2D companion to the 3D piezo inkjet scene: the same lumped-parameter physics, read off a live side-view cross-section and a synced drive-waveform strip chart instead of a rendered nozzle. A drop-on-demand piezo printhead fires with a bipolar "pull-push" waveform: the piezo actuator first bows outward, expanding the ink chamber and retracting the meniscus (pull); it holds this state for a dwell time td tuned near the channel's acoustic resonance; then it snaps back inward (push), driving a pressure wave that ejects a jet of ink through the nozzle.
Ohnesorge number: Oh = η / √(ρ·γ·a)
Printability number: Z = 1/Oh
Weber number: We = ρ·v²·a / γ
Here η is ink viscosity, ρ its density, γ its surface tension, a the nozzle radius, and v the jet velocity. Z compares inertia to the competing effects of viscosity (which damps the jet) and surface tension (which drives pinch-off). The commonly cited printable window (Reis & Derby; Derby, Annu. Rev. Mater. Res. 2010) is:
- Z < 4 — viscous damping dominates; the ligament never pinches off and the meniscus retracts back into the nozzle with no droplet ejected.
- 4 ≤ Z ≤ 14 — inertia, viscosity and surface tension are balanced; the ligament pinches cleanly into a single droplet.
- Z > 14 — surface tension can't damp the elongating ligament fast enough before pinch-off, so it breaks into a main droplet plus one or more trailing satellite droplets, which degrade print placement accuracy.
The jet velocity used here follows a simplified lumped-parameter model: v scales with drive voltage V, is damped by viscosity, and peaks when the dwell time matches the channel's acoustic resonance — exactly the tuning real printhead firmware performs per ink formulation. This nozzle is modelled at 40 µm diameter, ink density 1000 kg/m³, values typical of industrial piezo DOD heads. The strip chart below traces the same bipolar drive signal the 3D scene animates on the chamber and piezo ring; the vertical marker shows where the cross-section currently sits in that cycle.