Modeling and Stability of Flows in Compliant Microchannels

Fluids conveyed in deformable conduits are often encountered in  microfluidic applications, which makes fluid--structure interactions (FSIs) an unavoidable phenomenon. In particular, experiments reported the existence of FSI instabilities in compliant microchannels at low Reynolds numbers, Re, well below the established values for rigid conduits. This observation has significant implications for new strategies for mixing at the microscale, which might harness FSI instabilities in the absence of  turbulence. In this thesis, we conduct research on the modeling and stability of microscale FSIs. Understanding the steady response, the dynamics and the stability of these FSIs are the three major objectives. This thesis begins with the analysis of the steady-state scalings and the linear stability of a previously derived mathematical model, through which we emphasize the power of reduced modeling in making the FSI problems tractable. Next, we turn to a more realistic problem regarding FSIs in a common configuration of low-Re flows through long, shallow rectangular three-dimensional microchannels. Through a scaling analysis, which takes advantage of the geometric separation of scales, we find that the flow can be simplified under the lubrication approximation, while the wall deforms like a variable-stiffness Winkler foundation at the leading order. Coupling these dominant effects, we obtain a new fitting-parameter-free flow rate--pressure drop relation for a thick-walled microchannel, which rationalizes previous experiments. Then, we derive a one-dimensional (1D) steady model, at both vanishing and finite Re, by coupling the reduced flow and deformation models. To satisfy the displacement constraints along the channel edges, weak tension is introduced to regularize the underlying Winkler-foundation-like mechanism. This model is then made dynamic by introducing flow unsteadiness and the elastic wall's inertia. We conduct a global stability analysis of this system by perturbing the non-flat steady state with infinitesimal perturbations. We identify the existence of globally unstable modes, typically in the weakly inertial flow regime, whose features are consistent with experimental observations. The unstable eigenmodes oscillate at frequencies close to the natural frequency of the wall, suggesting that the instabilities are resonance phenomena. We also capture the transient energy amplification of perturbations through a linear non-normality analysis of the proposed reduced 1D FSI model.