We study the buckling of a clamped beam immersed in a creeping flow within a rectangular channel. Via a combination of precision experiments, simulations, and theoretical modeling, we show how the instability depends on a pressure feedback mechanism and rationalize it in terms of dimensionless parameters. As the beam can bend until touching the wall above a critical flow rate, we finally demonstrate how the system can be used as a tunable passive flow selector, effectively redirecting the flow within a designed hydraulic circuit.