Theory and simulation of elastoinertial rectification of oscillatory flows in two-dimensional deformable rectangular channels

Abstract

A slender two-dimensional (2D) channel bounded by a rigid bottom surface and a slender elastic layer above deforms when a fluid flows through it. Hydrodynamic forces cause deformation at the fluid--solid interface, which in turn changes the cross-sectional area of the fluidic channel. The nonlinear coupling between flow and deformation, along with the attendant asymmetry in geometry caused by flow-induced deformation, produces a streaming effect (a nonzero cycle-average despite time-periodic forcing). Surprisingly, flow inertia provides another nonlinear coupling, tightly connected to deformation, that enhances streaming, termed ``elastoinertial rectification'' by Zhang and Rallabandi [J.\ Fluid Mech.\ \textbf{996}, A16 (2024)]. We adapt the latter theory of how two-way coupled fluid--structure interaction (FSI) produces streaming to a 2D rectangular configuration, specifically taking care to capture the deformations of the nearly incompressible slender elastic layer via the combined foundation model of Chandler and Vella [Proc.\ R.\ Soc.\ A \textbf{476}, 20200551 (2020)]. We supplement the elastoinertial rectification theory with direct numerical simulations performed using a stabilized, conforming arbitrary Lagrangian--Eulerian (ALE) FSI formulation, implemented via the open-source computing platform FEniCS. We examine the axial variation of the cycle-averaged pressure as a function of key dimensionless groups of the problem: the Womersley number, the elastoviscous number, and the compliance number. Assuming a small compliance number, we find excellent agreement between theory and simulations for the leading-order pressure and deformation across a range of conditions. At the next order, the cycle-averaged pressures agree well. Finally, the theory also predicts nontrivial cycle-averaged vertical and horizontal displacements, in agreement with the simulations.