Mathematical Analysis of a Model of Blood Flow through a Channel with Flexible Walls

TL;DR

This paper analyzes the stability of blood flow in a flexible-walled channel, deriving a mathematical model that captures fluid-structure interactions and provides a series solution for the flow stability problem.

Contribution

It develops a mathematical framework for analyzing fluid flow stability in channels with flexible walls, incorporating traveling wave effects and deriving a closed-form series solution.

Findings

The model captures energy transfer between fluid and walls.

A boundary value problem for the stream function is formulated.

A closed-form series solution for the flow stability is obtained.

Abstract

The present research is devoted to the problem of stability of the fluid flow moving in a channel with flexible walls and interacting with the walls, which are subject to traveling waves. Experimental data shows that the energy of the flowing fluid can be transferred and consumed by the structure (the walls), which induces "traveling wave flutter." The problem of stability of fluid-structure interaction splits into two parts: (i) stability of fluid flow in the channel with harmonically moving walls and (ii) stability of solid structure participating in the energy exchange with the flow. Stability of fluid flow is the main focus of the research. It is shown that using the mass conservation and the incompressibility condition one can obtain the initial boundary value problem for the stream function. The boundary conditions reflect the facts that (i) for the axisymmetrical flow, there is no movement in the vertical direction along the axis of symmetry, and (ii) there is no relative movement between the near boundary flow and the structure ("no-slip" condition). The closed form solution is derived and represented in the form of an infinite functional series.