SlideShare uma empresa Scribd logo
1 de 7
Baixar para ler offline
Mathematical Theory and Modeling
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.11, 2013

www.iiste.org

Oscillatory Flow and Particle Suspension in a Fluid Through an
Elastic Tube
P. N. Habu* M. A. Mbah
Department of Mathematics, Federal University Lafia, P. M. B 146, Lafia, Nigeria.
* E-mail: peterhabu68@gmail.com
Abstract
Womersley gave a solution for the case of a thin-walled elastic tube, it being assumed that the effect of the
inertia term in the equations of viscous fluid motion can be neglected. He did not consider the presence of
particles, to account for the blood cells in the blood, within the viscous flow through the tube (artery).
In this paper, the corresponding solution for an oscillatory flow and particle suspension in a fluid (blood), to
account for blood cells, through an elastic tube is obtained. This solution is the frequency equation as it was
obtained by Womersley but it has a different structure. If the volume fraction particle density , is removed from
this solution it collapses to give the same equation as Womersley’s case, without particles.
Keywords: Oscillatory flow, Particle suspension, Elastic tube, Periodic function
1. Introduction:
The problem of blood flow and wave propagation in the arterial system has stimulated the interest of
physiologists and mathematicians for years. Its fundamental importance for present day research can be traced to
Witzig and later to the works of Womersley [10]. The contributions of Womersley [10] represent the best
attempt to date in developing a complete, practical, unified theory of arterial blood flow and pressure
propagation [6].
A survey of the published literature on the propagation of waves in the arterial system will show a large variation
of assumptions on the nature of flow conditions, fluid properties, and types of vessel walls. The analytical
solutions for pulsatile flow in rigid tubes are therefore based on a combination of different assumptions, just the
same as those of oscillatory flow through elastic tube.
The model used in this paper consists of three phases as against two phases by Womersley. The three phases are
the fluid phase, particulate phase and the equations of motion of the tube.
According to Womersley, the simple solution for the oscillatory motion of viscous liquid in a rigid tube, under a
simple-harmonic pressure gradient, was given by Lambossay who gave the formula for velocity and viscous drag
solely concerned with the effect of the viscous drag on the frequency response of pressure recording instruments.
Womersley obtained the same result independently, in a different form and derived the expression for the rate of
flow.
For the equations of the elastic tube, we adopt Oslen J.W et al [7] and Womersley [10]. All the assumptions by
Womersley are adopted and applied in this paper alongside his method of solution. The only variation between
Womersley and this approach is the introduction of the particulate phase, with , as the volume fraction particle
density, so that we now simulate blood properly, with proper consideration to the existence of blood cells, as the
particles.
2. Methods
2.1 Assumptions and Nomenclature
( , ) denote fluid phase velocities, ( , ) denote particulate phase velocities,
and,
are the actual
is the fluid phase
densities of the materials constituting fluid and particulate phase respectively, (1 − )
density,
the particulate phase density, P denotes the pressure, denotes the volume fraction density of the
particles,
is the particle fluid mixture viscosity and S is the drag coefficient of interaction for the force
exerted by one phase on the other. Inertia terms of the equations of motion are neglected, diffusivity terms are
also neglected. is chosen as constant, and we assume that the pressure wave is harmonic in time having
frequency n and wave velocity c.
2.2 Formulation of the Model
A porous tapered elastic tube filled with a viscous fluid is considered as a model of a vascular bed in which
successive branching of the blood cells leads to a rapid decrease in diameter with distance. The porosity of the
tapered tube is adjusted to simulate the effect of branching
The motion of the fluid is assumed to be cycisymetric and governed by the Navier-Stokes equations for an
incompressible fluid, now modified to simulate the effect of particle suspension as follows:
(1− )[

+

+

] = −(1− )

+(1− ) [
109

+

+

]+

(

−

)

(1)
Mathematical Theory and Modeling
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.11, 2013

(1− )[
+
] = −(1− )
With equation of continuity as,

www.iiste.org

+(1− ) [

+

−

+

]+

(

−

)

(2)

[(1− ) ]+ [(1− ) ]+
= 0,
(3)
where
!
are the axial and radial components of velocity, P is the pressure and and are the density
and viscosity of the fluid respectively.
The problem of determining the motion of a liquid in an elastic tube when subjected to a pressure – gradient
which is a periodic function of the time arises in connection with the flow of blood in the larger arteries. The
equations of motion of the tube from [6] are:
(

"

+

=

=

#$ "

#

%&

−

,

%#

[

'

.

-

+

* "

#

&

& /

+

+

&

]'( +
0

)

)

#

[

"

+

* +
&

]

(4)
(5)

Together with the boundary conditions for the motion of the fluid,
+
=
12=1

=
12=1
The particulate phase: The equations of motion of the particles are:
"

[

+

3

3

+

3

]=−

[
+
+
]=−
and their equation of continuity is:
[
]+ [
]+ 3 = 0
With boundary conditions
+
=
12=1
3

3

+

(

+

(

−

−

)

(7)

)

(8)
(9)

12=1

"

=

3

(6)

(10)
A

3. Solution: The expression for the drag coefficient for the present study is selected as,
=

4 6$ 9
8 (:)
57

, where 89 (:) =

;<=>?; =

[5 = ]

@

+3

Where C is the fluid viscosity, and ‘a’ is the radius of the particle. Relation where 89 (:) represents the classical
Stokes’ drag for small particle Reynolds number, modified to account for the finite particulate fractional volume
through the function 89 (:), obtained by [9].
In order to investigate the Pulsatile flow along the axis of the tube, we assume that the pressure wave is harmonic
in time having frequency n and wave velocity c. We therefore assume that,
(F), 5 (F), 5 (F)] exp [in (t− )]
(D, , , , , )= [E (F), (F),
(11)
G
We shall be concerned with the motion in which
H&
,
, 3 , 3 , are small, i.e when the wave velocity, c is very large, as compared to , , ,
, and nR.
G
G
G
G
G
From (11) we can write
P = E (r)exp[in(t− )]
I =

J=

I =

J =

=

G

G

(r)exp[in(t− )]

=

(12)

5 (r)exp[in(t− G )]

5 (r)exp[in(t− G )]

K
=0
KL
= −
=

G

(r)exp[in(t− )]

A

G

in exp[in(t− )]
A

G

G

exp[in(t− )]
A

G

exp[in(t− )]

(13)

G

exp[in(t− )]

110
Mathematical Theory and Modeling
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.11, 2013

=

A

=

A NH

G

=

=

G

exp[in(t− )]

M exp[in(t− )]

=−
=

www.iiste.org

A

G

A

G

G

exp[in(t− )]
G

(13 cont.)

exp[in(t− )]
A

G

exp[in(t− )]
G

exp[in(t− )]

=−
in exp[in(t− )]
G
G
Neglecting the inertia terms in (1), we can write equation (1) as
= −(1− )

(1− )

+(1− ) [

+

+

]+

(

−

)

(14)

Now substituting equations (12) and (13) into equation (14) and neglecting the diffusivity terms
Equation (14) simplifies to
'

A

+

'

'

+

'

'

A

−

=−

NH&

Let P = Q( ) R5 ⇒ P 5 =
"
"
Substituting (16) in (15), we get
'

A

H

A

− MP 5

=−

−

& ,A NH

6G
& H

"

& ,A NH
6G

−

&

O

6(
&

6(

O

)

)

[

[

−

]

−

]

5

5

(15)
(16)
(17)

(1− )
=−(1− ) +(1− ) [
+
− +
Now substituting equations (12) and (13) into (18), we get

]+

Similarly in equation (2), neglecting inertia terms, can be written as
M =−

Let 2 =
⇒

A

'

&

+

,A

#

+

"

A

+

"

A

−P

+

# (

O

)

[

5

−

]

T

−

U

⇒ F = 2Q, therefore equation (18) can be written as
A

'

− MP 5

'
& H

−

'

A

=

& ,A
6

'

−

&

6(

V

)

[

5

we have,

−

]

(18)
(19)

(20)

Where P 5 =
"
For equations (19) and (20), the conservation of mass equation can be written using (12) and (13) and
simplifying, we get
( A ')
NH& A
=
(21)
'

G

'

Equations (19) and (20) are now the equations of motion of the fluid and equation (21) is the conservation of
mass equation.
3.1 Evaluating the particulate phase
Now neglecting the inertia terms in equation (7), we can write equation (7) as

[ 3] = −
+ + ( − )
Now substituting (12) and (13) into (22), we have
V A
A
=
−
&[#3 NH<V]

#3 NH<V

(22)

'

(23)

=−
+ ( − )
Substituting equations (12) and (13) into (24), we get
V A
A NH
=
+

Also equation (8) can similarly be written as
3

G[#3 NH<V]

(24)

#3 NH<V

(25)

The equation of conservation of mass of the particles can be written using (12) and (13) as
( ')
NH&
=
'

'

G

(26)

We can now re-arrange the equations of motion of the particles to be written from (23), (25) and (26) to have:
V A
A
=
−
#3 NH<V
V A

=

&>#3 NH<V@
A NH

G[#3 NH<V]

+

#3 NH<V

'

(27)

with the equation of continuity as
111
Mathematical Theory and Modeling
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.11, 2013
(

'

')

'

=

www.iiste.org

NH&

G

Now to solve equations (19) and (20) [i.e. equations of motion of the fluid]
From equation (23), equation (19) can be written as
+

A

'

+

A

'

'

'

'

−

'

−

+ WM = P 5 +

A

+ M =P 5Y5

6(

&

=

Equation (28), can be written as
A

'

A

'

A

where Y 5 = 1 +
+

−

'

−

'

&

N Z "A (

O

&

6&

)T#3 NH<VU

+ M =[5

=

Equation (29) can be written as
A

'

A

'

'

A

Where [ = P Y

O

)T#3 NH<VU

& 
6&

!] =1+(

[1 +

−

−

6(
& V

N Z "A (

A

'

& V

6(

)

V

X

)

]

)(#3 NH<V)

=

&

6&

A

[1 +

6(

V

)

−

#3 NH<V

]

A

'

(28)

'

(29)

(30)

V

)(#3 NH<V)

Using Bessel function, the solution of equation (30), expressing E = ^ _C (`2), where k is to be determined.
Then A = ^ a_ (a2), therefore equation (29) can be written as
+

A

'

=

'

A

'

'

+ M =[5

A

−

bA c$ (N Z d ')

=

&

6&

]^ a_ (a2)

With the solution

complementary and particular solutions respectively, giving,
=

c$ (N Z d )

b c$ (N Z d ')

−

& efA cA (e')

