Propagation of Nonlinear Pressure Waves in Blood

The propagation of weakly nonlinear pressure waves in a fluid-filled elastic tube has been investigated. The reductive perturbation method has been employed to derive the Korteweg-de Vries equation for small but finite amplitude. The effect of the final inner radius of the tube on the basic properties of the soliton wave was discussed. Moreover, the conditions of stability and the soliton existence via the potential and the corresponding phase portrait were computed. The applicability of the present investigation to flow problems in arteries is discussed. 1. Introduction The intermittent ejection of blood from the left ventricle produces pressure waves that flow in the arterial tree. Experimental data found that the flow velocity in blood vessels largely depends on the elastic properties of the vessel wall and they propagate towards the periphery with a characteristic diagram [1]. In arterial mechanics, the propagation of pressure waves in fluid-filled distensible tubes has been theoretically studied by several researchers [2–4]. Experimental observation for the simultaneous changes in amplitude of the pressure waves at five sites from the ascending aorta to the saphenous artery in dogs showed that the pulsatile character of the blood is soliton waves [5]. Yomosa [6] investigated the nonlinear propagation of solitary waves in large blood vessels. He found that the pulse waves of pressure and flow propagating through the arteries can be described as solitary waves excited by cardiac ejections of blood, and the features of the pulse wave such as “peaking” and “steepening” are interpreted in the viewpoint of soliton. Later, R. M. Shoucri and M. M. Shoucri studied the application of the method of characteristics of shock waves in models of blood flow in the Aorta [7]. Recently, Gaik and Demiray [8] treated the arteries as an incompressible prestressed thin walled elastic tube with a stenosis and the blood as a Newtonian fluid with variable viscosity, which vanishes on the arterial wall, and it takes the maximum value at the center of the artery. They studied the propagation of weakly nonlinear waves in the long wave approximation by the use of the perturbation methods [9]. Many authors examined the stability and the soliton existence condition via the potential and the corresponding phase portrait [10–12]. Elwakil et al. [10] studied the propagation of solitary electron acoustic waves in unmagnetized collisionless plasma. They found that, there are saddle and two-center equilibrium state in the phase plane and there is two-finite separatrix going from a