Contents

Capillary number

Pronunciation: /kəˈpɪləri ˈnʌmbə/

The capillary number (Ca) is a dimensionless ratio that compares viscous forces with surface or interfacial tension: Ca = μU/γ. A small Ca means tension controls the shape of the interface; a large Ca means flow can stretch, break or push it.

Also called

Ca Nca

Related terms

Interfacial tension (IFT) Spreading coefficient

Topics

Interfacial tension Fluid flow Dimensionless numbers
Book a 15 minute demo Read the surface tension measurement guide

At a glance

What the capillary number compares, where it shows up, and what values mean.

What it is

Flow drags on an interface; tension pulls it back into the smallest area. The capillary number puts both on one scale. It has no units, so the same value means the same balance in a pore, a pipe or a microchannel.

  • μ: viscosity of the displacing or surrounding liquid
  • U: a typical speed
  • γ: surface or interfacial tension

Why it matters

Ca predicts when drops break up in a microchannel, how thick a film a moving bubble leaves behind, and how much oil stays trapped in rock. Raising Ca, often by lowering interfacial tension with surfactant, is the idea behind chemical enhanced oil recovery, as our note on wettability alteration screening with surfactants and chemicals explains.

Worked example

Waterflood in a reservoir: water viscosity μ = 1 mPa s = 0.001 Pa s, frontal speed about 1 ft per day ≈ 3.5 × 10−6 m/s, oil water IFT γ = 30 mN/m = 0.030 N/m.

  • Ca = 0.001 × 3.5 × 10−6 / 0.030 ≈ 1.2 × 10−7
  • Lower γ to 0.001 mN/m: Ca ≈ 3.5 × 10−3
  • Ca rises by a factor of 30,000, enough to mobilise trapped oil

Expert view

“Plot surface tension against surfactant concentration and the interface tells you when it is saturated: the curve bends at the point where the surface can hold no more.”

Dr. Alidad Amirfazli, Co-founder and Scientific Advisor, Droplet Lab

What the terms mean

Three regimes, from tension dominated to flow dominated.

Ca much less than 1 (capillary dominated)

Tension wins. Drops stay round, menisci keep their static shape, and oil stays trapped in small pores.

Ca around 10⁻³ to 10⁻¹ (transition)

Flow starts to deform interfaces. Drops elongate, films left by moving bubbles thicken, and some trapped oil starts to move.

Ca of order 1 or more (viscous dominated)

Flow wins. Drops stretch and break apart, and interfaces follow the flow instead of resisting it.

The thresholds between regimes depend on the system, so treat the ranges as rough guides and check the literature for your geometry.

Ca Capillary number, dimensionless
μ Dynamic viscosity of the relevant liquid, Pa s
U Characteristic speed, m/s
γ Surface or interfacial tension, N/m
h Film thickness left by a moving bubble, m
R Tube radius, m

Key equations

The definition and one classic result that uses it.

Equation: Ca = μ U / γ

Variable legend:

  • μ: viscosity (Pa s)
  • U: speed (m/s)
  • γ: surface or interfacial tension (N/m)

Check the units: (Pa s × m/s) / (N/m) = (N/m) / (N/m), so Ca has no units.

Interpretation: Ca grows with viscosity and speed and falls with tension. Halving γ doubles Ca, just as doubling the speed does.

When to use: Use it to compare flows of different liquids, speeds or scales, or to decide whether tension or flow will control an interface.

Assumptions:

  • Newtonian liquid with a well defined viscosity
  • A single representative speed

Common mistakes:

  • Using γ in mN/m without converting to N/m, which makes Ca 1000 times too small
  • Mixing up which liquid's viscosity to use

Equation: h ≈ 1.34 R Ca^(2/3)

Variable legend:

  • h: thickness of the liquid film left on the wall (m)
  • R: tube radius (m)
  • Ca: capillary number based on bubble speed and liquid viscosity

Example: R = 0.5 mm and Ca = 10−3 give Ca^(2/3) = 0.01, so h ≈ 1.34 × 0.5 mm × 0.01 ≈ 6.7 µm.

