Bond (Eotvos) Number & Ohnesorge Number for Droplet & Inkjet Physics

Keep density difference separate from liquid density

Bond number Bo=abs(rhoLiquid−rhoAmbient)×g×L²/gamma compares gravity with surface tension. Ohnesorge number Oh=mu/sqrt(rhoLiquid×gamma×L) instead uses liquid density, not the density difference. B7 supplies ambient density separately.

Use densities in kg/m³, acceleration in m/s², characteristic length in metres, surface tension in N/m and dynamic viscosity in Pa·s. All outputs are dimensionless. The initial ambient density of zero is an explicit negligible-ambient-density approximation.

Calculate Bond number with a declared length

E2 uses B4 as the declared diameter scale. Initially L=0.002 m gives Bo approximately0.5374. That is below one, not much smaller than one. It does not by itself prove that a falling drop remains spherical.

For an inkjet comparison use the nozzle diameter appropriate to the chosen convention; do not silently substitute droplet radius. Halving L reduces Bo by four. Equal densities or zero gravity give zero Bo. Aerodynamic forces, inertia and boundaries can still deform a drop.

=IF(COUNT(B2:B5,B7)<>5,"Enter numeric densities,g,L,tension",IF(OR(B2<=0,B7<0,B3<0,B4<=0,B5<=0),"Require liquid density,L,tension>0; ambient density,g>=0",ABS(B2-B7)*B3*B4^2/B5))

Calculate Ohnesorge and its reciprocal

E3 uses liquid density and viscosity. E4 computes Z=1/Oh from the unrounded value. With the initial inputs, Oh rounds to0.00262. Changing L to0.00005 m and viscosity to0.015 Pa·s gives Z about4.02.

These groups describe a balance of viscous, inertial and capillary effects, but omit jet velocity. Zero viscosity is accepted as a mathematical Oh=0 limit; the reciprocal then has no finite value and is explicitly reported as undefined.

=IF(COUNT(B2,B4:B6)<>4,"Enter numeric density,L,tension,viscosity",IF(OR(B2<=0,B4<=0,B5<=0,B6<0),"Require density,L,tension>0; viscosity>=0",B6/SQRT(B2*B5*B4)))

Use Z as guidance, not a printability guarantee

Ranges such as 1<Z<10 are screening heuristics. Experimental work has reported different usable ranges, including4–14; successful ejection also depends on the printhead, drive waveform and fluid behavior. A value inside a range does not certify printability, and a value outside does not prove satellite formation.

This worksheet therefore reports numbers without an “ideal ink” or “will shatter” classification. It does not assess cell viability in bioprinting or the behavior of a non-Newtonian formulation.

=IF(NOT(ISNUMBER(E3)),E3,IF(E3=0,"Z undefined at zero viscosity",1/E3))

Try independent changes

Change only ambient density: Bo changes, while Oh and Z stay fixed. Double only viscosity: Oh doubles, Z halves and Bo stays fixed. Double L: Bo increases fourfold and Oh decreases by sqrt(2).

Numeric guards reject missing inputs, nonpositive liquid density, length or tension, and negative viscosity, gravity or ambient density. A valid number is still not proof that the selected scales model the physical situation.

Further reading

• COMSOL — Other Dimensionless Numbers
• Jang, Kim and Moon — Influence of Fluid Physical Properties on Ink-Jet Printability
• Predicting pharmaceutical inkjet printing outcomes using machine learning