Weber Number Simulator — Droplet Inertia vs Surface Tension
Compute the Weber number We = rho V^2 L / gamma in real time from density rho, velocity V, characteristic length L and surface tension gamma. The tool classifies the droplet regime (sphere preserved when We below 1, oscillation and deformation between 1 and 12, breakup above 12) and reports the critical breakup velocity Vc and the Bond number Bo, turning sprays, fuel atomization and raindrop fragmentation into something you can see.
Parameters
Fluid density rho
kg/m³
Velocity V
m/s
Characteristic length L
mm
Surface tension gamma
mN/m
Defaults: water at 20 C (rho = 1000 kg/m³, gamma = 73 mN/m), L = 2 mm raindrop and V = 10 m/s airflow. Gravitational acceleration g = 9.81 m/s². The threshold We_c = 12 is the bag-breakup onset of Pilch and Erdman.
Results
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Weber number We
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Regime
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Critical velocity Vc
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Bond number Bo
Droplet schematic by regime
Blue = surface tension dominated (sphere preserved). Orange = oscillation and deformation (ellipsoidal vibration). Red = breakup regime (bag/shear). Arrow length scales with V; droplet size scales with L.
Weber number We vs velocity V (log)
X = V [m/s] (0.1 to 100, linear). Y = We (log). Blue = We(V) = rho V^2 L / gamma. Grey dashed = We = 1 (sphere limit). Red dashed = We = 12 (breakup threshold). Yellow dot = current (V, We).
Theory & Key Formulas
The ratio of droplet inertia to surface tension is the Weber number.
Weber number (dimensionless):
$$\mathrm{We} = \frac{\rho\,V^{2}\,L}{\gamma}$$
Critical breakup velocity (the V at We = 12):
$$V_{c} = \sqrt{\frac{12\,\gamma}{\rho\,L}}$$
Bond number (gravity vs surface tension):
$$\mathrm{Bo} = \frac{\rho\,g\,L^{2}}{\gamma}$$
$\rho$ is fluid density [kg/m³], $V$ is the relative velocity [m/s], $L$ is the droplet characteristic length [m], $\gamma$ is surface tension [N/m] and $g = 9.81$ m/s². We below 1 keeps the droplet spherical, 1 to 12 gives oscillation and deformation, We above 12 enters bag/shear breakup.
What is the Weber Number Simulator
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The default We comes out at 2740 — that sounds enormous. Does shooting a 2 mm water droplet at 10 m/s really shatter it?
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Good catch. Weber number We = rho V^2 L / gamma compares the dynamic pressure rho V^2 (which deforms the droplet) with the surface-tension pressure gamma / L (which pulls it back to a sphere). With water, L = 2 mm and V = 10 m/s the tool gives We = 2740, more than 200 times the breakup threshold of 12. Imagine the same droplet hitting the windshield of a fast car: it cannot stay round, it puffs into a bag and bursts. Drop V to about 0.66 m/s in this tool and you will see We hit exactly 12 and the regime label switch from breakup to oscillation.
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The critical velocity Vc = 0.662 m/s is shown — does that mean the droplet definitely shatters above this speed? It feels surprisingly slow.
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Right. Vc = sqrt(12 gamma / (rho L)) is the speed where We hits 12. For a 2 mm water droplet that is only 0.66 m/s, slower than walking. This is why naturally falling raindrops larger than 5 mm in diameter, with terminal velocity near 9 m/s, all sit at We above 12 and break apart in mid-air. Shrink L to 0.2 mm (a fine mist) and Vc rises by sqrt(10) to about 2.1 m/s, so mist droplets survive higher winds. Spray nozzle designers usually pick exit speeds a few times Vc to control the resulting droplet size.
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The right plot shows We rising with V squared. Does that mean velocity matters more than the other parameters?
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Exactly. We scales like V squared, so doubling V multiplies We by 4 and increasing V by 10x multiplies We by 100. Density rho and length L only enter linearly, while surface tension gamma sits in the denominator. That is why "atomize finer" almost always means "spray faster" — it is the basis of high-pressure common-rail diesel injection at 200 MPa. Conversely, to keep droplets intact you raise gamma (for example by removing surfactants) or shrink L. In this tool, push gamma from 73 to 500 mN/m and We drops by a factor of about 7, sometimes pulling the regime out of breakup.
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The last value Bo = 0.538 is the Bond number, right? Why do we need both Bo and We if both compare against surface tension?
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Same denominator, different numerator. Bo = rho g L^2 / gamma is gravity vs surface tension; We = rho V^2 L / gamma is inertia vs surface tension. The Bond number sets the shape of a droplet at rest (small ones round, big ones flatten), while the Weber number sets whether a droplet survives a flow. Our 2 mm water droplet has Bo = 0.54 and stays beautifully round, but the same droplet thrown into a 10 m/s wind has We = 2740 and is destroyed. In microgravity (g ≈ 0) Bo goes to zero and giant water blobs float, but accelerate the spacecraft and We jumps and the same blobs disintegrate.
Physical model and key equations
The Weber number is a dimensionless ratio of inertial force to surface-tension force on a droplet.
$$\mathrm{We} = \frac{\rho\,V^{2}\,L}{\gamma}$$
$\rho$ is the surrounding (or droplet) fluid density [kg/m³], $V$ is the relative velocity between droplet and surrounding flow [m/s], $L$ is a characteristic length (typically the droplet diameter) [m], and $\gamma$ is the surface tension [N/m]. The numerator $\rho V^2$ is dynamic pressure (deforms the droplet), and the denominator $\gamma / L$ is the curvature-dependent Laplace pressure (restores the sphere).