& 6 NZd

=

G

+

, where

G

!

e

are the
(31)

Similarly equation (20) has the solution,
c$ (N Z d )

NH& fA c$ (e')
NZd

e

(32)

Now using the approximation, Womersley [10], to equations (31) and (32), where from the equation of
continuity of the fluid we get the identities,
GA
NH&
=
G

d N G
N H & Z gfA

=

G 6
NH&gA⁄

and

&e fA
6

⇒a=
, so that _C (a2) becomes _C i
G
Also, we use the approximation
= _C i

H&gA⁄
G

H&g A⁄
G

NH&gA⁄

j=1

G

2j = 1 and _ (a2) becomes M_ i

H&gA⁄
G

2j =

NH&'gA⁄
5G

And
=_ i
j.
5G
G
Inserting these approximations in equations (31) and (32) we get
H&'g A⁄

=
=

−

bA cA (N Z d ')
c$ (N Z d )
b c$ (N Z d ')
c$ (N Z d )

−

H&g A⁄

& efA cA (e')

& 6 NZd
N& HgfA
6GN Z d
H&'g A⁄

Since _C (a2) = _C i
=

b c$ (N Z d ')
c$ (N Z d )

+

G

gfA

# G-

e

. On further simplification we get

j = 1 and M = [ 5 − a 5 = M = [ 5 similarly,

, after simplification

At the inner surface of the tube, i.e. when y=1, equation (33) and (34) become respectively,
NH&
NH& gfA
=
k5 l C ([) +
5 G

5 G # G-

Where, l C ([) =
And

=

5cA (N Z d )
d N Z c$ (N Z d )
gfA

= k5 +
[

# G-

+

=

#$ "

#

%#

%&

−

)

#

[

'

+

&

]'( +

* "
&

+

+

&

]

(34)
(35)

(36)
[

+

The equations of motion of the tube are: "

(33)

)

#

"

* +
&

]

(37)
(38)

Together with the boundary conditions for the motion of the liquid,
+
=
12=1
112
Mathematical Theory and Modeling
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.11, 2013

www.iiste.org

=
12=1
If it is now assumed that from (12) and (13)
8 = m exp [M (1 − q⁄k )] and P = r exp [M (1 − q⁄k )], where m and r are arbitrary constants, the boundary
conditions for
and , using the boundary conditions for the motion of the liquid we have:
+
= m inexp[M (1 − q⁄k )] =
(39)
But
= m exp [M (1 − q⁄k )] from (39)
Therefore = m M
(40)
From (36), we can write (37) as
NH&
gfA
M m =
[k5 l C ([) +
]
(41)
"

5 G

# G-

Also = r inexp[M (1 − q⁄k )] =
But from (12),
= exp [M (1 − q⁄k )], this implies
"

M r =

= k5 +

gfA

# G-

Therefore M r = k5 +

]

gfA

# G-

]

(43)

From (37) and (40), the equations of the tube become
f
) c
NHu
v
m 5 = A − [ i− Aj + A

−r

(42)

%#
# &
G
&
# "
5
= 5
=
W− M [ k5 l C ([)
%#&
5

+

H & g

G # -

X + .−
)
#

H uA
G

+ i−
c

&

NHvA
G

(44)

j0,

(45)

Equations (41), (43), (44) and (45) are four homogeneous equations in the arbitrary constants w, ^ ,k5 , m and r .
Eliminating them will give a frequency equation, which will determine the wave velocity c, in terms of the
elastic properties of the tube and the non-dimensional parameter [. The result of the elimination is:
x
kY 5

1

M Ql C
2k

M Qx
2k 5 Y 5
−[

P

5

1
ℎ

−M
5

0

Q5 x + 5 Px 5 Q
]
2k = Y 5 ℎQ

−M

0

N# &H-

−

A$

5%#

−

0
)

#&

NHc{
#&G

NHc{
5

(1 −

#&G
)

#G

= 0 (46)

)

H &

Operating on the rows and columns of equation (46) and neglecting
and continuing operating on rows and
G d
columns until we get
x
1
0
1
g
A$
−1
0 |
5
5
|
}
*}
=0
(47)
0
|e
|
e
e
*}
}
1−
0
− l CY5

where a =

%#

# &

5

,

!L=

e)

# G

Equation (47) can be written as
x
1
x
lC
| 5
0
| Y

e

0
−2
−L
−~L

e

1
0
|
−~L | = 0
(a − L)

0
5
Whose solution is
g
[x(1 − ~ 5 )(1 − l C )]L 5 − .2Y 5 + ax(1 − l C ) + l C Y 5 i − ~ − ~xj0 L + 2aY 5 + l C Y ; = 0
5
This reduces to
A$ - e

113

(48)

(49)
Mathematical Theory and Modeling
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.11, 2013

www.iiste.org

(1 − ~ 5 )(1 − l C )L 5 − .2 + a(1 − l C ) + l C i − 2~j0 L + 2a + l C = 0
5
I.e. when x = 1 and Y 5 = 1 in (48), i.e., when = 0, since
x=(

< V

)(#3 NH<V)

and Y 5 =

G Z "A (

&

V

)(#3 NH<V)

−

(

(50)

& V

)N Z "A

Using Womersley [10], the roots of equation (49) are given by
(1 − ~ 5 )L = • ± [• 5 − (1 − ~ 5 )•] R5

where • =

‚ A
W ƒ <„ *X

(

A$ )

+ +
e
5

*W

When Y 5 = 1 and x = 1, i.e.

A$ ‚ i < A jX AT
ƒ
„

(

A$ )

(51)

A$ - U

= 0 equation (51) reduces to • =

[10] solution for the case without suspension and the same as [6].
4. Conclusion
In terms of the notation used in the case of the solid case,

…A$

A
. < *0
„

(

A$ )

(52)

+ + ~ − , which is Womersley’s
e
5

;

= exp(−M†) /]9 C so that the quantities required

to compute the roots of equation (49) are already available. When ˆ C ([) is complex, L is always complex and
the motion is either damped or unstable.
If we write ((1 − ~ 5 ) LR2) R5 = ‰ − MŠ and denote by wC the velocity for the perfect fluid, then if w is the
w
wave-velocity, CRw = ‰ and over a distance of one wavelength the amplitude will be reduced to the ratio
exp (−2‹ ŠR‰ ) [6] [10]. Since Womersley’s case without suspension is considered as the first best description
of blood flow in arteries, we have now further proved that our model is in line with this best description of blood
flow and the propagation of waves in the arterial system.
Acknowledgement
We wish to acknowledge the contribution of Dr. L. M. Srivastava towards his guide in the M.Sc thesis (1986),
“Flow of Particle-Fluid Suspension in Elastic Tube”, at the Department of Mathematics, A.B.U, Zaria, Nigeria.
This contribution is one of the two major sources, which made this publication possible.
Also, to Associate Professor M. E. Nja of the Mathematics Department, Federal University Lafia,
Nigeria. For his great motivation and encouragement for this research. We thank you all.
References
1. Atabek, H.B. (1996). Wave propagation through a viscous incompressible fluid contained in an initially
stressed elastic tube. Biophysical Journal volume 6, pp 481-501
2. Cox, R.H. (1968) “Wave propagation through a Newtonian fluid contained within a thick-walled,
viscoelastic tube”, Biophysical Journal, volume 8, pp 691-708.
3. Evans, R.L. “Pulsatile flow in vessels whose distinsibility and size vary with site”. Physics in Medicine
and Biology, volume 7, pp 105-115.
4. Iberall, S.A. (1964) “study of the General Dynamics of the physical-chemical systems in Mammals”,
NASA CR-120
5. Lambert, J. W. (1958). “On the non-linearities of fluid flow in nonrigid tubes”, J. Franklin inst. 266, 83.
6. Ngbo, P.H, (1986), “Flow of Particle-Fluid Suspension in Elastic Tube”, M.Sc Thesis (unpublished),
Department of Mathematics, A.B.U, Zaria, Nigeria.
7. Oslen, J.H. et al (1967) “Large- amplitude unsteady flow in liquid filled elastic tubes”, J. Fluid mech,
vol 33, pp 513.
8. Pennisi, L.L. (1976) “Elements of complex variables”, Holt, Rinehart and Winston, pp 17-23.
9. Tam, C.K.W. (1969) “The drag on a cloud of spherical particles in low Reynolds number flow”, J. Fluid
Mech, volume 38, pp 537-546.
10. Womersley, J.R. (1955) Oscillatory motion of a viscous liquid in a thin-walled elastic tube 1. The linear
approximation for long waves, philosophical mag, volume 46, pp 199-215.

114
This academic article was published by The International Institute for Science,
Technology and Education (IISTE). The IISTE is a pioneer in the Open Access
Publishing service based in the U.S. and Europe. The aim of the institute is
Accelerating Global Knowledge Sharing.
More information about the publisher can be found in the IISTE’s homepage:
http://www.iiste.org
CALL FOR JOURNAL PAPERS
The IISTE is currently hosting more than 30 peer-reviewed academic journals and
collaborating with academic institutions around the world. There’s no deadline for
submission. Prospective authors of IISTE journals can find the submission
instruction on the following page: http://www.iiste.org/journals/
The IISTE
editorial team promises to the review and publish all the qualified submissions in a
fast manner. All the journals articles are available online to the readers all over the
world without financial, legal, or technical barriers other than those inseparable from
gaining access to the internet itself. Printed version of the journals is also available
upon request of readers and authors.
MORE RESOURCES
Book publication information: http://www.iiste.org/book/
Recent conferences: http://www.iiste.org/conference/
IISTE Knowledge Sharing Partners
EBSCO, Index Copernicus, Ulrich's Periodicals Directory, JournalTOCS, PKP Open
Archives Harvester, Bielefeld Academic Search Engine, Elektronische
Zeitschriftenbibliothek EZB, Open J-Gate, OCLC WorldCat, Universe Digtial
Library , NewJour, Google Scholar

Mais conteúdo relacionado

Mais procurados

A solution of the Burger’s equation arising in the Longitudinal Dispersion Ph...
A solution of the Burger’s equation arising in the Longitudinal Dispersion Ph...A solution of the Burger’s equation arising in the Longitudinal Dispersion Ph...
A solution of the Burger’s equation arising in the Longitudinal Dispersion Ph...IOSR Journals
 
FUNDAMENTALS of Fluid Mechanics (chapter 01)
FUNDAMENTALS of Fluid Mechanics (chapter 01)FUNDAMENTALS of Fluid Mechanics (chapter 01)
FUNDAMENTALS of Fluid Mechanics (chapter 01)Shekh Muhsen Uddin Ahmed
 
