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Weber Number Calculator

Weber number equals density times velocity squared times length divided by surface tension

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Weber Number

The Weber number compares the disruptive inertial forces in a flow to the cohesive surface-tension forces that hold a droplet or bubble together. At low We, surface tension wins and droplets remain spherical. At high We, inertial forces dominate and the droplet shatters.

We = ρv²L / σ

How It Works

The Weber number compares the disruptive inertial forces in a flow to the cohesive surface-tension forces that hold a droplet or bubble together. At low We, surface tension wins and droplets remain spherical. At high We, inertial forces dominate and the droplet deforms, oscillates, or shatters into smaller fragments. This makes We the key parameter for fuel injector design, spray coating, inkjet printing, and any process where controlling droplet size matters.

Example Problem

A 2 mm water droplet (ρ = 998 kg/m³, σ = 0.0728 N/m) travels at 5 m/s through air. What is the Weber number?

  1. Identify the known variables: ρ = 998 kg/m³, v = 5 m/s, L = 0.002 m, σ = 0.0728 N/m
  2. Write the Weber number formula: We = ρv²L / σ
  3. Square the velocity: v² = 5² = 25 m²/s²
  4. Calculate the numerator: ρ × v² × L = 998 × 25 × 0.002 = 49.9 N/m
  5. Divide by surface tension: We = 49.9 / 0.0728 = 685
  6. Interpret: We = 685 far exceeds the critical breakup threshold (≈ 12), so the droplet will shatter into a fine spray almost immediately

We = 685 far exceeds the critical breakup threshold (≈ 12), so the droplet will shatter into a fine spray almost immediately.

When to Use Each Variable

  • Solve for Weber Number — when you know density, velocity, length, and surface tension, e.g., checking whether a droplet will break up in a given flow.
  • Solve for Density — when you know the Weber number, velocity, length, and surface tension, e.g., identifying what fluid density would produce a critical Weber number.
  • Solve for Velocity — when you know the Weber number, density, length, and surface tension, e.g., finding the speed at which droplet breakup begins.
  • Solve for Characteristic Length — when you know the Weber number, density, velocity, and surface tension, e.g., predicting the maximum stable droplet diameter in a spray.
  • Solve for Surface Tension — when you know the Weber number, density, velocity, and length, e.g., determining what surface tension is needed to prevent atomization.

Key Concepts

The Weber number is a dimensionless ratio of disruptive inertial forces to restoring surface tension forces. Below the critical Weber number (approximately 12 for drops in gas), surface tension holds droplets together. Above it, inertial forces overcome cohesion and the droplet fragments through bag breakup, shear stripping, or catastrophic shattering depending on how far We exceeds the threshold.

Applications

  • Fuel injection: designing diesel and gasoline injectors to achieve fine atomization for efficient combustion
  • Spray coating: controlling droplet size in paint, pharmaceutical, and agricultural spray systems
  • Inkjet printing: tuning nozzle geometry and fluid properties for consistent droplet formation
  • Meteorology: predicting raindrop breakup size limits in turbulent atmospheric flows
  • Nuclear reactor safety: modeling coolant spray behavior during emergency core cooling

Common Mistakes

  • Using the wrong characteristic length — for droplets it is the droplet diameter, for jets it is the jet diameter, and for films it is the film thickness
  • Ignoring the Ohnesorge number — high-viscosity fluids resist breakup even at high Weber numbers because viscous forces dampen deformation
  • Forgetting that surface tension decreases with temperature — the same flow conditions produce a higher Weber number in hot fluids
  • Applying the critical We threshold for drops in gas (about 12) to drops in liquid — immiscible liquid-liquid systems have different breakup criteria

Frequently Asked Questions

What does the Weber number tell you about droplets and bubbles?

The Weber number indicates whether inertial forces or surface tension forces dominate a droplet or bubble. At low Weber numbers (below about 12), surface tension holds the droplet in a stable spherical shape. At high Weber numbers, the fluid's momentum overwhelms surface tension and the droplet deforms, oscillates, and eventually fragments into smaller drops.

At what Weber number do droplets break apart?