The empirical threshold We_c about 12 marks the onset of bag breakup. Increasing V further moves through bag-and-stamen (We about 50), sheet stripping (We about 100) and catastrophic breakup (We above 350). This tool uses only the first threshold to compute the critical velocity $V_c = \sqrt{12\gamma/(\rho L)}$.
The companion Bond number $\mathrm{Bo} = \rho g L^2/\gamma$ also sits over surface tension, but its numerator is gravitational pressure. Bo less than 1 means the droplet stays round, Bo greater than 1 means gravity flattens it. Use We for dynamic stability and Bo for static shape.
Real-world applications
Diesel fuel atomization: Common-rail diesel engines push fuel through a 0.1 mm nozzle at 200 MPa, producing exit speeds of order 500 m/s. Plug rho = 830 kg/m³ (diesel), gamma = 25 mN/m, L = 0.1 mm and V = 500 m/s into this tool and you get We about 8.3e5, deep in catastrophic breakup. The droplets shatter within millimetres of the nozzle, exposing a huge surface area for combustion. Engine designers work backwards from a target Sauter mean diameter (SMD) to the required We and choose the injection pressure and orifice diameter accordingly.
Agricultural and paint sprays: If droplets are too large, they fall before reaching the leaves; if too small, the wind carries them away as drift. Practical SMDs are 100 to 300 microns, which usually corresponds to We in the 10 to 50 range. This tool gives We about 60 for L = 0.3 mm, V = 20 m/s and gamma = 40 mN/m (with a wetting agent), right at the oscillation/breakup boundary that gives a stable droplet size distribution.
Natural raindrop fragmentation: Large raindrops (above 6 mm in diameter) fall at about 9 m/s near the ground. Plugging in rho = 1000, L = 6 mm, gamma = 73 mN/m and V = 9 m/s yields We about 670, well into bag breakup. High-speed cameras confirm that such drops deform from a "hamburger" into a "parachute" and split into smaller fragments. Atmospheric scientists use this fragmentation to set the maximum-diameter cutoff of the Marshall-Palmer raindrop distribution.
Aero-engines and aircraft icing: Jet-engine fuel atomizers drive droplet We into the 100 to 1000 range to optimize combustion. Conversely, supercooled large droplets (SLD) in icing conditions have low We; they do not shatter on impact but spread as a thin sheet that freezes into "runback ice" and ruins the wing aerodynamics. Try L = 2 mm, V = 80 m/s and gamma = 73 mN/m here and you get We about 1.75e5, showing how violently inlet droplets are torn apart in front of the engine.
Common misconceptions and caveats
The most common pitfall is treating Weber number as if it captured viscous effects too. Weber compares only inertia and surface tension; viscosity is handled separately by the Ohnesorge number $\mathrm{Oh} = \mu/\sqrt{\rho L \gamma}$. For Oh below 0.1 the threshold We_c about 12 holds, but for silicone oils or molten glass with larger Oh, We_c can shift upward by a factor of 2 to 10. This tool assumes Oh is small; for high-viscosity liquids you need an additional correction.
Next is the assumption that any V above Vc guarantees breakup. Vc is the threshold for steady, sufficiently long-duration airflow. In a brief gust the droplet may only oscillate without splitting. Real spray design considers not just the peak We but also the residence time at high We; the breakup mode is judged for the entire downstream flow field, not just the local instant. This tool reports an instantaneous We, so additional time-averaged analysis may be needed for transient cases.
Lastly, do not assume L is always the droplet diameter. Some references use the droplet radius as L, in which case the same physical situation yields a Weber number that is 2x different. This tool follows the Pilch & Erdman convention of L = diameter, but always check the definition when comparing to a paper or textbook. Likewise, distinguish surface tension (liquid-gas) from interfacial tension (liquid-liquid) — gamma here refers to a liquid-air surface.
Frequently Asked Questions
We = rho V^2 L / gamma is the ratio of inertial pressure (the dynamic pressure rho V^2 that tries to deform and tear the droplet apart) to the surface-tension pressure gamma / L (the Laplace pressure that pulls the droplet back to a sphere). Both numerator and denominator have units of pressure, which is why the ratio is dimensionless. We below 1 means surface tension wins and the droplet stays nearly spherical, while We above 12 means inertia wins and the droplet breaks up. With the defaults (water, V = 10 m/s, L = 2 mm) the tool returns We = 2740, deep inside the breakup regime.
Classical experiments by Hinze, Pilch and Erdman and others on spherical droplets in steady airflow place the onset of bag breakup at about We_c approximately 12. Above that, the regime evolves through bag-and-stamen breakup, sheet stripping and finally catastrophic breakup. The 12 figure assumes a low-viscosity (low Ohnesorge number, Oh below 0.1) liquid; for higher Oh the threshold drifts upward. Sweep V in this tool and watch the regime label switch from oscillation to breakup as We crosses 12.
Vc = sqrt(12 gamma / (rho L)) is the relative velocity at which We equals 12, the breakup threshold. For water with gamma = 73 mN/m and L = 2 mm the tool returns Vc about 0.66 m/s, slower than walking pace. Drop L to 0.2 mm (a fine mist) and Vc rises by sqrt(10) to about 2.1 m/s, showing why small droplets are harder to break. Spray nozzle designers pick exit speeds a few times Vc to get controlled atomization without catastrophic shattering.
Both share gamma in the denominator. Bo = rho g L^2 / gamma compares gravity to surface tension (a static balance), while We = rho V^2 L / gamma compares inertia to surface tension (a dynamic balance). The shape of a sitting droplet is set by Bo (small droplets stay round, big ones flatten), but whether a droplet survives a flow is set by We. With the defaults this tool reports Bo = 0.538 (gravity loses, droplet stays spherical) yet We = 2740 (inertia wins decisively), showing that the same droplet can be statically stable but dynamically destroyed.