DEMIRAY. zyxwvutsrqp H.; ANTAR, N.: Wave Characteristics in Elastic Tubes zyxwvut 52 1 ZAMM zyxwvutsrqp . zyxwvutsrqp Z. angew. Math. Mech. 76 (1996) 9, 521 -530 DEMIRAY, H.; ANTAR, N Effects of Initial in Elastic Tubes Stresses and Wall Thickness on Wave Akadernie Verlag Characteristics Unter der Verwmiung der Theorie der kleinen Deformationen, die groJen statischen Deformationen ant Eingung iiberlagert lcerden, wird die Ausbreitung harmonischer Wellen in einer vorgespannten dicken elastischen Rohre, die mit einenl ciskosen Fluid gefllt ist, untersucht. Aufgrund der Veranderlichkeit der KoefSienten der resultierenden Differentialgleichung der Rollre werden die Feldgleichungen mit einer Potenzreihenmethode gelost. Unter Verwendung der richtig gestellten Grenzbedingungen, die die Reaktion zyxwvutsr des Fluids init der Rohrenwand charakterisieren, wird die Dispersionsbezieliung als Funktion der Anfangs- deformationen und der geometrischen Kennzeichen erhalten. Die Dispersionsbeziehung wird numerisch und - wenn moglich - analytisch gepriift, und die Resultate werden graphisch dargestellt. Es zeigt sich, daJ die WellenReschivindigkeiien tnit dem Dickenparameter wachsen. Employing the theory of small deformations superimposed on large initial static deformations, the propagations zyx qf harmonic waves in an initially stressed thick elastic tube filled with a viscous fluid is studied. Due to variability of the coefficients of the resulting dfferentiaf equation of the tube, theJeld equations are solved by u power series method. Utilizing the properly posed boundary conditions that characterize the reaction offzuid with the tube wall, the dispersion relation is obtained as a junction of initial deformations and the geometrical characteristics. The dispersion equation is examined analjtically, whenever it is possible, and numerically, and the results are depicted on some graphs. It is observed that wave speeds increase with thickness parameter. MSC (1991): 73D15 1. Introduction Propagation of harmonic waves in initially stressed (or unstressed) elastic (or viscoelastic) tubes filled with viscous (or inviscid) fluids is a problem of interest since the time of THOMAS YOUNG who first determined the speed of pulse waves in human arteries. This subject, in particular, had received considerable interest and activity among the research workers in the last two decades or so. The current literature on the subject is so rich that it is almost impossible to cite all those contributions here. The historical evolution of the subject may be found in the books by ATTINGER [l], MCDONALU [2] and FUNG [3] and in the papers by LAMBOSSY [4] and SKALAK [5]. Significant contributions on wave motion in the elastic tubes filled with a viscous fluid have been made by WITZIG [6], MORGAN and KIELY [7], WOMERSLEY [8], ATABEK and LEW [9], MIRSKY [lo] and more recently by RACHEV [ll] and KUIKEN [12]. Nearly, in the all these works either the initial stress had been neglected and/or the artery had been treated as a thin walled cylindrical membrane. As is well known thin shell theories are applicable only when the ratio of thckness to the mean radius is less than 1/20. For a healthy young human being the systolic pressure is about 120 mm Hg, diastolic pressure is around 80 mm Hg and the mean pressure is 100 mm Hg. Moreover, in physiological conditions, the arteries are subjected to an axial stretch which is about 1.5. Considering these observations one may assume that the arteries are subjected to a large static inner pressure and an axial stretch, and in the course of blood flow, +_20 mm Hg pressure increment is added on this initial static field. Furthermore, even for large blood vessels the ratio of thickness to mean radius changes between 1/4 - 1/6. Therefore, the arteries are also thick walled, and thin shell theories cannot be applied to arterial mechanics. Considering the above described physiological conditions, in the present work we shall study the propagation of harmonic waves in a fluid filled and prestressed thick elastic tube by making use of the field equations and the boundary conditions of the theory so called “small deformations superimposed on large initial static deformations”. For simplicity in mathematical analysis, the arterial wall is taken as an isotropic and incompressible elastic material whereas the blood as an incompressible Newtonian fluid. The governing differential equations of incremental motion of the both fluid and solid body are obtained in the cylindrical polar coordinates. Due to variability of the initial stress through the thickness, the coefficients of the resulting differential equation of the tube are also variable. Although a closed form of solution to differential equations governing the fluid body is possible, due to variable character of the coefficients of the differential equations of solid body, a truncated power series solution is presented in terms of thickness coordinate (y). Keeping the terms up to (yz) and utilizing the boundary conditions, the dispersion relation is obtained as a function of inner pressure, axial stretch, thickness ratio, frequency, and the wave number. After examining some special cases, the general dispersion equation is analysed by numerical means and the results are depicted on some graphs. 2. Basic equation Due to the interaction of blood with its container, the pulsatile motion of heart leads to wave phenomena both in blood and arteries. Therefore, the governing equations and the boundary conditions should include these interactions. 35a 2. angew. Math. Mech., 6d. 76, H. 9