For a liquid drop falling through a gas, the critical Weber number is approximately 12. Above this value the drop begins to deform and fragment. The exact threshold depends on the Ohnesorge number (viscosity effects) and the breakup regime: bag breakup occurs around We ≈ 12–50, multimode breakup at We ≈ 50–100, and catastrophic shattering above We ≈ 350.

What Weber number causes droplet breakup?

For a liquid drop in a gas stream, the critical Weber number is about 12. Above this value the drop deforms and fragments. The exact threshold depends on the Ohnesorge number (viscosity effects) and the type of breakup (bag, shear-stripping, or catastrophic).

How is the Weber number used in fuel injector design?

Fuel injectors must atomize liquid fuel into fine droplets for efficient combustion. Engineers design nozzle geometry and injection pressure to produce a Weber number well above 12, ensuring rapid breakup. A typical diesel injector produces We on the order of 10,000 to 100,000 at the nozzle exit.

What is the difference between the Weber number and the Bond number?

The Weber number compares inertial forces to surface tension, while the Bond number (or Eotvos number) compares gravitational forces to surface tension. Use We when the flow velocity is the driving mechanism and Bo when gravity (buoyancy) drives the motion.

Does surface tension depend on temperature?

Yes. Surface tension generally decreases with increasing temperature. Water drops from 0.0756 N/m at 0°C to 0.0589 N/m at 100°C. This means the Weber number for the same flow conditions increases at higher temperatures, making atomization easier.

What role does the Ohnesorge number play alongside Weber number?

The Ohnesorge number (Oh) accounts for viscous damping that the Weber number ignores. High-viscosity fluids resist deformation even at high We. The breakup regime is best described on a We-Oh map: at low Oh, the critical We is about 12, but at high Oh (above 0.1), significantly higher Weber numbers are needed for breakup.

Weber Number Formula

The Weber number is a dimensionless ratio of disruptive inertial forces to cohesive surface tension forces:

We = ρv²L / σ

Where:

  • We — Weber number (dimensionless)
  • ρ — fluid density, measured in kg/m³
  • v — flow velocity, measured in m/s
  • L — characteristic length (e.g., droplet diameter), measured in meters (m)
  • σ — surface tension, measured in N/m

When We is below the critical threshold (≈ 12 for drops in gas), surface tension holds the droplet together. Above this value, inertial forces overwhelm surface tension and the droplet fragments through bag breakup, shear stripping, or catastrophic shattering.

Worked Examples

Spray Systems

Will a diesel fuel injector atomize fuel into a fine spray?

Diesel fuel (ρ = 850 kg/m³, σ = 0.026 N/m) exits a nozzle at 200 m/s with a jet diameter of 0.2 mm. What is the Weber number?

  • We = ρv²L / σ = 850 × 40,000 × 0.0002 / 0.026
  • We = 6,800 / 0.026
  • We = 261,538

We ≫ 12, so the fuel jet will atomize into extremely fine droplets, which is exactly what diesel engines need for efficient combustion.

Meteorology

What is the maximum stable raindrop size at terminal velocity?

A raindrop falls at terminal velocity of 9 m/s through air. Water has ρ = 998 kg/m³ and σ = 0.0728 N/m. At the breakup threshold We = 12, what is the maximum diameter?

  • L = We × σ / (ρ × v²) = 12 × 0.0728 / (998 × 81)
  • L = 0.8736 / 80,838
  • L = 0.0000108 m (0.011 mm)

This gives the aerodynamic Weber number limit. In practice, raindrops reach 4-5 mm because the relevant velocity is the relative velocity between the drop and surrounding air disturbances, not the terminal fall speed.

Inkjet Printing

Is the ink droplet velocity optimized for clean droplet formation?

An inkjet nozzle ejects ink (ρ = 1,050 kg/m³, σ = 0.035 N/m) at 8 m/s with droplet diameter L = 0.05 mm. Calculate the Weber number.

  • We = ρv²L / σ = 1,050 × 64 × 0.00005 / 0.035
  • We = 3.36 / 0.035
  • We = 96

We = 96 is well above the breakup threshold, meaning satellite droplets may form. Inkjet designers aim for We between 4 and 50 for clean single-droplet jetting.

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