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)

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]

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.

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