean-velocity distribution, but at R=3, the effect of
Knudsen number Kn vanishes, but at R > 3 the presence of Knudsen number Kn results
decreases in the mean-velocity distribution and increase in the reversal flow.
The reversal flow increases with increasing particle concentration C and R.
Interpreted physiologically, this means under same conditions, urine in which solute
particles are suspended (i.e., urine from a diseased kidney) is more susceptible to
reversal flow in ureter, in comparison to pure urine without solute particles. Also under
the same conditions, urine in which solute particles are suspended with no-slip
condition is more susceptible to reversal flow in ureter, in comparison to with slip
condition. The reversal flow increases with decreasing wave number α but the
decrement in the reversal flow is very small. Also we notice that at α < 0.6, the
presence of Knudsen number Kn results an increase in the reversal flow, but at α=0.6,
the effect of Knudsen number Kn vanishes, but at α > 0.6, we observed that the
presence of Knudsen number Kn results a decrease in the reversal flow.
It is of interest to note that the result of the present model includes results of different
mathematical models such as:
1. The results of Srivastava and Srivastava [5] have been recovered by taking
compressibility number Kn=0.0 (no-slip).
2. The results of Fung and Yih [24] have been recovered by taking the fluid is
particles free, i.e., (C=0.0).
In Chapter (3), the purpose of this chapter is to study the effect of compressibility of
the liquid on the peristaltic pumping of a compressible Liquid with suspended particles
in a two-dimensional channel. There is no any attempt to study this problem.
Our results declared the dependency of the constant W, the perturbation function
Jz(y) and the net axial velocity on the compressibility number χ, the Reynolds
number R and the particle concentration C. We found that the compressibility number χ
has a significant influence on W, and Reynolds number R plays a more significant role
in the flow of a compressible liquid than of an incompressible one, the constant W
decreases with increasing χ for low values of Reynolds number R, Further, for higher
values of R, W attains a maximum for a certain value of χ. Also we notice that W
increases with increasing particle concentration C and α. Further, W is greatly affected
by C for higher values of α and lower value of R.
We observed that the perturbation function Jz(y) at lower value of R, increases with
increasing particle concentration C in the reversal flow. But at higher value of R and
lower value of χ, we observed that Jz(y) decreases with increasing C until C=0.2, then
Jz(y) increases with increasing, with increasing χ at higher value of R, we observed that
Jz(y) increases with increasing C in the reversal flow until C=0.1, then Jz(y) decreases
with increasing C in reversal flow until C=0.5, then Jz(y) increases with increasing C in
the forward flow. The perturbation function Jz(y), we observed that Jz(y) decreases with
increasing R in the reversal flow.
The net axial velocity, we observed that at lower value of R, increases with
increasing C, but with increasing R, increases with increasing C until C=0.4,
then for C > 0.4, decreases with increasing C. Also we observed that
decreases with increasing χ at lower value of R, but at higher value of R, we observed
that increases with increasing χ. Also, we observed that increases
with increasing R.
It is of interest to note that the result of the present model includes results of different
mathematical models such as:
3. The results of Srivastava and Srivastava [5] have been recovered by taking
compressibility number χ=0.0 (incompressible liquid).
4. The results of Fung and Yih [24] have been recovered by taking compressibility
number (χ=0.0) (incompressible liquid) and the fluid is particles free, i.e.,
(C=0.0).
The second part in chapters (4, 5 and 6) the numerical study by using a new
technique of GDQM: a new technique of differential quadrature method is introduced
to find numerical solution of nonlinear partial differential equations such as the Navier-
Stokes equations. The presence of the nonlinearity in the problem leads to severe
difficulties in the solution approximation. In construction of the numerical scheme of a
new technique, a GDQM is to use for derivatives with respect to space variables of
differential equations and for the time derivative applying 4th order RKM. The GDQM
changed the nonlinear partial differential equations into a system of nonlinear ordinary
differential equations (ODEs). The obtained system of ODEs is solved by 4th order
RKM. This combination of DQM and 4th order RKM gives a very good numerical
technique for solving time dependent problems.
In Chapter (4), in the present mathematical model, the unsteady pulsatile blood
flow through porous medium in the presence of magnetic field with periodic body
acceleration through a rigid straight circular tube (artery) has been studied. It is of
interest to note that the axial velocity increases with increasing of the permeability
parameter of porous medium k and Womersley parameter α whereas it decreases with
increasing the Hartmann number Ha, frequency of body acceleration b.
The present model gives a numerical solution of velocity distribution with pipe
radius and time. It is of interest to note that the result of the present model includes
results of different mathematical models such as:
- The results of [24], [25],
- the results of [23] have been recovered by taking the permeability of porous
medium (k →∞) without stochastic and no body acceleration (b=0.0),
- The results of [19] have been recovered by taking (no magnetic field) Hartmann
number (Ha=0.0),
- the results of [18] have been recovered by taking the permeability of porous
medium (k →∞) and (no magnetic field) Hartmann number (Ha=0.0).
In Chapter (5), in the present mathematical model, the unsteady pulsatile blood
flow through porous medium in the presence of magnetic field with periodic body
acceleration and slip condition through a rigid straight circular tube (artery) have been
studied. It is of interest to note that the axial velocity increases with increasing of the
permeability parameter of porous medium k, Womersley parameter α and Knudsen
number Kn whereas it decreases with increasing the Hartmann number Ha and
frequency of body acceleration b.
It is of interest to note that the result of the present model includes results of
different mathematical models such as:
- The results of Eldesoky [32] have been recovered,
- The results of Eldesoky et al. [39] and Megahed et al. [45], have been recovered
by taking Knudsen number (kn = 0.0) (no slip condition),
- The results of Kamel and El-Tawil [43] have been recovered by taking Knudsen
number (Kn=0) (no slip condition), the permeability of porous medium (k→ ∞)
without stochastic and no body acceleration (b=0.0),
- The results of El-Shahed [44] have been recovered by taking Knudsen number
(kn=0.0) (no slip condition) and Hartmann number (no magnetic field)
(Ha=0.0),
- The results of Chaturani and Palanisamy [42] have been recovered by taking
Knudsen number (Kn=0.0) (no slip condition), the permeability of porous
medium (k→∞) and (no magnetic field) Hartmann number (Ha=0.0).
In Chapter (6), in the present mathematical model, the unsteady pulsating flow of
an incompressible couple stress fluid between permeable beds through Porous Medium
in channel under the influence of periodic body acceleration in the presence of
magnetic field. The slip condition on the wall artery has been considered.
We found that the unsteady velocity increases with increasing of the Couple stress
parameter β and the permeability parameter of porous medium k whereas it decreases
with increasing the Reynolds number R and the Hartmann number Ha. Also the
velocity increases with increasing the Knudsen number Kn for the region to
[ ]. But for to
, the velocity decrease with increasing Kn.
It is observed that the presence of couple stresses results in a decrease in the
velocity. Also it is observed that when the couple stresses are present, the Reynolds
number seems to have no influence on the unsteady velocity component. This is in
contrast with the disturbance we see in the absence of couple stresses.
It is observed that when the couple stresses are present:
- As the Reynolds number R increases, the maximum velocity is attained nearer
to the upper permeable bed plate.
- As the Knudsen number Kn decreases, the maximum velocity is attained
nearing to the upper permeable bed plate slightly (slowly).
- As the Hartmann number Ha increases, the maximum velocity region increase.
- As the permeability parameter of porous medium k increases, the maximum
velocity region increase.