We analyze the linear stability of the base state of the problem of coupled flow and deformation in a long and shallow rectangular soft hydraulic conduit with a thick top wall. Specifically, the steady base state is computed at low but finite Reynolds number. Then, we show that with the upstream flux fixed and the outlet pressure set to gauge, the flow is linearly stable to infinitesimal flow-wise perturbations. Multiple oscillatory but stable eigenmodes are computed in a range of the reduced Reynolds number, $hat{Re}$, and the so-called fluid--structure interaction (FSI) parameter, $lambda$, indicating the stiffness of this FSI system. These results provide a framework to address, in future work, the individual effects of various aspects of two-way FSI coupling on instability and flow transition in soft hydraulic conduits.
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