A 1D model for unsteady fluid--structure interactions in a soft-walled microchannel

A one-dimensional model for the transient fluid--structure interaction (FSI) between a soft-walled microchannel and viscous fluid flow within it is developed. An Euler--Bernoulli beam, with transverse bending rigidity and nonlinear axial tension, is coupled to a 1D fluid model obtained by depth-averaging 2D incompressible Navier--Stokes equations in the lubrication scaling. The resulting set of coupled nonlinear PDEs are solved numerically through a segregated approach employing fully-implicit time stepping and second-order finite-difference discretizations. The Strouhal number is fixed at unity, while the Reynolds number $Re$ and a dimensionless Young's modulus $\Sigma$ are varied independently to explore the parameter space. A critical $Re$ is defined by determining when the maximum steady-state deformation exceeds a certain threshold. It is shown that the critical $Re\propto\Sigma^{3/4}$, a scaling that indicates ``wall modes'' play a role in the evolution of the system away from an initially flat state. The maximum wall displacement at steady state correlates with a single dimensionless group, namely $Re/\Sigma^{0.9}$ for both pure bending and bending with tension.