Ohnesorge Number (Viscosity against Inertia and Surface Tension)

Ohnesorge is the group with no velocity in it, and that absence is the point. Write out the definition and the reason appears immediately: $Oh = \mu/\sqrt{\rho \sigma L}$, and $\sqrt{We}/Re = \sqrt{\rho v^2 L/\sigma} \cdot \mu/(\rho v L)$. The $v$ cancels exactly. So

$Oh = \sqrt{We}\,/\,Re$, exactly, always, with no approximation anywhere.

Which means Ohnesorge carries no information that Weber and Reynolds did not already carry. Feed a 2 mm water drop at 10 m/s through all three: $We = 2778$, $Re = 20000$, and $\sqrt{2778}/20000 = 0.00264$, which is what the direct formula gives to every digit. The third number is a rearrangement of the first two.

This is the clearest illustration on this site of something that trips up nearly everyone who meets dimensional analysis. Buckingham's Pi theorem tells you HOW MANY independent dimensionless groups a problem has. It does not tell you WHICH ones. A problem with five variables and three dimensions has two independent groups — that is fixed, and no amount of cleverness changes it. But any two independent combinations will do, and there are infinitely many. $\{Re, We\}$, $\{Re, Oh\}$ and $\{We, Oh\}$ all span the same space. Choosing among them is a matter of convenience, taste and trade custom, not of physics.

Ohnesorge's convenience is separation of concerns. It contains only fluid properties and a length — nothing about how fast you are throwing the liquid. A formulator can compute it from a data sheet. So the standard atomisation map plots Ohnesorge against Reynolds number, and the breakup regimes fall into bands on it: Rayleigh breakup into uniform drops at low values of both, first and second wind-induced regimes above, full atomisation at high Reynolds number. Weber does not appear on the map at all, because it is implied by the axes.

What Ohnesorge actually measures is how effectively viscosity damps capillary waves. Below about 0.1 it barely does: a jet pinches off cleanly, satellite drops are few, and Weber alone predicts the outcome. Between 0.1 and 1 viscosity stretches the ligaments, delays pinch-off and raises the critical Weber number above the usual 12. Above 1 viscosity dominates and an atomiser sized on Weber number will not produce the spray it was designed for. Ink-jet practice uses the reciprocal, $Z = 1/Oh$, and quotes a printable window of roughly $1 < Z < 10$ — which is another convention with a good deal of published disagreement at both ends.

Two traps. Because $Oh \propto L^{-1/2}$, the same liquid has a different Ohnesorge number at every drop size; viscosity matters more the smaller the drop, which is why a fluid that atomises beautifully through a coarse nozzle can refuse through a fine one. And the identity only holds if all three groups use the same characteristic length. Use a diameter for Weber and a radius for Ohnesorge and it breaks by $\sqrt{2}$ — silently, with three plausible numbers on the page describing three different problems.