Flow of the Viscous-Elastic Liquid in the Non- Homogeneous Tube

TL;DR

This paper models wave propagation of viscous-elastic liquids in inhomogeneous deformable tubes, addressing mathematical complexities through integral equations and sequential approximation methods to analyze fluid-structure interactions.

Contribution

It introduces a mathematical framework for viscous-elastic liquid flow in inhomogeneous tubes, reducing the problem to a solvable integral equation with a novel approach.

Findings

Derived a Volterra integral equation for the problem

Solved the equation using sequential approximations

Analyzed wave flow in inhomogeneous deformable tubes

Abstract

A problem on propagation of waves in deformable shells with flowing liquid is very urgent in connection with wide use of liquid transportation systems in living organisms and technology. It is necessary to consider shell motion equations for influence of moving liquid in cavity on the dynamics of a shell by solving such kind problems. Nowadays a totality of such problems is a widely developed field of hydrodynamics. However, a number of peculiarities connected with taking into account viscous-elastic properties of the liquid and inhomogeneity of the shell material generates considerable mathematical difficulties connected with integration of boundary value problems with variable coefficients. In the paper we consider wave flow of the liquid enclosed in deformable tube. The used mathematical model is described by the equation of motion of incompressible viscous elastic liquid combined with equation of continuity and dynamics equation for a tube inhomogeneous in length. It is accepted that the tube is cylindric, semi-infinite and rigidly fastened to the environment. At the infinity the tube is homogeneous. As a final result, the problem is reduced to the solution of Volterra type integral equation that is solved by sequential approximations method. Pulsating pressure is given at the end of the tube to determine the desired hydrodynamic functions.}