A Homogenised Model of Fluid-String Interaction

TL;DR

This paper develops a homogenised multiscale model describing the interaction between aligned flexible strings and viscous fluid flow, deriving simplified equations and analyzing shear stress behavior under harmonic forcing.

Contribution

It introduces a systematic homogenised approach to model fluid-structure interaction involving many flexible strings, resulting in a simplified PDE framework.

Findings

Derived a modified Darcy law for fluid flow with microscale averaging.

Obtained analytical solutions for cylindrical geometries with harmonic string motion.

Identified scaling laws for shear stress variation with forcing frequency.

Abstract

A homogenised model is developed to describe the interaction between aligned strings and an incompressible, viscous, Newtonian fluid. In the case of many strings, the ratio of string separation to domain width gives a small parameter which can be exploited to simplify the problem. Model derivation using multiscale asymptotics results in a modified Darcy law for fluid flow, with coefficients determined by averaged solutions to microscale problems. Fluid flow is coupled to solid deformation via a homogenised force balance obtained by coarse-graining the balance on each string. This approach offers an alternative method to systematically derive the equations governing the interaction of Stokes flow with many flexible structures. The resulting model of fluid-structure interaction is reduced to a single scalar, linear, partial differential equation by introducing a potential for the pressure. Analytical solutions are presented for a cylindrical geometry subject to time harmonic motion of the string ends. Scaling laws are identified that describe the variation of shear stress exerted on the string surface with the forcing frequency.