An Integrability Technique for Fluid Flow Induced Deformation of a Boundary Hair*

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.