Passive Control of Viscous Flow via Elastic Snap-Through

We demonstrate the passive control of viscous flow in a channel by using an elastic arch embedded in the flow. Depending on the fluid flux, the arch may “snap” between two states—constricting and unconstricting—that differ in hydraulic conductivity by up to an order of magnitude. We use a combination of experiments at a macroscopic scale and theory to study the constricting and unconstricting states, and determine the critical flux required to transition between them. We show that such a device may be precisely tuned for use in a range of applications, and, in particular, has potential as a passive microfluidic fuse to prevent excessive fluxes in rigid-walled channels.

Figure 1

Viscous flow through a channel containing a flexible wall. (a) A thin elastic strip, buckled into an arch, initially constricts part of a channel (red shape). At higher flow rates, the arch rapidly snaps through (blue shape); the flow is then unconstricted and the channel’s conductivity increases. (b) Three-dimensional view showing the finite channel depth. (c) Shapes of the arch during a snapping experiment ($h=0.25\mathrm{mm}$, $w_0=4.7\mathrm{mm}$, $\eta =1.60\pm 0.10\mathrm{Pa}\mathrm{s}$), together with the shapes predicted by our beam-lubrication model (red dashed curves).

Figure 2

Pressure-flux relations for a flexible channel ($h=0.25\mathrm{mm}$, $\eta =1.60\pm 0.10\mathrm{Pa}\mathrm{s}$). (a) Evolution of the upstream pressure, $p(0)$, for different channel blocking parameters $W_0=w_0/d$ (indicated by the colorbar). For each $W_0$, three data sets through the snapping transition are shown, together with a fourth in which the arch remains in the snapped configuration throughout (symbols). Predictions from the beam-lubrication model, (6), are also shown (solid curves). (The snapping transition appears continuous in experiments because the arch motion is overdamped.) (b) The hysteresis loop highlighted for intermediate $W_0$. (c) The effective hydraulic conductivity $k=q_{\text{in}}/p(0)$ is plotted for the same data (with $p(0)>140\mathrm{Pa}$, to avoid noise due to inaccurate readings at low pressure).

Figure 3

Critical flux for snap-through. Inset: Experimentally measured snap-through flux, $q_{\text{snap}}$ (averaged over three runs), as a function of the initial midpoint height, $w_0$. Data are shown for $h=0.25\mathrm{mm}$ with $\eta =1.38\pm 0.17\mathrm{Pa}\mathrm{s}$ (blue circles) and $\eta =1.61\pm 0.18\mathrm{Pa}\mathrm{s}$ (red squares); and for $h=0.1\mathrm{mm}$ with $\eta =1.20\pm 0.10\mathrm{Pa}\mathrm{s}$ (green diamonds) and $\eta =1.33\pm 0.08\mathrm{Pa}\mathrm{s}$ (magenta triangles, increasing $q_{\text{in}}$ in steps of $0.25\mathrm{mL}{\min }^{-1}$ every minute rather than ramping). Horizontal error bars correspond to the $\pm 0.2\mathrm{mm}$ uncertainty in $w_0$; vertical error bars give the standard deviation of the measured values. Main plot: Rescaling to plot the dimensionless flux $Q_{\text{snap}}=\eta L^5{q}_{\text{snap}}/(Bb{d}^4)$ in terms of the channel blocking parameter $W_0=w_0/d$; the data collapse onto the prediction of our numerical analysis (solid black curve). Vertical error bars here also account for uncertainties in the bending stiffness $B$ and viscosity $\eta$. Also plotted is the asymptotic result $Q_{\text{snap}}\approx 16W_0$ valid for $W_0\ll 1$ [21] (black dotted line).

Figure 4

Flow limiting using snap-through. Inset: A channel with a flexible wall is connected in parallel to a rigid channel of constant conductivity (schematics not drawn to scale). Main figure: For low total fluxes $Q_{\text{total}}=\eta L^5{q}_{\text{total}}/(Bb{d}^4)$ with the flexible channel constricting (red curves), almost all of the flow is directed through the rigid channel, i.e., $q_r/q_{\text{total}}\approx 1$. This is diverted through the flexible channel as soon as the arch snaps (blue curves). Here $W_0=0.99$ and numerical results are shown for different conductivity ratios $\lambda$ between the channels: $\lambda ={10}^{-2}$ (solid curves), $\lambda ={10}^{-1}$ (dashed curves), and $\lambda =1$ (dotted curves).