76201951
7620195176201951
76201951IJRAT
 
Critical analysis of zwietering correlation for solids
Critical analysis of zwietering correlation for solidsCritical analysis of zwietering correlation for solids
Critical analysis of zwietering correlation for solidsfabiola_9
 
Pressure Gradient Influence on MHD Flow for Generalized Burgers’ Fluid with S...
Pressure Gradient Influence on MHD Flow for Generalized Burgers’ Fluid with S...Pressure Gradient Influence on MHD Flow for Generalized Burgers’ Fluid with S...
Pressure Gradient Influence on MHD Flow for Generalized Burgers’ Fluid with S...IJERA Editor
 
Solution of unsteady rolling
Solution of unsteady rollingSolution of unsteady rolling
Solution of unsteady rollingcsandit
 
Comparison of flow analysis of a sudden and gradual change of pipe diameter u...
Comparison of flow analysis of a sudden and gradual change of pipe diameter u...Comparison of flow analysis of a sudden and gradual change of pipe diameter u...
Comparison of flow analysis of a sudden and gradual change of pipe diameter u...eSAT Journals
 
Comparison of flow analysis of a sudden and gradual change
Comparison of flow analysis of a sudden and gradual changeComparison of flow analysis of a sudden and gradual change
Comparison of flow analysis of a sudden and gradual changeeSAT Publishing House
 
Exact Solutions for MHD Flow of a Viscoelastic Fluid with the Fractional Bur...
Exact Solutions for MHD Flow of a Viscoelastic Fluid with the  Fractional Bur...Exact Solutions for MHD Flow of a Viscoelastic Fluid with the  Fractional Bur...
Exact Solutions for MHD Flow of a Viscoelastic Fluid with the Fractional Bur...IJMER
 
Analytic Solutions to the Darcy-Lapwood-Brinkman Equation with Variable Perme...
Analytic Solutions to the Darcy-Lapwood-Brinkman Equation with Variable Perme...Analytic Solutions to the Darcy-Lapwood-Brinkman Equation with Variable Perme...
Analytic Solutions to the Darcy-Lapwood-Brinkman Equation with Variable Perme...IJERA Editor
 
Mathematical modelling and analysis of three dimensional darcy
Mathematical modelling and analysis of three dimensional darcyMathematical modelling and analysis of three dimensional darcy
Mathematical modelling and analysis of three dimensional darcyIAEME Publication
 
Blood flow through stenosed inclined tubes with
Blood flow through stenosed inclined tubes withBlood flow through stenosed inclined tubes with
Blood flow through stenosed inclined tubes witheSAT Publishing House
 
Effects of wall properties and heat transfer on the peristaltic
Effects of wall properties and heat transfer on the peristalticEffects of wall properties and heat transfer on the peristaltic
Effects of wall properties and heat transfer on the peristalticAlexander Decker
 
A numerical analysis of three dimensional darcy model in an inclined rect
A numerical analysis of three dimensional darcy model in an inclined rectA numerical analysis of three dimensional darcy model in an inclined rect
A numerical analysis of three dimensional darcy model in an inclined rectIAEME Publication
 
Finite Element Analysis of Convective Micro Polar Fluid Flow through a Porou...
Finite Element Analysis of Convective Micro Polar Fluid Flow  through a Porou...Finite Element Analysis of Convective Micro Polar Fluid Flow  through a Porou...
Finite Element Analysis of Convective Micro Polar Fluid Flow through a Porou...IJMER
 
The International Journal of Engineering and Science (The IJES)
The International Journal of Engineering and Science (The IJES)The International Journal of Engineering and Science (The IJES)
The International Journal of Engineering and Science (The IJES)theijes
 

Mais procurados (19)

A solution of the Burger’s equation arising in the Longitudinal Dispersion Ph...
A solution of the Burger’s equation arising in the Longitudinal Dispersion Ph...A solution of the Burger’s equation arising in the Longitudinal Dispersion Ph...
A solution of the Burger’s equation arising in the Longitudinal Dispersion Ph...
 
FUNDAMENTALS of Fluid Mechanics (chapter 01)
FUNDAMENTALS of Fluid Mechanics (chapter 01)FUNDAMENTALS of Fluid Mechanics (chapter 01)
FUNDAMENTALS of Fluid Mechanics (chapter 01)
 
K0736871
K0736871K0736871
K0736871
 
76201951
7620195176201951
76201951
 
Critical analysis of zwietering correlation for solids
Critical analysis of zwietering correlation for solidsCritical analysis of zwietering correlation for solids
Critical analysis of zwietering correlation for solids
 
Pressure Gradient Influence on MHD Flow for Generalized Burgers’ Fluid with S...
Pressure Gradient Influence on MHD Flow for Generalized Burgers’ Fluid with S...Pressure Gradient Influence on MHD Flow for Generalized Burgers’ Fluid with S...
Pressure Gradient Influence on MHD Flow for Generalized Burgers’ Fluid with S...
 
Solution of unsteady rolling
Solution of unsteady rollingSolution of unsteady rolling
Solution of unsteady rolling
 
Comparison of flow analysis of a sudden and gradual change of pipe diameter u...
Comparison of flow analysis of a sudden and gradual change of pipe diameter u...Comparison of flow analysis of a sudden and gradual change of pipe diameter u...
Comparison of flow analysis of a sudden and gradual change of pipe diameter u...
 
Comparison of flow analysis of a sudden and gradual change
Comparison of flow analysis of a sudden and gradual changeComparison of flow analysis of a sudden and gradual change
Comparison of flow analysis of a sudden and gradual change
 
Exact Solutions for MHD Flow of a Viscoelastic Fluid with the Fractional Bur...
Exact Solutions for MHD Flow of a Viscoelastic Fluid with the  Fractional Bur...Exact Solutions for MHD Flow of a Viscoelastic Fluid with the  Fractional Bur...
Exact Solutions for MHD Flow of a Viscoelastic Fluid with the Fractional Bur...
 
Analytic Solutions to the Darcy-Lapwood-Brinkman Equation with Variable Perme...
Analytic Solutions to the Darcy-Lapwood-Brinkman Equation with Variable Perme...Analytic Solutions to the Darcy-Lapwood-Brinkman Equation with Variable Perme...
Analytic Solutions to the Darcy-Lapwood-Brinkman Equation with Variable Perme...
 
Mathematical modelling and analysis of three dimensional darcy
Mathematical modelling and analysis of three dimensional darcyMathematical modelling and analysis of three dimensional darcy
Mathematical modelling and analysis of three dimensional darcy
 
Blood flow through stenosed inclined tubes with
Blood flow through stenosed inclined tubes withBlood flow through stenosed inclined tubes with
Blood flow through stenosed inclined tubes with
 
Effects of wall properties and heat transfer on the peristaltic
Effects of wall properties and heat transfer on the peristalticEffects of wall properties and heat transfer on the peristaltic
Effects of wall properties and heat transfer on the peristaltic
 
A numerical analysis of three dimensional darcy model in an inclined rect
A numerical analysis of three dimensional darcy model in an inclined rectA numerical analysis of three dimensional darcy model in an inclined rect
A numerical analysis of three dimensional darcy model in an inclined rect
 
Finite Element Analysis of Convective Micro Polar Fluid Flow through a Porou...
Finite Element Analysis of Convective Micro Polar Fluid Flow  through a Porou...Finite Element Analysis of Convective Micro Polar Fluid Flow  through a Porou...
Finite Element Analysis of Convective Micro Polar Fluid Flow through a Porou...
 
The International Journal of Engineering and Science (The IJES)
The International Journal of Engineering and Science (The IJES)The International Journal of Engineering and Science (The IJES)
The International Journal of Engineering and Science (The IJES)
 
12 chapter 3
12 chapter 312 chapter 3
12 chapter 3
 
Jh3616111616
Jh3616111616Jh3616111616
Jh3616111616
 

Destaque

John Scott IT skills resume
John Scott IT skills resumeJohn Scott IT skills resume
John Scott IT skills resumeJohn Scott
 
Documentos primaria-sesiones-unidad02-integradas-tercer grado-sesion14-integr...
Documentos primaria-sesiones-unidad02-integradas-tercer grado-sesion14-integr...Documentos primaria-sesiones-unidad02-integradas-tercer grado-sesion14-integr...
Documentos primaria-sesiones-unidad02-integradas-tercer grado-sesion14-integr...Teresa Clotilde Ojeda Sánchez
 
Detección de la evolución adaptativa linaje específica de los genes expresado...
Detección de la evolución adaptativa linaje específica de los genes expresado...Detección de la evolución adaptativa linaje específica de los genes expresado...
Detección de la evolución adaptativa linaje específica de los genes expresado...David Muñoz
 
Optimization of pelagic fishing efforts in muncar area , indonesia
Optimization of pelagic fishing efforts  in muncar area , indonesiaOptimization of pelagic fishing efforts  in muncar area , indonesia
Optimization of pelagic fishing efforts in muncar area , indonesiaAlexander Decker
 
Documentos primaria-sesiones-unidad02-integradas-tercer grado-sesion09-integr...
Documentos primaria-sesiones-unidad02-integradas-tercer grado-sesion09-integr...Documentos primaria-sesiones-unidad02-integradas-tercer grado-sesion09-integr...
Documentos primaria-sesiones-unidad02-integradas-tercer grado-sesion09-integr...Teresa Clotilde Ojeda Sánchez
 
MS Dynamics AX 2012 R3 on Azure
MS Dynamics AX 2012 R3 on AzureMS Dynamics AX 2012 R3 on Azure
MS Dynamics AX 2012 R3 on AzureJuan Fabian
 
On improved estimation of population mean using qualitative auxiliary informa...
On improved estimation of population mean using qualitative auxiliary informa...On improved estimation of population mean using qualitative auxiliary informa...
On improved estimation of population mean using qualitative auxiliary informa...Alexander Decker
 
Orm presentation slide show
Orm presentation slide showOrm presentation slide show
Orm presentation slide showPastor Bermudez
 
Metodologia Yuuki Furuya
Metodologia Yuuki FuruyaMetodologia Yuuki Furuya
Metodologia Yuuki FuruyaYuuki Furuya
 
Evolución político-territorial de la República Mexicana
Evolución político-territorial de la República MexicanaEvolución político-territorial de la República Mexicana
Evolución político-territorial de la República MexicanaChristianovl
 
Oxide film growth on copper in neutral aqueous solutions
Oxide film growth on copper in neutral aqueous solutionsOxide film growth on copper in neutral aqueous solutions
Oxide film growth on copper in neutral aqueous solutionsAlexander Decker
 
