Nonlinear body to course and transport supplements, oxygen, carbon

Nonlinear Schrodinger Equation In An
Inviscid Fluid-filled Thick Elastic Tube

Nur Fara Adila Binti Ahmad1, a), Choy Yaan Yee2, b), and Tay Kim Gaik 3, c)

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1,2 Department of Mathematics, Faculty of Science, Technology and Human
Development,

Universiti
Tun Hussein Onn Malaysia, 86400 Parit Raja, Batu Pahat, Johor, Malaysia.

3 Department of Communication Engineering, Faculty of Electric and
Electronic,

Universiti
Tun Hussein Onn Malaysia, 86400 Parit Raja, Batu Pahat, Johor, Malaysia.

 

a) [email protected]
b)[email protected]

c)[email protected]

 

Abstract. In this
research, by employing the approximation equations of an incompressible
inviscid fluid and non-linear equations of an incompressible, isotropic and
thick elastic tube, the modulation of nonlinear wave modulation wave in thick
elastic tube filled with inviscid fluid is studied. By use of reductive
perturbation method, we obtained Nonlinear Schrodinger (NLS) type equation as
the evolution equation.

Keywords: inviscid fluid; isotropic; thick elastic tube; nonlinear wave
modulation.

1.  Introduction

In the study of biology, scientists have found that human
blood circulatory system is complex. The circulatory framework is focused on
the heart, that functioning as a muscular organ that rhythmically pumps. To
give support and help in battling sicknesses, the circulatory framework permits
blood in human body to course and transport supplements, oxygen, carbon dioxide,
hormones and platelets to human body. Blood in human body moves travels through
veins, and heart funtioning as a pump for the blood to moves. Blood exists to provide oxygen that human body need and nourishment to
the organs and tissue and to collect waste susbtances. Human circulatory
system or cardiovascular system is closed, which is the blood always be in the
network of arteries, vein and cappilaries.

In 1578 to 1628, William Harvey was an English physician
that first known to potray totally and in detail the deliberate course and
properties of blood being pumped to the body by the heart. In his exploration,
Harvey concentrated more on mechanics of blood stream in the human
body. By observing the motion of the heart in living animal, Harvey able to see
that systole was the dynamic period of the heart’s development, pumping out the
blood by its solid withdrawal. Harvey additionally utilized numerical
information to demonstrate that the blood was not being devoured. At long last,
Harvey proposed the presence of little fine anastomoses amongst supply routes
and veins, yet these were not found until 1661 by Marcello Malpighi.

Arterial wall is involving the
transport phenomena of blood solutes 1. The scientific conditions of liquid
elements are the key segments of haemodynamics demonstrating. Blood is a
suspension of particles in a liquid called plasma, that made up from water
(90-92%), proteins (7%) and inorganic constituents. Velocity and pressure are
the principal quatities that describe blood flow in arteries 2. Weight, speed
and vessel divider removal will be capacity of time and the spatial position. A
feature of blood flow in human body is represented by its pulsatality. With
some approximation the blood flow to be periodic in time. The size of arteries
have its own effect on sheer stress, for example the rate of sheer stresses
will be very low and the blood in arteries is treated as a non-Newtonion fluid
3.

The propagation of pressure pulses in fluid-filled has been
studied by several research workers (Pedley 4 and Fung 5). The blood wall material is known to be incompressible, anisotropic and
viscoelastic. However, for its mathematical simplicity in nonlinear analysis,
the arterial wall material will be assumed to be incompressible, homogenous,
isotropic and elastic 6.

