Dripping-Jetting Transitions in a Dripping Faucet

Fascinating dynamics is known to result when the flow rate $Q$ at which water drips from a faucet varies. Starting with simple (period-1) dripping, the system transitions as $Q$ increases to complex dripping, where it exhibits period-$n$ ($n=2,4,\dots$) and chaotic responses, and then jets once $Q$ exceeds a threshold. New experiments and simulations show that high viscosity ($\mu$) liquids, e.g., syrup, transition directly from simple dripping to jetting as $Q$ increases. Phase diagrams showing transitions between simple and complex dripping and jetting in $(Q,\mu )$ space are developed. Values of $Q$ for transition from dripping to jetting are estimated from scaling arguments and shown to accord well with simulations.

Figure 1

Regimes of drop formation. (a) Dripping with satellite formation, (b) dripping without satellite formation, and (c) jetting. The flow rate increases from left to right.

Figure 2

Computed variation with We of three quantitative measures of the dynamics for detecting transition from dripping to jetting. Here $\mathrm{O}\mathrm{h}=0.01$ and $G=0.5$.

Figure 3

Phase diagram that shows computationally and experimentally obtained curves, indicated by (Comp) and (Exp), respectively, that identify the values of We where the dynamics transitions from simple to complex dripping, ${\mathrm{W}\mathrm{e}}_d$, and from dripping to jetting, ${\mathrm{W}\mathrm{e}}_j$. Here $0.31\le G\le 0.97$.

Figure 4

A computed phase diagram in $(\mathrm{W}\mathrm{e},\mathrm{O}\mathrm{h})$ space when $G=0.5$. The curves ${\mathrm{W}\mathrm{e}}_d$ and ${\mathrm{W}\mathrm{e}}_j$ have the same meanings as in Fig. 3. When $\mathrm{O}\mathrm{h}>{\mathrm{O}\mathrm{h}}_c$, the leaky faucet transitions directly from simple dripping to jetting.

Figure 5

Computed snapshots of the dynamics when $G=0.5$. (a) $\mathrm{O}\mathrm{h}=0.01$ and $\mathrm{W}\mathrm{e}=0.73$, dripping; (b) $\mathrm{O}\mathrm{h}=0.01$ and $\mathrm{W}\mathrm{e}=0.98$, jetting; (c) $\mathrm{O}\mathrm{h}=0.5$ and $\mathrm{W}\mathrm{e}=0.07$, dripping; and (d) $\mathrm{O}\mathrm{h}=0.5$ and $\mathrm{W}\mathrm{e}=0.071$, jetting.