Public private partnership in land management a learning strategy for improvi...
Public private partnership in land management a learning strategy for improvi...Public private partnership in land management a learning strategy for improvi...
Public private partnership in land management a learning strategy for improvi...Alexander Decker
 
Paradigmatic appraisal of techniques and technology of adire in the last five...
Paradigmatic appraisal of techniques and technology of adire in the last five...Paradigmatic appraisal of techniques and technology of adire in the last five...
Paradigmatic appraisal of techniques and technology of adire in the last five...Alexander Decker
 
Planning and assessment
Planning and assessmentPlanning and assessment
Planning and assessmenttamarafleming
 
How To Create Funny Images
How To Create Funny ImagesHow To Create Funny Images
How To Create Funny Imageswindkidney20
 
Observable effects of developing mathematical skills of students through team...
Observable effects of developing mathematical skills of students through team...Observable effects of developing mathematical skills of students through team...
Observable effects of developing mathematical skills of students through team...Alexander Decker
 
Organisational climate and corporate performance
Organisational climate and corporate performanceOrganisational climate and corporate performance
Organisational climate and corporate performanceAlexander Decker
 

Destaque (20)

John Scott IT skills resume
John Scott IT skills resumeJohn Scott IT skills resume
John Scott IT skills resume
 
Documentos primaria-sesiones-unidad02-integradas-tercer grado-sesion14-integr...
Documentos primaria-sesiones-unidad02-integradas-tercer grado-sesion14-integr...Documentos primaria-sesiones-unidad02-integradas-tercer grado-sesion14-integr...
Documentos primaria-sesiones-unidad02-integradas-tercer grado-sesion14-integr...
 
Detección de la evolución adaptativa linaje específica de los genes expresado...
Detección de la evolución adaptativa linaje específica de los genes expresado...Detección de la evolución adaptativa linaje específica de los genes expresado...
Detección de la evolución adaptativa linaje específica de los genes expresado...
 
Optimization of pelagic fishing efforts in muncar area , indonesia
Optimization of pelagic fishing efforts  in muncar area , indonesiaOptimization of pelagic fishing efforts  in muncar area , indonesia
Optimization of pelagic fishing efforts in muncar area , indonesia
 
Documentos primaria-sesiones-unidad02-integradas-tercer grado-sesion09-integr...
Documentos primaria-sesiones-unidad02-integradas-tercer grado-sesion09-integr...Documentos primaria-sesiones-unidad02-integradas-tercer grado-sesion09-integr...
Documentos primaria-sesiones-unidad02-integradas-tercer grado-sesion09-integr...
 
MS Dynamics AX 2012 R3 on Azure
MS Dynamics AX 2012 R3 on AzureMS Dynamics AX 2012 R3 on Azure
MS Dynamics AX 2012 R3 on Azure
 
Actividad 2
Actividad 2Actividad 2
Actividad 2
 
On improved estimation of population mean using qualitative auxiliary informa...
On improved estimation of population mean using qualitative auxiliary informa...On improved estimation of population mean using qualitative auxiliary informa...
On improved estimation of population mean using qualitative auxiliary informa...
 
Orm presentation slide show
Orm presentation slide showOrm presentation slide show
Orm presentation slide show
 
Metodologia Yuuki Furuya
Metodologia Yuuki FuruyaMetodologia Yuuki Furuya
Metodologia Yuuki Furuya
 
Evolución político-territorial de la República Mexicana
Evolución político-territorial de la República MexicanaEvolución político-territorial de la República Mexicana
Evolución político-territorial de la República Mexicana
 
Oxide film growth on copper in neutral aqueous solutions
Oxide film growth on copper in neutral aqueous solutionsOxide film growth on copper in neutral aqueous solutions
Oxide film growth on copper in neutral aqueous solutions
 
Public private partnership in land management a learning strategy for improvi...
Public private partnership in land management a learning strategy for improvi...Public private partnership in land management a learning strategy for improvi...
Public private partnership in land management a learning strategy for improvi...
 
protocolos de red
protocolos de red protocolos de red
protocolos de red
 
Paradigmatic appraisal of techniques and technology of adire in the last five...
Paradigmatic appraisal of techniques and technology of adire in the last five...Paradigmatic appraisal of techniques and technology of adire in the last five...
Paradigmatic appraisal of techniques and technology of adire in the last five...
 
Planning and assessment
Planning and assessmentPlanning and assessment
Planning and assessment
 
How To Create Funny Images
How To Create Funny ImagesHow To Create Funny Images
How To Create Funny Images
 
Presentación slideshare
Presentación slidesharePresentación slideshare
Presentación slideshare
 
Observable effects of developing mathematical skills of students through team...
Observable effects of developing mathematical skills of students through team...Observable effects of developing mathematical skills of students through team...
Observable effects of developing mathematical skills of students through team...
 
Organisational climate and corporate performance
Organisational climate and corporate performanceOrganisational climate and corporate performance
Organisational climate and corporate performance
 

Semelhante a Oscillatory flow and particle suspension in a fluid through an elastic tube

Peristaltic flow of blood through coaxial vertical channel with effect of mag...
Peristaltic flow of blood through coaxial vertical channel with effect of mag...Peristaltic flow of blood through coaxial vertical channel with effect of mag...
Peristaltic flow of blood through coaxial vertical channel with effect of mag...ijmech
 
Non- Newtonian behavior of blood in very narrow vessels
Non- Newtonian behavior of blood in very narrow vesselsNon- Newtonian behavior of blood in very narrow vessels
Non- Newtonian behavior of blood in very narrow vesselsIOSR Journals
 
An exact solution of einstein equations for interior field of an anisotropic ...
An exact solution of einstein equations for interior field of an anisotropic ...An exact solution of einstein equations for interior field of an anisotropic ...
An exact solution of einstein equations for interior field of an anisotropic ...eSAT Journals
 
A study-to-understand-differential-equations-applied-to-aerodynamics-using-cf...
A study-to-understand-differential-equations-applied-to-aerodynamics-using-cf...A study-to-understand-differential-equations-applied-to-aerodynamics-using-cf...
A study-to-understand-differential-equations-applied-to-aerodynamics-using-cf...zoya rizvi
 
INRIA-USFD-KCL- Identification of artery wall stiffness - 2014
INRIA-USFD-KCL- Identification of artery wall stiffness - 2014INRIA-USFD-KCL- Identification of artery wall stiffness - 2014
INRIA-USFD-KCL- Identification of artery wall stiffness - 2014Cristina Staicu
 
Effect of Magnetic Field on Peristaltic Flow of Williamson Fluid in a Symmetr...
Effect of Magnetic Field on Peristaltic Flow of Williamson Fluid in a Symmetr...Effect of Magnetic Field on Peristaltic Flow of Williamson Fluid in a Symmetr...
Effect of Magnetic Field on Peristaltic Flow of Williamson Fluid in a Symmetr...IOSRJM
 
Fluctuating Flow of Vescoelastic Fluids between Two Coaxial Cylinders
Fluctuating Flow of Vescoelastic Fluids between Two Coaxial CylindersFluctuating Flow of Vescoelastic Fluids between Two Coaxial Cylinders
Fluctuating Flow of Vescoelastic Fluids between Two Coaxial Cylindersinventionjournals
 
Effect of Magnetic Field on Blood Flow (Elastico- Viscous) Under Periodic Bod...
Effect of Magnetic Field on Blood Flow (Elastico- Viscous) Under Periodic Bod...Effect of Magnetic Field on Blood Flow (Elastico- Viscous) Under Periodic Bod...
Effect of Magnetic Field on Blood Flow (Elastico- Viscous) Under Periodic Bod...IOSR Journals
 
Annals of Limnology and Oceanography
Annals of Limnology and OceanographyAnnals of Limnology and Oceanography
Annals of Limnology and Oceanographypeertechzpublication
 
Mathematical Hydraulic Models of One-Dimensional Unsteady Flow in Rivers
Mathematical Hydraulic Models of One-Dimensional Unsteady Flow in Rivers Mathematical Hydraulic Models of One-Dimensional Unsteady Flow in Rivers
Mathematical Hydraulic Models of One-Dimensional Unsteady Flow in Rivers IJAEMSJORNAL
 
Mathematical Modeling of Bingham Plastic Model of Blood Flow Through Stenotic...
Mathematical Modeling of Bingham Plastic Model of Blood Flow Through Stenotic...Mathematical Modeling of Bingham Plastic Model of Blood Flow Through Stenotic...
Mathematical Modeling of Bingham Plastic Model of Blood Flow Through Stenotic...IJERA Editor
 
CFD simulation of Lid driven cavity flow
CFD simulation of Lid driven cavity flowCFD simulation of Lid driven cavity flow
CFD simulation of Lid driven cavity flowIJSRD
 
Critical Overview of Some Pumping Test Analysis Equations
Critical Overview of Some Pumping Test Analysis EquationsCritical Overview of Some Pumping Test Analysis Equations
Critical Overview of Some Pumping Test Analysis EquationsScientific Review SR
 
Critical Overview of Some Pumping Test Analysis Equations
Critical Overview of Some Pumping Test Analysis EquationsCritical Overview of Some Pumping Test Analysis Equations
Critical Overview of Some Pumping Test Analysis EquationsScientific Review SR
 
Application of DRP scheme solving for rotating disk-driven cavity
Application of DRP scheme solving for rotating disk-driven cavityApplication of DRP scheme solving for rotating disk-driven cavity
Application of DRP scheme solving for rotating disk-driven cavityijceronline
 
Tryggvason
TryggvasonTryggvason
TryggvasonMing Ma
 

Semelhante a Oscillatory flow and particle suspension in a fluid through an elastic tube (20)

Peristaltic flow of blood through coaxial vertical channel with effect of mag...
Peristaltic flow of blood through coaxial vertical channel with effect of mag...Peristaltic flow of blood through coaxial vertical channel with effect of mag...
Peristaltic flow of blood through coaxial vertical channel with effect of mag...
 
Non- Newtonian behavior of blood in very narrow vessels
Non- Newtonian behavior of blood in very narrow vesselsNon- Newtonian behavior of blood in very narrow vessels
Non- Newtonian behavior of blood in very narrow vessels
 
An exact solution of einstein equations for interior field of an anisotropic ...
An exact solution of einstein equations for interior field of an anisotropic ...An exact solution of einstein equations for interior field of an anisotropic ...
An exact solution of einstein equations for interior field of an anisotropic ...
 