In the previous years, Hilmi Demiray who graduated
from Istanbul Technical University have done many researches regarding to
arterial waves. In 1998, Demiray done 
his research on investigation of the nonlinear waves in a fluid-filled
thick elastic tube. In this research, Demiray used reductive perturbation
method to solve the fluid and tube equations and it is demonstrate that the
abundacy tweak of these waves is administered by a nonlinear Schrodinger (NLS)
equation 7. Then, in 1999 Demiray studied a research on
propagation of weakly nonlinear waves in fluid-filled thick viscoleastic tube.
To investigate the propagation of weakly nonlinear wave in the long-wave
approximation, he used reductive perturbation method. By a legitimate scaling,
its demonstrated that the general equation lessens to advancement conditions,
for example, Korteweg-de Vries equation, Burgers’ equation, Korteweg-de
Vries-Burgers’ (KdVB) equation and the generalized Burgers’ equation 8.

 In 2001, Demiray investigated on modulation of
non-linear waves in a viscous fluid contained in a elastic tube. He utilized
the reductive perturbation technique for this model and then the dissipative
non-linear Schrodinger equation is obtained. Next, in 2005 the research of
head-on collision of solitary waves in fluid-filled elastic tube was done by
Demiray. In this research he used extended Poincare-Lighthill-Kuo (PLK)
perturbation method. Result of this research showed that the head-on collision
of solitary waves is elastic 9. After that, in 2007 Demiray investigated
doing research on waves in a fluid-filled elastic tube with a stenosis for the
variable coefficient KdV equations. By utilizing the reductive perturbation
technique, the variable coefficients KdV and changed KdV equations are acquired
10.

In the present work, treating the arteries as a thick elastic tube and
the blood as full inviscid fluid, we have studied the amplitude modulation of
nonlinear waves in a thick elastic  tube
filled with full inviscid fluid by using the method of reductive perturbation.
The governing evolution equation for this solution is nonlinear Schrodinger
equation with variable coefficients.

 

2. Basic equations and theoretical preliminaries

Human blood circulatory system is a part of knowledge that
need to be explore and biology scientist have found that human blood
circulatory system is complex. Because of
complex of blood flow phenomena, it is described analytically and numerically
by many researchers. Human blood is known to be
Newtonion fluid but in this research, as simplicity the blood is assumed to be
inviscid fluid. The equation that represent the
conservation of mass may be given by

                                                      

                                                                   (1)                                                                        

where

 is the
inner cross-sectional area of the tube,

is the axial velocity of the fluid,

is the parameter of time and

 stand
for the axial coordinate for a point in the cylindrical coordinate system.The
balance of the linear momentum in the axial direction equation may be given by

                                                    

                                                               (2)

where

 stand for the
mass of density and

 stand for the
pressure of the fluid medium.

In this research, the tube is
assumed to be thick elastic, isotropic and incompressible. The equation of
thick elastic tube is defined as                        

                                  (3) where

 used as
the mass density of the tube material,

 stands for radial acceleration
component,

is defined as cylindrical polar coordinates of
material and

as an unknown function.The
solution of the differential equation (3) must satisfy the given boundary
conditions

                               

                                                    (4)                                                          

where

 and

 is the
inner and outer radii of the tube after the deformation of finite static and

 is given by

                                                        

                                                                          (5)

Next, the following dimensionless
quantities are introduce

  

                                                       (6)

where

 stand for the
inner radius in the undeformed configuration of tube,

is the speed of wave and

stand for the ratio of the deformed inner
cross-sectional area after static deformation to the undeformed cross-sectional
area. Firstly we need to introduce equation (6) into equations
(1) – (5), we will obtain

                                                                                                                                                (7)                                                                               

                                                    
                                                                                                        (8)

                                                                          (9)

                                                                             (10)

                                                                                                                                             (11)

The tube pressure can be expressed
as follows:

                                                       (12)       

where

 (i = 1, 2, …, 14) coefficients are to be determined from the
differential equations (16) and the boundary conditions equations (10).

 

3. Long wave approximation

The amplitude of weakly nonlinear wave modulation in a
fluid-filled thick elastic tube will be examine in this section.