Interpretation: Faster bubbles or lower tension leave thicker films. The film grows slower than Ca itself, as its two thirds power.

When to use: Use it for long bubbles moving slowly through tubes or microchannels, such as in segmented flow.

Assumptions:

  • Small Ca, roughly below 10⁻² to 10⁻³
  • Clean interface, negligible inertia and gravity

Common mistakes:

  • Applying it with surfactant present, which can thicken the film

How to read it

What a calculated capillary number tells you about your system.

Ca tells you which force is in charge, not the exact outcome. Pair it with real IFT and viscosity data; our surface tension of common liquids dataset is a starting point for γ, and our article on why water forms droplets in oil but oil spreads in water shows tension at work.

From Ca to meaning

If you see It usually means Check next
Ca around 10⁻⁷ to 10⁻⁶ in a waterflood Capillary forces hold residual oil in place Whether IFT can be lowered by surfactant
Ca rising by several orders after adding surfactant Trapped oil may start to move Core flood tests at the new Ca
Drops breaking up in a microchannel Local Ca has passed the breakup threshold Flow rate, viscosity ratio and IFT
Ca changes but behaviour does not Another effect, such as wettability or gravity, dominates Contact angle and Bond number

Common misconceptions

Where the capillary number is most often misapplied.

Raising the flow rate is the easy way to raise Ca

In a reservoir, speed and viscosity can change by small factors at best. Interfacial tension can fall by thousands of times with the right surfactant, so lowering γ is usually the bigger lever. Drilling and completion fluids tested under API RP 13B field testing practice show how much attention viscosity already gets.

There is one universal critical capillary number

The value at which trapped oil starts to move depends on rock structure and wettability. Published thresholds span a range, often quoted around 10⁻⁵ to 10⁻⁴ for sandstones. Treat any single number as a rough guide.

The capillary number replaces measuring tension

Ca needs a reliable γ, and a guessed value can be off by an order of magnitude once surfactants are present. Measure IFT for your actual liquids. Teaching labs can compare instruments in our guide to the best tensiometer for teaching labs, and our free contact angle tool covers the wetting side.

Related terms

Where the capillary number sits among other interfacial concepts.

Broader term

  • Interfacial tension (IFT)
  • Surface tension

Related terms

  • Spreading coefficient
  • Surfactant
  • Bond number
  • Wettability

Often confused with

  • Bond number
  • Weber number

FAQ

Short answers to the questions people ask most about the capillary number.

What does the capillary number tell you?

It tells you whether viscous flow or surface tension controls the shape and motion of an interface. Below about 10⁻³, tension usually dominates and interfaces keep their static shapes. Near or above 1, flow dominates and drops stretch and break. It is a quick way to compare very different systems on one scale.

Why is the capillary number important in enhanced oil recovery?

After waterflooding, oil is trapped in pores by capillary forces, and Ca is typically around 10⁻⁷ to 10⁻⁶. Raising Ca by several orders of magnitude can mobilise that oil. Since speed and viscosity are hard to change in a reservoir, chemical floods aim to cut interfacial tension with surfactants instead.

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

Both compare a force with surface tension. The capillary number compares viscous forces from flow with tension. The Bond number compares gravity with tension, using density difference and size. A drop at rest has a Bond number but no capillary number, because nothing is moving.

References

Primary sources and further reading.

1. Bretherton, F. P. (1961). The motion of long bubbles in tubes. Journal of Fluid Mechanics, 10(2), 166 to 188. https://doi.org/10.1017/S0022112061000160
2. Chatzis, I., and Morrow, N. R. (1984). Correlation of capillary number relationships for sandstone. Society of Petroleum Engineers Journal, 24(5), 555 to 562. https://doi.org/10.2118/10114-PA
3. de Gennes, P. G., Brochard-Wyart, F., and Quéré, D. (2004). Capillarity and Wetting Phenomena: Drops, Bubbles, Pearls, Waves. Springer, New York. https://doi.org/10.1007/978-0-387-21656-0
4. Surface tension measurement guide
5. Interfacial tension (IFT)
6. Spreading coefficient