A study-to-understand-differential-equations-applied-to-aerodynamics-using-cf...
A study-to-understand-differential-equations-applied-to-aerodynamics-using-cf...A study-to-understand-differential-equations-applied-to-aerodynamics-using-cf...
A study-to-understand-differential-equations-applied-to-aerodynamics-using-cf...
 
INRIA-USFD-KCL- Identification of artery wall stiffness - 2014
INRIA-USFD-KCL- Identification of artery wall stiffness - 2014INRIA-USFD-KCL- Identification of artery wall stiffness - 2014
INRIA-USFD-KCL- Identification of artery wall stiffness - 2014
 
Effect of Magnetic Field on Peristaltic Flow of Williamson Fluid in a Symmetr...
Effect of Magnetic Field on Peristaltic Flow of Williamson Fluid in a Symmetr...Effect of Magnetic Field on Peristaltic Flow of Williamson Fluid in a Symmetr...
Effect of Magnetic Field on Peristaltic Flow of Williamson Fluid in a Symmetr...
 
PASTICCIO
PASTICCIOPASTICCIO
PASTICCIO
 
PASTICCIO
PASTICCIOPASTICCIO
PASTICCIO
 
Fluctuating Flow of Vescoelastic Fluids between Two Coaxial Cylinders
Fluctuating Flow of Vescoelastic Fluids between Two Coaxial CylindersFluctuating Flow of Vescoelastic Fluids between Two Coaxial Cylinders
Fluctuating Flow of Vescoelastic Fluids between Two Coaxial Cylinders
 
TWO-DIMENSIONAL IDEAL FLOW (Chapter 6)
TWO-DIMENSIONAL IDEAL FLOW (Chapter 6)TWO-DIMENSIONAL IDEAL FLOW (Chapter 6)
TWO-DIMENSIONAL IDEAL FLOW (Chapter 6)
 
Effect of Magnetic Field on Blood Flow (Elastico- Viscous) Under Periodic Bod...
Effect of Magnetic Field on Blood Flow (Elastico- Viscous) Under Periodic Bod...Effect of Magnetic Field on Blood Flow (Elastico- Viscous) Under Periodic Bod...
Effect of Magnetic Field on Blood Flow (Elastico- Viscous) Under Periodic Bod...
 
Poster format
Poster formatPoster format
Poster format
 
Annals of Limnology and Oceanography
Annals of Limnology and OceanographyAnnals of Limnology and Oceanography
Annals of Limnology and Oceanography
 
Mathematical Hydraulic Models of One-Dimensional Unsteady Flow in Rivers
Mathematical Hydraulic Models of One-Dimensional Unsteady Flow in Rivers Mathematical Hydraulic Models of One-Dimensional Unsteady Flow in Rivers
Mathematical Hydraulic Models of One-Dimensional Unsteady Flow in Rivers
 
Mathematical Modeling of Bingham Plastic Model of Blood Flow Through Stenotic...
Mathematical Modeling of Bingham Plastic Model of Blood Flow Through Stenotic...Mathematical Modeling of Bingham Plastic Model of Blood Flow Through Stenotic...
Mathematical Modeling of Bingham Plastic Model of Blood Flow Through Stenotic...
 
CFD simulation of Lid driven cavity flow
CFD simulation of Lid driven cavity flowCFD simulation of Lid driven cavity flow
CFD simulation of Lid driven cavity flow
 
Critical Overview of Some Pumping Test Analysis Equations
Critical Overview of Some Pumping Test Analysis EquationsCritical Overview of Some Pumping Test Analysis Equations
Critical Overview of Some Pumping Test Analysis Equations
 
Critical Overview of Some Pumping Test Analysis Equations
Critical Overview of Some Pumping Test Analysis EquationsCritical Overview of Some Pumping Test Analysis Equations
Critical Overview of Some Pumping Test Analysis Equations
 
Application of DRP scheme solving for rotating disk-driven cavity
Application of DRP scheme solving for rotating disk-driven cavityApplication of DRP scheme solving for rotating disk-driven cavity
Application of DRP scheme solving for rotating disk-driven cavity
 
Tryggvason
TryggvasonTryggvason
Tryggvason
 

Mais de Alexander Decker

Abnormalities of hormones and inflammatory cytokines in women affected with p...
Abnormalities of hormones and inflammatory cytokines in women affected with p...Abnormalities of hormones and inflammatory cytokines in women affected with p...
Abnormalities of hormones and inflammatory cytokines in women affected with p...Alexander Decker
 
A validation of the adverse childhood experiences scale in
A validation of the adverse childhood experiences scale inA validation of the adverse childhood experiences scale in
A validation of the adverse childhood experiences scale inAlexander Decker
 
A usability evaluation framework for b2 c e commerce websites
A usability evaluation framework for b2 c e commerce websitesA usability evaluation framework for b2 c e commerce websites
A usability evaluation framework for b2 c e commerce websitesAlexander Decker
 
A universal model for managing the marketing executives in nigerian banks
A universal model for managing the marketing executives in nigerian banksA universal model for managing the marketing executives in nigerian banks
A universal model for managing the marketing executives in nigerian banksAlexander Decker
 
A unique common fixed point theorems in generalized d
A unique common fixed point theorems in generalized dA unique common fixed point theorems in generalized d
A unique common fixed point theorems in generalized dAlexander Decker
 
A trends of salmonella and antibiotic resistance
A trends of salmonella and antibiotic resistanceA trends of salmonella and antibiotic resistance
A trends of salmonella and antibiotic resistanceAlexander Decker
 
A transformational generative approach towards understanding al-istifham
A transformational  generative approach towards understanding al-istifhamA transformational  generative approach towards understanding al-istifham
A transformational generative approach towards understanding al-istifhamAlexander Decker
 
A time series analysis of the determinants of savings in namibia
A time series analysis of the determinants of savings in namibiaA time series analysis of the determinants of savings in namibia
A time series analysis of the determinants of savings in namibiaAlexander Decker
 
A therapy for physical and mental fitness of school children
A therapy for physical and mental fitness of school childrenA therapy for physical and mental fitness of school children
A therapy for physical and mental fitness of school childrenAlexander Decker
 
A theory of efficiency for managing the marketing executives in nigerian banks
A theory of efficiency for managing the marketing executives in nigerian banksA theory of efficiency for managing the marketing executives in nigerian banks
A theory of efficiency for managing the marketing executives in nigerian banksAlexander Decker
 
A systematic evaluation of link budget for
A systematic evaluation of link budget forA systematic evaluation of link budget for
A systematic evaluation of link budget forAlexander Decker
 
A synthetic review of contraceptive supplies in punjab
A synthetic review of contraceptive supplies in punjabA synthetic review of contraceptive supplies in punjab
A synthetic review of contraceptive supplies in punjabAlexander Decker
 
A synthesis of taylor’s and fayol’s management approaches for managing market...
A synthesis of taylor’s and fayol’s management approaches for managing market...A synthesis of taylor’s and fayol’s management approaches for managing market...
A synthesis of taylor’s and fayol’s management approaches for managing market...Alexander Decker
 
A survey paper on sequence pattern mining with incremental
A survey paper on sequence pattern mining with incrementalA survey paper on sequence pattern mining with incremental
A survey paper on sequence pattern mining with incrementalAlexander Decker
 
A survey on live virtual machine migrations and its techniques
A survey on live virtual machine migrations and its techniquesA survey on live virtual machine migrations and its techniques
A survey on live virtual machine migrations and its techniquesAlexander Decker
 
A survey on data mining and analysis in hadoop and mongo db
A survey on data mining and analysis in hadoop and mongo dbA survey on data mining and analysis in hadoop and mongo db
A survey on data mining and analysis in hadoop and mongo dbAlexander Decker
 
A survey on challenges to the media cloud
A survey on challenges to the media cloudA survey on challenges to the media cloud
A survey on challenges to the media cloudAlexander Decker
 
A survey of provenance leveraged
A survey of provenance leveragedA survey of provenance leveraged
A survey of provenance leveragedAlexander Decker
 
A survey of private equity investments in kenya
A survey of private equity investments in kenyaA survey of private equity investments in kenya
A survey of private equity investments in kenyaAlexander Decker
 
A study to measures the financial health of
A study to measures the financial health ofA study to measures the financial health of
A study to measures the financial health ofAlexander Decker
 

Mais de Alexander Decker (20)

Abnormalities of hormones and inflammatory cytokines in women affected with p...
Abnormalities of hormones and inflammatory cytokines in women affected with p...Abnormalities of hormones and inflammatory cytokines in women affected with p...
Abnormalities of hormones and inflammatory cytokines in women affected with p...
 
A validation of the adverse childhood experiences scale in
A validation of the adverse childhood experiences scale inA validation of the adverse childhood experiences scale in
A validation of the adverse childhood experiences scale in
 
A usability evaluation framework for b2 c e commerce websites
A usability evaluation framework for b2 c e commerce websitesA usability evaluation framework for b2 c e commerce websites
A usability evaluation framework for b2 c e commerce websites
 
A universal model for managing the marketing executives in nigerian banks
A universal model for managing the marketing executives in nigerian banksA universal model for managing the marketing executives in nigerian banks
A universal model for managing the marketing executives in nigerian banks
 
A unique common fixed point theorems in generalized d
A unique common fixed point theorems in generalized dA unique common fixed point theorems in generalized d
A unique common fixed point theorems in generalized d
 
A trends of salmonella and antibiotic resistance
A trends of salmonella and antibiotic resistanceA trends of salmonella and antibiotic resistance
A trends of salmonella and antibiotic resistance
 
A transformational generative approach towards understanding al-istifham
A transformational  generative approach towards understanding al-istifhamA transformational  generative approach towards understanding al-istifham
A transformational generative approach towards understanding al-istifham
 
A time series analysis of the determinants of savings in namibia
A time series analysis of the determinants of savings in namibiaA time series analysis of the determinants of savings in namibia
A time series analysis of the determinants of savings in namibia
 
A therapy for physical and mental fitness of school children
A therapy for physical and mental fitness of school childrenA therapy for physical and mental fitness of school children
A therapy for physical and mental fitness of school children
 
A theory of efficiency for managing the marketing executives in nigerian banks
A theory of efficiency for managing the marketing executives in nigerian banksA theory of efficiency for managing the marketing executives in nigerian banks
A theory of efficiency for managing the marketing executives in nigerian banks
 
A systematic evaluation of link budget for
A systematic evaluation of link budget forA systematic evaluation of link budget for
A systematic evaluation of link budget for
 
A synthetic review of contraceptive supplies in punjab
A synthetic review of contraceptive supplies in punjabA synthetic review of contraceptive supplies in punjab
A synthetic review of contraceptive supplies in punjab
 
A synthesis of taylor’s and fayol’s management approaches for managing market...
A synthesis of taylor’s and fayol’s management approaches for managing market...A synthesis of taylor’s and fayol’s management approaches for managing market...
A synthesis of taylor’s and fayol’s management approaches for managing market...
 
