Modulation of Nonlinear Waves in Viscous Fluid Contained in Stenosed Elastic Tube

Abstract

In the past studies, most of the researchers were less focused on wave modulation compared to wave propagation in the artery. The study of wave modulation in the arteries is rather difficult to construct because the mathematical model involves complex solution. Therefore, the purpose of this study is to investigate the non-linear wave modulation in a stenosed elastic tube filled with viscous fluid. The artery is considered as an incompressible, pre-stressed, thin-walled and long elastic tube with a symmetrical stenosis while the blood is assumed as an incompressible viscous fluid. The stretched coordinates and asymptotic series were introduced to the non-dimensional equations of tube and fluid. By implementing the method of Reductive Perturbation (RPM), a sets of non-linear differential equations of various orders is obtained. Solving these differential equations will results in the dissipative non-linear Schrodinger (NLS) equation with variable coefficients. Analytical solutions for the dissipative NLS equation with variable coefficients will be carried out. Based on the graphical output, it is noticed that, when blood flowing in a stenosed elastic tube, the radial displacement decreases gradually due to the resistance of fluid flow. It is observed that increase the blood viscosity caused an increase in the pressure to walls of arteries consist stenosis. Besides, it is found that the wave amplitude decreases obviously when the viscous effect of fluid increases. Other than that, the wave speed also increase rapidly since the cross-sectional area of artery reduced due to the existence of the stenosis.