Reduced models for analyzing flow instability in compliant rectangular microchannels

Fluids conveyed in deformable conduits are often encountered in microfluidic applications, which makes fluid--structure interactions (FSIs) an inherent feature of these systems. Previous experiments reported the existence of FSI-induced instabilities in microchannel flows at low Reynolds number (Re). This observation suggests new strategies to enhance mixing at the microscale, where mixing is diffusion limited. To provide new understanding of these phenomena, we formulate an unsteady reduced model of flow in a long, shallow rectangular microchannel with a compliant top wall. Based on previous work, the slenderness of the compliant channel gives rise to a Winkler-foundation-like behavior of the fluid--solid interface; i.e., the interface deformation is fully determined by the local pressure. To apply this idea to flow stability, we first spanwise average the deformed channel. Second, to regularize the model, we introduce weak tension, which enables satisfaction of the displacement constraints along the edges of the channel. Third, the deformed channel height is introduced into a cross-sectionally averaged 1D flow model, which is obtained from the Navier--Stokes equations, keeping leading-order terms in Re ≠ 0. The steady response of the resulting reduced model is analyzed, and the global linear stability of the inflated base state is determined. It is shown that, for specific combination of the model's parameters, the base state is linearly unstable. Numerical simulations of the reduced model show that the unstable cases correspond to self-sustained oscillations of the channel wall. The proposed model can be useful for understanding flow instabilities observed in microchannels at Re as low as 200-300.