A survey paper on sequence pattern mining with incremental
A survey paper on sequence pattern mining with incrementalA survey paper on sequence pattern mining with incremental
A survey paper on sequence pattern mining with incremental
 
A survey on live virtual machine migrations and its techniques
A survey on live virtual machine migrations and its techniquesA survey on live virtual machine migrations and its techniques
A survey on live virtual machine migrations and its techniques
 
A survey on data mining and analysis in hadoop and mongo db
A survey on data mining and analysis in hadoop and mongo dbA survey on data mining and analysis in hadoop and mongo db
A survey on data mining and analysis in hadoop and mongo db
 
A survey on challenges to the media cloud
A survey on challenges to the media cloudA survey on challenges to the media cloud
A survey on challenges to the media cloud
 
A survey of provenance leveraged
A survey of provenance leveragedA survey of provenance leveraged
A survey of provenance leveraged
 
A survey of private equity investments in kenya
A survey of private equity investments in kenyaA survey of private equity investments in kenya
A survey of private equity investments in kenya
 
A study to measures the financial health of
A study to measures the financial health ofA study to measures the financial health of
A study to measures the financial health of
 

Último

ADOPTING WEB 3 FOR YOUR BUSINESS: A STEP-BY-STEP GUIDE
ADOPTING WEB 3 FOR YOUR BUSINESS: A STEP-BY-STEP GUIDEADOPTING WEB 3 FOR YOUR BUSINESS: A STEP-BY-STEP GUIDE
ADOPTING WEB 3 FOR YOUR BUSINESS: A STEP-BY-STEP GUIDELiveplex
 
AI You Can Trust - Ensuring Success with Data Integrity Webinar
AI You Can Trust - Ensuring Success with Data Integrity WebinarAI You Can Trust - Ensuring Success with Data Integrity Webinar
AI You Can Trust - Ensuring Success with Data Integrity WebinarPrecisely
 
Empowering Africa's Next Generation: The AI Leadership Blueprint
Empowering Africa's Next Generation: The AI Leadership BlueprintEmpowering Africa's Next Generation: The AI Leadership Blueprint
Empowering Africa's Next Generation: The AI Leadership BlueprintMahmoud Rabie
 
IESVE Software for Florida Code Compliance Using ASHRAE 90.1-2019
IESVE Software for Florida Code Compliance Using ASHRAE 90.1-2019IESVE Software for Florida Code Compliance Using ASHRAE 90.1-2019
IESVE Software for Florida Code Compliance Using ASHRAE 90.1-2019IES VE
 
UWB Technology for Enhanced Indoor and Outdoor Positioning in Physiological M...
UWB Technology for Enhanced Indoor and Outdoor Positioning in Physiological M...UWB Technology for Enhanced Indoor and Outdoor Positioning in Physiological M...
UWB Technology for Enhanced Indoor and Outdoor Positioning in Physiological M...UbiTrack UK
 
20230202 - Introduction to tis-py
20230202 - Introduction to tis-py20230202 - Introduction to tis-py
20230202 - Introduction to tis-pyJamie (Taka) Wang
 
activity_diagram_combine_v4_20190827.pdfactivity_diagram_combine_v4_20190827.pdf
activity_diagram_combine_v4_20190827.pdfactivity_diagram_combine_v4_20190827.pdfactivity_diagram_combine_v4_20190827.pdfactivity_diagram_combine_v4_20190827.pdf
activity_diagram_combine_v4_20190827.pdfactivity_diagram_combine_v4_20190827.pdfJamie (Taka) Wang
 
AI Fame Rush Review – Virtual Influencer Creation In Just Minutes
AI Fame Rush Review – Virtual Influencer Creation In Just MinutesAI Fame Rush Review – Virtual Influencer Creation In Just Minutes
AI Fame Rush Review – Virtual Influencer Creation In Just MinutesMd Hossain Ali
 
Salesforce Miami User Group Event - 1st Quarter 2024
Salesforce Miami User Group Event - 1st Quarter 2024Salesforce Miami User Group Event - 1st Quarter 2024
Salesforce Miami User Group Event - 1st Quarter 2024SkyPlanner
 
UiPath Studio Web workshop series - Day 6
UiPath Studio Web workshop series - Day 6UiPath Studio Web workshop series - Day 6
UiPath Studio Web workshop series - Day 6DianaGray10
 
Apres-Cyber - The Data Dilemma: Bridging Offensive Operations and Machine Lea...
Apres-Cyber - The Data Dilemma: Bridging Offensive Operations and Machine Lea...Apres-Cyber - The Data Dilemma: Bridging Offensive Operations and Machine Lea...
Apres-Cyber - The Data Dilemma: Bridging Offensive Operations and Machine Lea...Will Schroeder
 
Secure your environment with UiPath and CyberArk technologies - Session 1
Secure your environment with UiPath and CyberArk technologies - Session 1Secure your environment with UiPath and CyberArk technologies - Session 1
Secure your environment with UiPath and CyberArk technologies - Session 1DianaGray10
 
Designing A Time bound resource download URL
Designing A Time bound resource download URLDesigning A Time bound resource download URL
Designing A Time bound resource download URLRuncy Oommen
 
Igniting Next Level Productivity with AI-Infused Data Integration Workflows
Igniting Next Level Productivity with AI-Infused Data Integration WorkflowsIgniting Next Level Productivity with AI-Infused Data Integration Workflows
Igniting Next Level Productivity with AI-Infused Data Integration WorkflowsSafe Software
 
Anypoint Code Builder , Google Pub sub connector and MuleSoft RPA
Anypoint Code Builder , Google Pub sub connector and MuleSoft RPAAnypoint Code Builder , Google Pub sub connector and MuleSoft RPA
Anypoint Code Builder , Google Pub sub connector and MuleSoft RPAshyamraj55
 
Building Your Own AI Instance (TBLC AI )
Building Your Own AI Instance (TBLC AI )Building Your Own AI Instance (TBLC AI )
Building Your Own AI Instance (TBLC AI )Brian Pichman
 
UiPath Community: AI for UiPath Automation Developers
UiPath Community: AI for UiPath Automation DevelopersUiPath Community: AI for UiPath Automation Developers
UiPath Community: AI for UiPath Automation DevelopersUiPathCommunity
 
Building AI-Driven Apps Using Semantic Kernel.pptx
Building AI-Driven Apps Using Semantic Kernel.pptxBuilding AI-Driven Apps Using Semantic Kernel.pptx
Building AI-Driven Apps Using Semantic Kernel.pptxUdaiappa Ramachandran
 
UiPath Studio Web workshop series - Day 7
UiPath Studio Web workshop series - Day 7UiPath Studio Web workshop series - Day 7
UiPath Studio Web workshop series - Day 7DianaGray10
 
NIST Cybersecurity Framework (CSF) 2.0 Workshop
NIST Cybersecurity Framework (CSF) 2.0 WorkshopNIST Cybersecurity Framework (CSF) 2.0 Workshop
NIST Cybersecurity Framework (CSF) 2.0 WorkshopBachir Benyammi
 

Último (20)

ADOPTING WEB 3 FOR YOUR BUSINESS: A STEP-BY-STEP GUIDE
ADOPTING WEB 3 FOR YOUR BUSINESS: A STEP-BY-STEP GUIDEADOPTING WEB 3 FOR YOUR BUSINESS: A STEP-BY-STEP GUIDE
ADOPTING WEB 3 FOR YOUR BUSINESS: A STEP-BY-STEP GUIDE
 
AI You Can Trust - Ensuring Success with Data Integrity Webinar
AI You Can Trust - Ensuring Success with Data Integrity WebinarAI You Can Trust - Ensuring Success with Data Integrity Webinar
AI You Can Trust - Ensuring Success with Data Integrity Webinar
 
Empowering Africa's Next Generation: The AI Leadership Blueprint
Empowering Africa's Next Generation: The AI Leadership BlueprintEmpowering Africa's Next Generation: The AI Leadership Blueprint
Empowering Africa's Next Generation: The AI Leadership Blueprint
 
IESVE Software for Florida Code Compliance Using ASHRAE 90.1-2019
IESVE Software for Florida Code Compliance Using ASHRAE 90.1-2019IESVE Software for Florida Code Compliance Using ASHRAE 90.1-2019
IESVE Software for Florida Code Compliance Using ASHRAE 90.1-2019
 
UWB Technology for Enhanced Indoor and Outdoor Positioning in Physiological M...
UWB Technology for Enhanced Indoor and Outdoor Positioning in Physiological M...UWB Technology for Enhanced Indoor and Outdoor Positioning in Physiological M...
UWB Technology for Enhanced Indoor and Outdoor Positioning in Physiological M...
 
20230202 - Introduction to tis-py
20230202 - Introduction to tis-py20230202 - Introduction to tis-py
20230202 - Introduction to tis-py
 
activity_diagram_combine_v4_20190827.pdfactivity_diagram_combine_v4_20190827.pdf
activity_diagram_combine_v4_20190827.pdfactivity_diagram_combine_v4_20190827.pdfactivity_diagram_combine_v4_20190827.pdfactivity_diagram_combine_v4_20190827.pdf
activity_diagram_combine_v4_20190827.pdfactivity_diagram_combine_v4_20190827.pdf
 
AI Fame Rush Review – Virtual Influencer Creation In Just Minutes
AI Fame Rush Review – Virtual Influencer Creation In Just MinutesAI Fame Rush Review – Virtual Influencer Creation In Just Minutes
AI Fame Rush Review – Virtual Influencer Creation In Just Minutes
 
Salesforce Miami User Group Event - 1st Quarter 2024
Salesforce Miami User Group Event - 1st Quarter 2024Salesforce Miami User Group Event - 1st Quarter 2024
Salesforce Miami User Group Event - 1st Quarter 2024
 
UiPath Studio Web workshop series - Day 6
UiPath Studio Web workshop series - Day 6UiPath Studio Web workshop series - Day 6
UiPath Studio Web workshop series - Day 6
 
Apres-Cyber - The Data Dilemma: Bridging Offensive Operations and Machine Lea...
Apres-Cyber - The Data Dilemma: Bridging Offensive Operations and Machine Lea...Apres-Cyber - The Data Dilemma: Bridging Offensive Operations and Machine Lea...
Apres-Cyber - The Data Dilemma: Bridging Offensive Operations and Machine Lea...
 
