Nonlinear Schrodinger Equation In AnInviscid 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 HumanDevelopment,UniversitiTun Hussein Onn Malaysia, 86400 Parit Raja, Batu Pahat, Johor, Malaysia.3 Department of Communication Engineering, Faculty of Electric andElectronic,UniversitiTun Hussein Onn Malaysia, 86400 Parit Raja, Batu Pahat, Johor, Malaysia.
In thisresearch, by employing the approximation equations of an incompressibleinviscid fluid and non-linear equations of an incompressible, isotropic andthick elastic tube, the modulation of nonlinear wave modulation wave in thickelastic tube filled with inviscid fluid is studied. By use of reductiveperturbation method, we obtained Nonlinear Schrodinger (NLS) type equation asthe evolution equation. Keywords: inviscid fluid; isotropic; thick elastic tube; nonlinear wavemodulation.1.
IntroductionIn the study of biology, scientists have found that humanblood circulatory system is complex. The circulatory framework is focused onthe heart, that functioning as a muscular organ that rhythmically pumps. Togive support and help in battling sicknesses, the circulatory framework permitsblood in human body to course and transport supplements, oxygen, carbon dioxide,hormones and platelets to human body. Blood in human body moves travels throughveins, and heart funtioning as a pump for the blood to moves. Blood exists to provide oxygen that human body need and nourishment tothe organs and tissue and to collect waste susbtances. Human circulatorysystem or cardiovascular system is closed, which is the blood always be in thenetwork of arteries, vein and cappilaries.In 1578 to 1628, William Harvey was an English physicianthat first known to potray totally and in detail the deliberate course andproperties of blood being pumped to the body by the heart.
In his exploration,Harvey concentrated more on mechanics of blood stream in the humanbody. By observing the motion of the heart in living animal, Harvey able to seethat systole was the dynamic period of the heart’s development, pumping out theblood by its solid withdrawal. Harvey additionally utilized numericalinformation to demonstrate that the blood was not being devoured. At long last,Harvey proposed the presence of little fine anastomoses amongst supply routesand veins, yet these were not found until 1661 by Marcello Malpighi.Arterial wall is involving thetransport phenomena of blood solutes 1. The scientific conditions of liquidelements are the key segments of haemodynamics demonstrating. Blood is asuspension of particles in a liquid called plasma, that made up from water(90-92%), proteins (7%) and inorganic constituents.
Velocity and pressure arethe principal quatities that describe blood flow in arteries 2. Weight, speedand vessel divider removal will be capacity of time and the spatial position. Afeature of blood flow in human body is represented by its pulsatality. Withsome approximation the blood flow to be periodic in time. The size of arterieshave its own effect on sheer stress, for example the rate of sheer stresseswill be very low and the blood in arteries is treated as a non-Newtonion fluid3.
The propagation of pressure pulses in fluid-filled has beenstudied by several research workers (Pedley 4 and Fung 5). The blood wall material is known to be incompressible, anisotropic andviscoelastic. 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 graduatedfrom Istanbul Technical University have done many researches regarding toarterial waves. In 1998, Demiray done his research on investigation of the nonlinear waves in a fluid-filledthick elastic tube. In this research, Demiray used reductive perturbationmethod to solve the fluid and tube equations and it is demonstrate that theabundacy tweak of these waves is administered by a nonlinear Schrodinger (NLS)equation 7. Then, in 1999 Demiray studied a research onpropagation of weakly nonlinear waves in fluid-filled thick viscoleastic tube.
To investigate the propagation of weakly nonlinear wave in the long-waveapproximation, 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-deVries-Burgers’ (KdVB) equation and the generalized Burgers’ equation 8. In 2001, Demiray investigated on modulation ofnon-linear waves in a viscous fluid contained in a elastic tube.
He utilizedthe reductive perturbation technique for this model and then the dissipativenon-linear Schrodinger equation is obtained. Next, in 2005 the research ofhead-on collision of solitary waves in fluid-filled elastic tube was done byDemiray. In this research he used extended Poincare-Lighthill-Kuo (PLK)perturbation method. Result of this research showed that the head-on collisionof solitary waves is elastic 9. After that, in 2007 Demiray investigateddoing research on waves in a fluid-filled elastic tube with a stenosis for thevariable coefficient KdV equations. By utilizing the reductive perturbationtechnique, the variable coefficients KdV and changed KdV equations are acquired10.In the present work, treating the arteries as a thick elastic tube andthe blood as full inviscid fluid, we have studied the amplitude modulation ofnonlinear waves in a thick elastic tubefilled with full inviscid fluid by using the method of reductive perturbation.
