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Soft hydraulics, which addresses the interaction between an internal flow anda compliant conduit, is a central problem in microfluidics. We analyzeNewtonian fluid flow in a rectangular duct with a soft top wall at steadystate. The resulting fluid--structure interaction (FSI) is formulated for bothvanishing and finite flow inertia. At the leading-order in the small aspectratio, the lubrication approximation implies that the pressure only varies inthe streamwise direction. Meanwhile, the compliant wall's slenderness makes thefluid--solid interface behave like a Winkler foundation, with the displacementfully determined by the local pressure. Coupling flow and deformation andaveraging across the cross-section leads to a one-dimensional reduced model. Inthe case of vanishing flow inertia, an effective deformed channel height isdefined rigorously to eliminate the spanwise dependence of the deformation. Itis shown that a previously-used averaged height concept is an acceptableapproximation. From the one-dimensional model, a friction factor and thecorresponding Poiseuille number are derived. Unlike the rigid duct case, thePoiseuille number for a compliant duct is not constant but varies in thestreamwise direction. Compliance can increase the Poiseuille number by a factorof up to four. The model for finite flow inertia is obtained by assuming aparabolic vertical variation of the streamwise velocity. To satisfy thedisplacement constraints along the edges of the channel, weak tension isintroduced in the streamwise direction to regularize theWinkler-foundation-like model. Matched asymptotic solutions of the regularizedmodel are derived.

Reduced models of unidirectional flows in compliant rectangular ducts at finite Reynolds number