Fluid Flow Induced Deformation of a Boundary Hair

TL;DR

This paper derives an analytical solution for the nonlinear deformation of a hair boundary caused by fluid flow, compares it with numerical results, and explores the solution structure to aid understanding of fluid-structure interactions.

Contribution

It provides the first analytical solution to a nonlinear integro-differential equation modeling hair deformation under fluid flow, revealing integrability and solution branches.

Findings

Analytical solution matches numerical results.

Identified physically relevant solution branch.

Explored solution structure via conserved quantity.

Abstract

The deformation of a dense carpet of hair due to Stokes flow in a channel can be described by a nonlinear integro-differential equation for the shape of a single hair, which possesses several solutions for a given choice of parameters. While being posed in a previous study and bearing resemblance to the pendulum problem from mechanics, this equation has not been analytically solved until now. Despite the presence on an integral with a nonlinear functional dependence on the dependent variable, the system is integrable. We compare the analytically obtained solution to a finite-difference numerical approach, identify the physically realizable solution branch, and briefly study the solution structure through a conserved energy-like quantity. Time-dependent fluid-structure interactions are a rich and complex subject to investigate and we argue that the solution discussed herein can be used as a basis for understanding these systems.