Now, the following stretching
coordinate will be introduced

                                                                                              (13)

where

 stand for small
parameter,

 is a constant
that will be represent the group velocity. Here are the field quantities are
functions for both fast

 and slow

 variables. The
following substitution are introduced

            

                                                                                       (14)

Substitute equation (14) into the
field equations (7) and (8), we obtained

                                                                (15)

                                                                                     (16)

similar substitution for equation (12).

Then, the field quantities will be
expand into an asymptotic series of

 as follow:

                                                  
(17)

where

 stand for a
function of both fast and slow variables. By applying equation (17) into
equations (12), (15) and (16), the following sets of differential equations are
obtained:

 order equations:

                 
                                          (18)

 

 

 

order
equations:

 

                                                        (19)

 order equations:

                                    
(20)

3.1 Solution of the field equations

To obtain the solution of

 order equations,
we introduced a harmonic wave type solution as

                                   (21)

where

 and

stand for the complex amplitude functions of the slow
variables

 and

. Next, substitute equation (21) into equations (18)
and requiring non-vanishing solution for

 and

 , we obtained

                                                        (22)

and its provided
the following dispersion relation

                                                                                           (33)

where

 is the
frequency of angular and

 stand for the
wave number. The group of velocity will be given by

                                                                                            (24)

 is an unknown
function that will be obtained later.

For the solution of

 order
equations, it suggest us to looking for a solution of the following form

                                                               (25)

where

 are functions
of the slow variables of

 and

. By applying equation (25) into equations (19) we obtained

                                            (26)

   

                                                                 (27)

For the solution
of

 order
equations, we need to introduced equations 

 and

 as follow:

                                                                 (28)

To solve
equations (20), we need to substitute equation (28) into equation (20) and we
will obtain:

                                                             (29)

By eliminating

 and utilizing
equations (27) and (29) the equation of nonlinear Schrodinger equation is
obtained

                                                                                              (30)

 

 

 

The coefficients

and

 are defined by:

 

 

4. Conclusion

 In this research, the
modulation of fluid-filled thick elastic tube has been studied by applying the
reductive perturbation method analytically. We considered blood as
incompressible inviscid fluid and artery as thick elastic tube. At the end of
this study, it is shown that the modulation of these waves is governed by
nonlinear Schrodinger (NLS) equation.

 

References

1.       T. E. Diller, B. B.
Mikic, and P. A. Drinker, Shear-induced
augmentation of oxygen transfer in blood, J. Biomech. Engg., 102 (1980),
pp. 67-72.

2.       A. Quateroni, A.
Veneziani, and P. Zunino, Mathematical
and numerical modelling of solute dynamic in blood flow and arterial walls, SIAM
J. Sci. Comput., 39 (2001), pp. 1488-1511.

3.       Nichols, W. W. and O’
Rourke, M. F. McDonald’s Blood Flow in
Arteries: Theoretical, Experimantal and Clinical Principle. 5th
edtion. London: Hodder Amold, (2005).

4.       T. J. Pedley, Fluid machanics of large blood vessels, Cambridge
University Press, Cambridge, 1980.

5.       Y. C. Fung, Biodynamics: Circulation, Springer-Verlag,
New York, (1981).

6.       Lambossy, P. Apereu et historique sur le problem de la
propagationdes ondes dans un liquide compresible enferme dans untube elastique.
He/u/ Physiol. Acta 9 (1951), pp. 145-161.

7.       Demiray. H, Dost. S, Solitary waves in thick walled elastic tube.
Applied Mathematical Modeling, 22 (1998), pp. 83-99.

8.       Demiray. H, Propagation of weakly nonlinear waves in a
fluid-filled thick viscoelastic-tube. Applied Mathematical Moddeling, 23
(1999), pp. 779-798.

9.       Demiray. H, Head-on collision of solitary waves in
fluid-filled elastic tube. Applied Mathematics Latters., 18 (2005), pp.
941-950.

10.    Demiray. H, Waves in a fluid-filled elastic tube with a
stenosis: Variable coefficients KdV equations. Journal of Computational and
Applied Mathematics., 202 (2007), pp. 328-338.