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This paper presents a microfluidic control device that regulates nearly constant flow rates under variable driving pressures. This flow regulator is fabricated from PDMS and exploits the large compliance of PDMS to achieve passive flow regulation without consuming power or requiring active control circuitry. The device features a compliant flap and a rigid block that form a flow constriction in a microchannel within which they are embedded. Under driving pressure variations, the flow constriction is narrowed due to flap deflections, and flow rate changes that would result from the pressure variations are canceled by constriction narrowing effects. As a result, a nearly constant flow rate can be maintained, as will be demonstrated with prototype devices that regulate water flow rates from 0.15 to 1.2 ml/min under driving pressures varying from 100 to 200 kPa.