# A Review of Passive Constant Flow Regulators for Microfluidic Applications
## 3 Mechanical Passive Constant Flow Regulators
### Pillar Valves in Parallel
#### 3.1.2 Devices with Compliant Elastomeric Membrane(s)  
In 2005, [PERSON] et al. made early use of compliant PDMS membranes for the dynamic flow stabilization in microfluidic systems [36]. The device consists of a microchannel and PDMS membranes in series on one or more walls of the channel. The compliant membranes are equivalent to capacitors that are used to remove fluctuations in a pulsatile flow, and thus nearly constant outlet flow rates and pressure were obtained. To obtain a regulation of the flow rate at set pressures, other device structures were investigated. The present section of the review provides a detailed description of these passive devices that exploit the compliance of elastomeric membranes to achieve a constant flow rate under varying static pressure conditions. [PERSON]'s group developed several detour autoregulatory devices in PDMS [37; 38; 39]. A schematic description of the working principle of these devices is provided in Figure 7. Pressure on the working fluid is applied at the inlet to produce a flow in the main channel of the upper layer. A detour channel is in fluidic connection with the main channel and through a via with a pushup valve located in the lower layer. The fluid flow induces a decrease of the pressure along the main channel while the static pressure in the detour channel remains constant and equal to the pressure value at the split point of the two channels. The increase of the applied pressure generates a larger pressure drop along the main channel and thus the pushup valve will progressively obstruct the main channel and generate a nonlinear fluidic behavior for Newtonian fluids (see Reference [40] for the characterization of the mechanical behavior of microfluidic valves in PDMS and Reference [41] for the flow-induced deformation of shallow microchannels in PDMS). [PERSON] et al. proposed quantitative modeling of this microfluidic  
Figure 6: Schematic cross-section of a triple-stack microfluidic device dedicated to drug delivery at a constant flow rate of less than 10 mL/h, with a top SOI wafer (membrane), an intermediate borosilicate layer, a middle Si wafer (substrate) comprising pillars, through holes and microchannels, and a bottom borosilicate wafer with inlet and outlet ports. The top surface of the membrane and the inlet ports are in direct communication with the pressurized reservoir. The direction of the flow is indicated by red arrows. As the pressure increases, the gap between the membrane and the pillars (i.e., the radial diffusers) narrows and the hydraulic resistance of the device increases to keep the flow rate constant (adapted from Reference [9]). system in PDMS used as a current source [39]. The \(Q(\Delta p)\) characteristic shows, at low pressure, a linear behavior with a slope inversely proportional to the hydraulic resistance of the non-deformed main channel. Further increasing the applied pressure gradient \(\Delta p\) induces nonlinearity due to the activation of the pushup valve, and the flow rate remains constant beyond a saturation pressure defined as the point where \(\frac{\mathrm{d}Q(\Delta p)}{\mathrm{d}\Delta p}=0\). Several microfluidic devices made of a stack of PDMS layers (Dow Corning Sylgard 184) with vertical connections (vias) were fabricated and tested using water. Constant flow rates of 0.015, 0.0198, and 0.024 mL/min were measured in the pressure range \([620;1240]\) mbar for detour channel ratio L\({}_{2}\)/L of 0.8, 0.68, and 0.57 respectively, in good agreement with modeling data [37; 39]. [PERSON] et al. measured, for a fixed detour channel ratio of 0.66, the effect of the increase of the pushup valve width on saturation pressure and flow rate [38]. For pushup valve width of 100 and 110 \(\upmu\)m, the flow rate did not saturate, and the \(Q(\Delta p)\) showed a small and positive slope above 480 mbar.