Secure your environment with UiPath and CyberArk technologies - Session 1
Secure your environment with UiPath and CyberArk technologies - Session 1Secure your environment with UiPath and CyberArk technologies - Session 1
Secure your environment with UiPath and CyberArk technologies - Session 1
 
Designing A Time bound resource download URL
Designing A Time bound resource download URLDesigning A Time bound resource download URL
Designing A Time bound resource download URL
 
Igniting Next Level Productivity with AI-Infused Data Integration Workflows
Igniting Next Level Productivity with AI-Infused Data Integration WorkflowsIgniting Next Level Productivity with AI-Infused Data Integration Workflows
Igniting Next Level Productivity with AI-Infused Data Integration Workflows
 
Anypoint Code Builder , Google Pub sub connector and MuleSoft RPA
Anypoint Code Builder , Google Pub sub connector and MuleSoft RPAAnypoint Code Builder , Google Pub sub connector and MuleSoft RPA
Anypoint Code Builder , Google Pub sub connector and MuleSoft RPA
 
Building Your Own AI Instance (TBLC AI )
Building Your Own AI Instance (TBLC AI )Building Your Own AI Instance (TBLC AI )
Building Your Own AI Instance (TBLC AI )
 
UiPath Community: AI for UiPath Automation Developers
UiPath Community: AI for UiPath Automation DevelopersUiPath Community: AI for UiPath Automation Developers
UiPath Community: AI for UiPath Automation Developers
 
Building AI-Driven Apps Using Semantic Kernel.pptx
Building AI-Driven Apps Using Semantic Kernel.pptxBuilding AI-Driven Apps Using Semantic Kernel.pptx
Building AI-Driven Apps Using Semantic Kernel.pptx
 
UiPath Studio Web workshop series - Day 7
UiPath Studio Web workshop series - Day 7UiPath Studio Web workshop series - Day 7
UiPath Studio Web workshop series - Day 7
 
NIST Cybersecurity Framework (CSF) 2.0 Workshop
NIST Cybersecurity Framework (CSF) 2.0 WorkshopNIST Cybersecurity Framework (CSF) 2.0 Workshop
NIST Cybersecurity Framework (CSF) 2.0 Workshop
 

