A Mathematical Model and Analytical Solution for Newtonian Fluid Contained in Elastic Tube with Variable Radius

Abstract

Throughout the twentieth century, studies on arterial wave mechanics conducted by mathematicians were focused on wave propagation in a prestressed thin-walled elastic tube filled with different types of blood. In this paper, the main objective is to propose a mathematical model for wave propagation in blood flow. Here, the artery represents a prestressed thin-walled elastic tube with a variable radius and the blood is an incompressible Newtonian fluid. First, the dimensional equations of fluid and tube are reduced to non-dimensional equations by introducing non-dimensional quantities. Then, the dimensionless equations of tube and fluid are reduced to nonlinear differential equations in various orders through the reductive perturbation method. These differential equations are solved to obtain the Korteweg-de Vries (KdV) equation with a variable coefficient, and the analytical solution of the KdV equation is determined. As a result, the wavelength of radial displacement becomes narrower but remains unity when the radius of the artery increases. In addition to this, the wave speed decreases until the terminal velocity is achieved. Lastly, before the blood passes through the origin, the wave trajectory decreases along the space and increases slightly at a certain space as the fluid passes through the tube. After that, the wave trajectory continues decreasing. Finally, the wave speed and wave trajectory are influenced by various of radius of the artery.