The governing evolution equation for this solution is nonlinear Schrodingerequation with variable coefficients. 2. Basic equations and theoretical preliminariesHuman blood circulatory system is a part of knowledge thatneed to be explore and biology scientist have found that human bloodcirculatory system is complex. Because ofcomplex of blood flow phenomena, it is described analytically and numericallyby many researchers. Human blood is known to beNewtonion fluid but in this research, as simplicity the blood is assumed to beinviscid fluid. The equation that represent theconservation of mass may be given by (1) where is theinner cross-sectional area of the tube, is the axial velocity of the fluid, is the parameter of time and standfor the axial coordinate for a point in the cylindrical coordinate system.Thebalance of the linear momentum in the axial direction equation may be given by (2)where stand for themass of density and stand for thepressure of the fluid medium.
In this research, the tube isassumed to be thick elastic, isotropic and incompressible. The equation ofthick elastic tube is defined as (3) where used asthe mass density of the tube material, stands for radial accelerationcomponent, is defined as cylindrical polar coordinates ofmaterial and as an unknown function.Thesolution of the differential equation (3) must satisfy the given boundaryconditions (4) where and is theinner and outer radii of the tube after the deformation of finite static and is given by (5)Next, the following dimensionlessquantities are introduce (6)where stand for theinner radius in the undeformed configuration of tube, is the speed of wave and stand for the ratio of the deformed innercross-sectional area after static deformation to the undeformed cross-sectionalarea.
Firstly we need to introduce equation (6) into equations(1) – (5), we will obtain (7) (8) (9) (10) (11)The tube pressure can be expressedas follows: (12) where (i = 1, 2, …, 14) coefficients are to be determined from thedifferential equations (16) and the boundary conditions equations (10). 3. Long wave approximationThe amplitude of weakly nonlinear wave modulation in afluid-filled thick elastic tube will be examine in this section.Now, the following stretchingcoordinate will be introduced (13)where stand for smallparameter, is a constantthat will be represent the group velocity. Here are the field quantities arefunctions for both fast and slow variables. Thefollowing substitution are introduced (14)Substitute equation (14) into thefield equations (7) and (8), we obtained (15) (16)similar substitution for equation (12).Then, the field quantities will beexpand into an asymptotic series of as follow: (17)where stand for afunction of both fast and slow variables. By applying equation (17) intoequations (12), (15) and (16), the following sets of differential equations areobtained: order equations: (18) orderequations: (19) order equations: (20)3.
1 Solution of the field equationsTo 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 slowvariables and . Next, substitute equation (21) into equations (18)and requiring non-vanishing solution for and , we obtained (22)and its providedthe following dispersion relation (33)where is thefrequency of angular and stand for thewave number. The group of velocity will be given by (24) is an unknownfunction that will be obtained later.For the solution of orderequations, it suggest us to looking for a solution of the following form (25)where are functionsof the slow variables of and . By applying equation (25) into equations (19) we obtained (26) (27)For the solutionof orderequations, we need to introduced equations and as follow: (28)To solveequations (20), we need to substitute equation (28) into equation (20) and wewill obtain: (29)By eliminating and utilizingequations (27) and (29) the equation of nonlinear Schrodinger equation isobtained (30) The coefficients and are defined by: 4.
Conclusion In this research, themodulation of fluid-filled thick elastic tube has been studied by applying thereductive perturbation method analytically. We considered blood asincompressible inviscid fluid and artery as thick elastic tube. At the end ofthis study, it is shown that the modulation of these waves is governed bynonlinear Schrodinger (NLS) equation.
References1. T. E.
Diller, B. B.Mikic, and P.
A. Drinker, Shear-inducedaugmentation of oxygen transfer in blood, J. Biomech. Engg., 102 (1980),pp. 67-72.2. A.
Quateroni, A.Veneziani, and P. Zunino, Mathematicaland numerical modelling of solute dynamic in blood flow and arterial walls, SIAMJ.
Sci. Comput., 39 (2001), pp. 1488-1511.3. Nichols, W.
W. and O’Rourke, M. F. McDonald’s Blood Flow inArteries: Theoretical, Experimantal and Clinical Principle. 5thedtion.
London: Hodder Amold, (2005).4. T. J. Pedley, Fluid machanics of large blood vessels, CambridgeUniversity Press, Cambridge, 1980.5. Y.
C. Fung, Biodynamics: Circulation, Springer-Verlag,New York, (1981).6. Lambossy, P. Apereu et historique sur le problem de lapropagationdes 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 afluid-filled thick viscoelastic-tube. Applied Mathematical Moddeling, 23(1999), pp.
779-798.9. Demiray. H, Head-on collision of solitary waves influid-filled elastic tube. Applied Mathematics Latters., 18 (2005), pp.
941-950.10. Demiray. H, Waves in a fluid-filled elastic tube with astenosis: Variable coefficients KdV equations. Journal of Computational andApplied Mathematics., 202 (2007), pp.