Oscillatory flow and particle suspension in a fluid through an elastic tube

  • 1. Mathematical Theory and Modeling ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.3, No.11, 2013 www.iiste.org Oscillatory Flow and Particle Suspension in a Fluid Through an Elastic Tube P. N. Habu* M. A. Mbah Department of Mathematics, Federal University Lafia, P. M. B 146, Lafia, Nigeria. * E-mail: peterhabu68@gmail.com Abstract Womersley gave a solution for the case of a thin-walled elastic tube, it being assumed that the effect of the inertia term in the equations of viscous fluid motion can be neglected. He did not consider the presence of particles, to account for the blood cells in the blood, within the viscous flow through the tube (artery). In this paper, the corresponding solution for an oscillatory flow and particle suspension in a fluid (blood), to account for blood cells, through an elastic tube is obtained. This solution is the frequency equation as it was obtained by Womersley but it has a different structure. If the volume fraction particle density , is removed from this solution it collapses to give the same equation as Womersley’s case, without particles. Keywords: Oscillatory flow, Particle suspension, Elastic tube, Periodic function 1. Introduction: The problem of blood flow and wave propagation in the arterial system has stimulated the interest of physiologists and mathematicians for years. Its fundamental importance for present day research can be traced to Witzig and later to the works of Womersley [10]. The contributions of Womersley [10] represent the best attempt to date in developing a complete, practical, unified theory of arterial blood flow and pressure propagation [6]. A survey of the published literature on the propagation of waves in the arterial system will show a large variation of assumptions on the nature of flow conditions, fluid properties, and types of vessel walls. The analytical solutions for pulsatile flow in rigid tubes are therefore based on a combination of different assumptions, just the same as those of oscillatory flow through elastic tube. The model used in this paper consists of three phases as against two phases by Womersley. The three phases are the fluid phase, particulate phase and the equations of motion of the tube. According to Womersley, the simple solution for the oscillatory motion of viscous liquid in a rigid tube, under a simple-harmonic pressure gradient, was given by Lambossay who gave the formula for velocity and viscous drag solely concerned with the effect of the viscous drag on the frequency response of pressure recording instruments. Womersley obtained the same result independently, in a different form and derived the expression for the rate of flow. For the equations of the elastic tube, we adopt Oslen J.W et al [7] and Womersley [10]. All the assumptions by Womersley are adopted and applied in this paper alongside his method of solution. The only variation between Womersley and this approach is the introduction of the particulate phase, with , as the volume fraction particle density, so that we now simulate blood properly, with proper consideration to the existence of blood cells, as the particles. 2. Methods 2.1 Assumptions and Nomenclature ( , ) denote fluid phase velocities, ( , ) denote particulate phase velocities, and, are the actual is the fluid phase densities of the materials constituting fluid and particulate phase respectively, (1 − ) density, the particulate phase density, P denotes the pressure, denotes the volume fraction density of the particles, is the particle fluid mixture viscosity and S is the drag coefficient of interaction for the force exerted by one phase on the other. Inertia terms of the equations of motion are neglected, diffusivity terms are also neglected. is chosen as constant, and we assume that the pressure wave is harmonic in time having frequency n and wave velocity c. 2.2 Formulation of the Model A porous tapered elastic tube filled with a viscous fluid is considered as a model of a vascular bed in which successive branching of the blood cells leads to a rapid decrease in diameter with distance. The porosity of the tapered tube is adjusted to simulate the effect of branching The motion of the fluid is assumed to be cycisymetric and governed by the Navier-Stokes equations for an incompressible fluid, now modified to simulate the effect of particle suspension as follows: (1− )[ + + ] = −(1− ) +(1− ) [ 109 + + ]+ ( − ) (1)
  • 2. Mathematical Theory and Modeling ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.3, No.11, 2013 (1− )[ + ] = −(1− ) With equation of continuity as, www.iiste.org +(1− ) [ + − + ]+ ( − ) (2) [(1− ) ]+ [(1− ) ]+ = 0, (3) where ! are the axial and radial components of velocity, P is the pressure and and are the density and viscosity of the fluid respectively. The problem of determining the motion of a liquid in an elastic tube when subjected to a pressure – gradient which is a periodic function of the time arises in connection with the flow of blood in the larger arteries. The equations of motion of the tube from [6] are: ( " + = = #$ " # %& − , %# [ ' . - + * " # & & / + + & ]'( + 0 ) ) # [ " + * + & ] (4) (5) Together with the boundary conditions for the motion of the fluid, + = 12=1 = 12=1 The particulate phase: The equations of motion of the particles are: " [ + 3 3 + 3 ]=− [ + + ]=− and their equation of continuity is: [ ]+ [ ]+ 3 = 0 With boundary conditions + = 12=1 3 3 + ( + ( − − ) (7) ) (8) (9) 12=1 " = 3 (6) (10) A 3. Solution: The expression for the drag coefficient for the present study is selected as, = 4 6$ 9 8 (:) 57 , where 89 (:) = ;<=>?; = [5 = ] @ +3 Where C is the fluid viscosity, and ‘a’ is the radius of the particle. Relation where 89 (:) represents the classical Stokes’ drag for small particle Reynolds number, modified to account for the finite particulate fractional volume through the function 89 (:), obtained by [9]. In order to investigate the Pulsatile flow along the axis of the tube, we assume that the pressure wave is harmonic in time having frequency n and wave velocity c. We therefore assume that, (F), 5 (F), 5 (F)] exp [in (t− )] (D, , , , , )= [E (F), (F), (11) G We shall be concerned with the motion in which H& , , 3 , 3 , are small, i.e when the wave velocity, c is very large, as compared to , , , , and nR. G G G G G From (11) we can write P = E (r)exp[in(t− )] I = J= I = J = = G G (r)exp[in(t− )] = (12) 5 (r)exp[in(t− G )] 5 (r)exp[in(t− G )] K =0 KL = − = G (r)exp[in(t− )] A G in exp[in(t− )] A G G exp[in(t− )] A G exp[in(t− )] (13) G exp[in(t− )] 110
  • 3. Mathematical Theory and Modeling ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.3, No.11, 2013 = A = A NH G = = G exp[in(t− )] M exp[in(t− )] =− = www.iiste.org A G A G G exp[in(t− )] G (13 cont.) exp[in(t− )] A G exp[in(t− )] G exp[in(t− )] =− in exp[in(t− )] G G Neglecting the inertia terms in (1), we can write equation (1) as = −(1− ) (1− ) +(1− ) [ + + ]+ ( − ) (14) Now substituting equations (12) and (13) into equation (14) and neglecting the diffusivity terms Equation (14) simplifies to ' A + ' ' + ' ' A − =− NH& Let P = Q( ) R5 ⇒ P 5 = " " Substituting (16) in (15), we get ' A H A − MP 5 =− − & ,A NH 6G & H " & ,A NH 6G − & O 6( & 6( O ) ) [ [ − ] − ] 5 5 (15) (16) (17) (1− ) =−(1− ) +(1− ) [ + − + Now substituting equations (12) and (13) into (18), we get ]+ Similarly in equation (2), neglecting inertia terms, can be written as M =− Let 2 = ⇒ A ' & + ,A # + " A + " A −P + # ( O ) [ 5 − ] T − U ⇒ F = 2Q, therefore equation (18) can be written as A ' − MP 5 ' & H − ' A = & ,A 6 ' − & 6( V ) [ 5 we have, − ] (18) (19) (20) Where P 5 = " For equations (19) and (20), the conservation of mass equation can be written using (12) and (13) and simplifying, we get ( A ') NH& A = (21) ' G ' Equations (19) and (20) are now the equations of motion of the fluid and equation (21) is the conservation of mass equation. 3.1 Evaluating the particulate phase Now neglecting the inertia terms in equation (7), we can write equation (7) as [ 3] = − + + ( − ) Now substituting (12) and (13) into (22), we have V A A = − &[#3 NH<V] #3 NH<V (22) ' (23) =− + ( − ) Substituting equations (12) and (13) into (24), we get V A A NH = + Also equation (8) can similarly be written as 3 G[#3 NH<V] (24) #3 NH<V (25) The equation of conservation of mass of the particles can be written using (12) and (13) as ( ') NH& = ' ' G (26) We can now re-arrange the equations of motion of the particles to be written from (23), (25) and (26) to have: V A A = − #3 NH<V V A = &>#3 NH<V@ A NH G[#3 NH<V] + #3 NH<V ' (27) with the equation of continuity as 111
  • 4. Mathematical Theory and Modeling ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.3, No.11, 2013 ( ' ') ' = www.iiste.org NH& G Now to solve equations (19) and (20) [i.e. equations of motion of the fluid] From equation (23), equation (19) can be written as + A ' + A ' ' ' ' − ' − + WM = P 5 + A + M =P 5Y5 6( & = Equation (28), can be written as A ' A ' A where Y 5 = 1 + + − ' − ' & N Z "A ( O & 6& )T#3 NH<VU + M =[5 = Equation (29) can be written as A ' A ' ' A Where [ = P Y O )T#3 NH<VU & 6& !] =1+( [1 + − − 6( & V N Z "A ( A ' & V 6( ) V X ) ] )(#3 NH<V) = & 6& A [1 + 6( V ) − #3 NH<V ] A ' (28) ' (29) (30) V )(#3 NH<V) Using Bessel function, the solution of equation (30), expressing E = ^ _C (`2), where k is to be determined. Then A = ^ a_ (a2), therefore equation (29) can be written as + A ' = ' A ' ' + M =[5 A − bA c$ (N Z d ') = & 6& ]^ a_ (a2) With the solution complementary and particular solutions respectively, giving, = c$ (N Z d ) b c$ (N Z d ') − & efA cA (e') & 6 NZd = G + , where G ! e are the (31) Similarly equation (20) has the solution, c$ (N Z d ) NH& fA c$ (e') NZd e (32) Now using the approximation, Womersley [10], to equations (31) and (32), where from the equation of continuity of the fluid we get the identities, GA NH& = G d N G N H & Z gfA = G 6 NH&gA⁄ and &e fA 6 ⇒a= , so that _C (a2) becomes _C i G Also, we use the approximation = _C i H&gA⁄ G H&g A⁄ G NH&gA⁄ j=1 G 2j = 1 and _ (a2) becomes M_ i H&gA⁄ G 2j = NH&'gA⁄ 5G And =_ i j. 5G G Inserting these approximations in equations (31) and (32) we get H&'g A⁄ = = − bA cA (N Z d ') c$ (N Z d ) b c$ (N Z d ') c$ (N Z d ) − H&g A⁄ & efA cA (e') & 6 NZd N& HgfA 6GN Z d H&'g A⁄ Since _C (a2) = _C i = b c$ (N Z d ') c$ (N Z d ) + G gfA # G- e . On further simplification we get j = 1 and M = [ 5 − a 5 = M = [ 5 similarly, , after simplification At the inner surface of the tube, i.e. when y=1, equation (33) and (34) become respectively, NH& NH& gfA = k5 l C ([) + 5 G 5 G # G- Where, l C ([) = And = 5cA (N Z d ) d N Z c$ (N Z d ) gfA = k5 + [ # G- + = #$ " # %# %& − ) # [ ' + & ]'( + * " & + + & ] (34) (35) (36) [ + The equations of motion of the tube are: " (33) ) # " * + & ] (37) (38) Together with the boundary conditions for the motion of the liquid, + = 12=1 112
  • 5. Mathematical Theory and Modeling ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.3, No.11, 2013 www.iiste.org = 12=1 If it is now assumed that from (12) and (13) 8 = m exp [M (1 − q⁄k )] and P = r exp [M (1 − q⁄k )], where m and r are arbitrary constants, the boundary conditions for and , using the boundary conditions for the motion of the liquid we have: + = m inexp[M (1 − q⁄k )] = (39) But = m exp [M (1 − q⁄k )] from (39) Therefore = m M (40) From (36), we can write (37) as NH& gfA M m = [k5 l C ([) + ] (41) " 5 G # G- Also = r inexp[M (1 − q⁄k )] = But from (12), = exp [M (1 − q⁄k )], this implies " M r = = k5 + gfA # G- Therefore M r = k5 + ] gfA # G- ] (43) From (37) and (40), the equations of the tube become f ) c NHu v m 5 = A − [ i− Aj + A −r (42) %# # & G & # " 5 = 5 = W− M [ k5 l C ([) %#& 5 + H & g G # - X + .− ) # H uA G + i− c & NHvA G (44) j0, (45) Equations (41), (43), (44) and (45) are four homogeneous equations in the arbitrary constants w, ^ ,k5 , m and r . Eliminating them will give a frequency equation, which will determine the wave velocity c, in terms of the elastic properties of the tube and the non-dimensional parameter [. The result of the elimination is: x kY 5 1 M Ql C 2k M Qx 2k 5 Y 5 −[ P 5 1 ℎ −M 5 0 Q5 x + 5 Px 5 Q ] 2k = Y 5 ℎQ −M 0 N# &H- − A$ 5%# − 0 ) #& NHc{ #&G NHc{ 5 (1 − #&G ) #G = 0 (46) ) H & Operating on the rows and columns of equation (46) and neglecting and continuing operating on rows and G d columns until we get x 1 0 1 g A$ −1 0 | 5 5 | } *} =0 (47) 0 |e | e e *} } 1− 0 − l CY5 where a = %# # & 5 , !L= e) # G Equation (47) can be written as x 1 x lC | 5 0 | Y e 0 −2 −L −~L e 1 0 | −~L | = 0 (a − L) 0 5 Whose solution is g [x(1 − ~ 5 )(1 − l C )]L 5 − .2Y 5 + ax(1 − l C ) + l C Y 5 i − ~ − ~xj0 L + 2aY 5 + l C Y ; = 0 5 This reduces to A$ - e 113 (48) (49)
  • 6. Mathematical Theory and Modeling ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.3, No.11, 2013 www.iiste.org (1 − ~ 5 )(1 − l C )L 5 − .2 + a(1 − l C ) + l C i − 2~j0 L + 2a + l C = 0 5 I.e. when x = 1 and Y 5 = 1 in (48), i.e., when = 0, since x=( < V )(#3 NH<V) and Y 5 = G Z "A ( & V )(#3 NH<V) − ( (50) & V )N Z "A Using Womersley [10], the roots of equation (49) are given by (1 − ~ 5 )L = • ± [• 5 − (1 − ~ 5 )•] R5 where • = ‚ A W ƒ <„ *X ( A$ ) + + e 5 *W When Y 5 = 1 and x = 1, i.e. A$ ‚ i < A jX AT ƒ „ ( A$ ) (51) A$ - U = 0 equation (51) reduces to • = [10] solution for the case without suspension and the same as [6]. 4. Conclusion In terms of the notation used in the case of the solid case, …A$ A . < *0 „ ( A$ ) (52) + + ~ − , which is Womersley’s e 5 ; = exp(−M†) /]9 C so that the quantities required to compute the roots of equation (49) are already available. When ˆ C ([) is complex, L is always complex and the motion is either damped or unstable. If we write ((1 − ~ 5 ) LR2) R5 = ‰ − MŠ and denote by wC the velocity for the perfect fluid, then if w is the w wave-velocity, CRw = ‰ and over a distance of one wavelength the amplitude will be reduced to the ratio exp (−2‹ ŠR‰ ) [6] [10]. Since Womersley’s case without suspension is considered as the first best description of blood flow in arteries, we have now further proved that our model is in line with this best description of blood flow and the propagation of waves in the arterial system. Acknowledgement We wish to acknowledge the contribution of Dr. L. M. Srivastava towards his guide in the M.Sc thesis (1986), “Flow of Particle-Fluid Suspension in Elastic Tube”, at the Department of Mathematics, A.B.U, Zaria, Nigeria. This contribution is one of the two major sources, which made this publication possible. Also, to Associate Professor M. E. Nja of the Mathematics Department, Federal University Lafia, Nigeria. For his great motivation and encouragement for this research. We thank you all. References 1. Atabek, H.B. (1996). Wave propagation through a viscous incompressible fluid contained in an initially stressed elastic tube. Biophysical Journal volume 6, pp 481-501 2. Cox, R.H. (1968) “Wave propagation through a Newtonian fluid contained within a thick-walled, viscoelastic tube”, Biophysical Journal, volume 8, pp 691-708. 3. Evans, R.L. “Pulsatile flow in vessels whose distinsibility and size vary with site”. Physics in Medicine and Biology, volume 7, pp 105-115. 4. Iberall, S.A. (1964) “study of the General Dynamics of the physical-chemical systems in Mammals”, NASA CR-120 5. Lambert, J. W. (1958). “On the non-linearities of fluid flow in nonrigid tubes”, J. Franklin inst. 266, 83. 6. Ngbo, P.H, (1986), “Flow of Particle-Fluid Suspension in Elastic Tube”, M.Sc Thesis (unpublished), Department of Mathematics, A.B.U, Zaria, Nigeria. 7. Oslen, J.H. et al (1967) “Large- amplitude unsteady flow in liquid filled elastic tubes”, J. Fluid mech, vol 33, pp 513. 8. Pennisi, L.L. (1976) “Elements of complex variables”, Holt, Rinehart and Winston, pp 17-23. 9. Tam, C.K.W. (1969) “The drag on a cloud of spherical particles in low Reynolds number flow”, J. Fluid Mech, volume 38, pp 537-546. 10. Womersley, J.R. (1955) Oscillatory motion of a viscous liquid in a thin-walled elastic tube 1. The linear approximation for long waves, philosophical mag, volume 46, pp 199-215. 114
  • 7. This academic article was published by The International Institute for Science, Technology and Education (IISTE). The IISTE is a pioneer in the Open Access Publishing service based in the U.S. and Europe. The aim of the institute is Accelerating Global Knowledge Sharing. More information about the publisher can be found in the IISTE’s homepage: http://www.iiste.org CALL FOR JOURNAL PAPERS The IISTE is currently hosting more than 30 peer-reviewed academic journals and collaborating with academic institutions around the world. There’s no deadline for submission. Prospective authors of IISTE journals can find the submission instruction on the following page: http://www.iiste.org/journals/ The IISTE editorial team promises to the review and publish all the qualified submissions in a fast manner. All the journals articles are available online to the readers all over the world without financial, legal, or technical barriers other than those inseparable from gaining access to the internet itself. Printed version of the journals is also available upon request of readers and authors. MORE RESOURCES Book publication information: http://www.iiste.org/book/ Recent conferences: http://www.iiste.org/conference/ IISTE Knowledge Sharing Partners EBSCO, Index Copernicus, Ulrich's Periodicals Directory, JournalTOCS, PKP Open Archives Harvester, Bielefeld Academic Search Engine, Elektronische Zeitschriftenbibliothek EZB, Open J-Gate, OCLC WorldCat, Universe Digtial Library , NewJour, Google Scholar