Figure 1

(a) An optical micrograph of the FF bubble generator comprising a planar network of channels fabricated in poly(dimethylsiloxane) and having a uniform height of $25\mu \mathrm{m}$. The relevant dimensions are plotted in the figure—the width of the inlet channel ($w_{\mathrm{in}}=30\mu \mathrm{m}$), the distance between the inlet channel and the orifice ($l_{\mathrm{in}}=100\mu \mathrm{m}$), the width of the orifice ($w_{\mathrm{or}}=30\mu \mathrm{m}$), and the length of the orifice ($l_{\mathrm{or}}=100\mu \mathrm{m}$). The original micrograph is shown in black; the gray lines extend the contours of the walls. (b) Bifurcation diagram showing the diameters of the bubbles as a function of the flow rate $Q$ of the liquid ($\mu =0.9\mathrm{mPa}\mathrm{s}$) for constant pressure ($p=76\mathrm{kPa}$) of gas. The first bifurcation occurs at $Q=1.06\mu \mathrm{L}/\mathrm{s}$. It is followed by period halving at $Q=1.44\mu \mathrm{L}/\mathrm{s}$. At $Q=2.8\mu \mathrm{L}/\mathrm{s}$, the first of a cascade of period-doubling bifurcations occurs. The cascade leads to chaos which, upon further increase of $Q$, gives way to a stable period-3 cycle. The solid line shows an outline of the bifurcation diagram (not to scale). The letters mark the rates of flow at which the micrographs of the period-1, period-2, period-4, and period-3 bubbling are shown in insets (c) to (f), respectively.Reuse & Permissions

Figure 2

(a) Phase diagram of the dynamics of the bubble generator in the $p,Q$ space ($\mu =0.9\mathrm{mPa}\mathrm{s}$) ○, period-1 cycles, and ●, higher-order periodic or chaotic behavior. Below the long-dashed diagonal line, $p$ is insufficient to force the gaseous thread through the orifice and no bubbling occurs. The horizontal, short-dashed line indicates the cut through the phase diagram (for $p=76\mathrm{kPa}$) illustrated in Fig. 1b. The solid line yields a constant frequency, $f=5\mathrm{KHz}$. (b) Frequency of bubbling plotted against the product ($pQ$). Different symbols correspond to different viscosities $\mu$ of the liquid [values in (mPa s) given in the legend]. The solid line shows the fit $f=\alpha (Qp{)}^{\beta }$, with $\alpha =53.3$ and $\beta =1.04$ (regression coefficient $R=0.94$). (c) A log-log plot of the critical capillary (○) and Weber (●) numbers as a function of the Ohnesorge number.Reuse & Permissions