Capillary Action Calculator

The capillary action calculator solves Jurin's law for the height a liquid rises (or falls) in a narrow vertical tube. The underlying physics is the same whether you're studying how water climbs the xylem of a tree, why mercury is depressed in a glass barometer, how a paper towel wicks up a spill, or how a thin capillary tube draws a liquid column several centimetres above the free surface. The formula is simple, the inputs are well-defined, and the result has a clear physical meaning. This page explains the theory, gives the standard reference values for common liquids, and walks through worked examples.

The calculator accepts five inputs: surface tension γ, contact angle θ, liquid density ρ, gravitational acceleration g, and the inner radius of the capillary tube r. The output is the capillary height h, signed so that positive values mean a rise and negative values mean a depression. The calculator includes preset values for water (on clean glass and on PTFE), ethanol, mercury, glycerol, olive oil, hexane, and blood plasma, so you can see realistic results without digging through a data table.

How to use the capillary action calculator

1. Enter the surface tension. γ is in newtons per metre (N/m). For water at 20 °C use 0.0728 N/m. Load a preset to fill γ, θ, and ρ together.
2. Enter the contact angle. θ is in degrees between 0° and 180°. 0° = complete wetting (the liquid climbs right up the wall). 90° = neutral (no rise, no depression). 180° = total non-wetting (maximum depression). For water on clean glass the angle is very close to 0°; for mercury on glass it is about 140°.
3. Enter the density. ρ is in kg/m³. Water at 20 °C is 998 kg/m³; mercury is 13,534 kg/m³.
4. Enter the gravity. g is in m/s². Earth at sea level is 9.80665 m/s². The Moon is about 1.62 m/s² and Mars is about 3.71 m/s², useful for planetary-science examples.
5. Enter the capillary radius. r is the inner radius in metres. A typical laboratory glass tube has r ≈ 0.5 mm = 0.0005 m. A fine xylem vessel has r ≈ 20 µm = 0.000020 m.
6. Click Calculate. The result panel shows the capillary height in metres and in millimetres, plus the wetting classification: capillary rise (h > 0), capillary depression (h < 0), or neutral (h = 0).

The classic demonstration is a clean glass tube of radius 0.5 mm dipped in water: the water climbs to about 2.97 cm. The same tube in mercury gives a depression of about 1.12 cm. Both numbers come out of the same formula with the same inputs, only the contact angle and the density change.

Jurin's law

The formula is derived from a force balance on the lifted column. The upward force is the vertical component of the surface tension acting along the tube's inner circumference, 2πr · γ · cos θ. The downward force is the weight of the lifted column, πr² · ρ · g · h. Equating them and solving for h gives:

h = (2γ · cos θ) / (ρ · g · r)

where:

• h, capillary rise (m). Positive = rising, negative = depressed.
• γ, surface tension of the liquid (N/m).
• θ, contact angle between the liquid and the tube wall (degrees).
• ρ, density of the liquid (kg/m³).
• g, gravitational acceleration (m/s²).
• r, inner radius of the capillary tube (m).

The sign of h is governed by cos θ. When the liquid wets the tube (θ < 90°), cos θ > 0 and h is positive. When the liquid does not wet the tube (θ > 90°), cos θ < 0 and h is negative. When θ = 90°, h = 0 exactly, the liquid is neither pulled up nor pushed down by the tube wall.

The formula is exact for a cylindrical tube in the limit that the meniscus curvature is small compared to the tube radius. For most laboratory tubes (r > 0.1 mm) the error is well under 1%. For very thin tubes (r < 1 µm) the linear-theory approximation breaks down and the full Young-Laplace equation must be used, but the Jurin formula remains a useful engineering estimate.

Why h is inversely proportional to r

The capillary height scales as 1/r. Halving the radius doubles the height. This is why a thin glass tube draws water up several centimetres while a wide cup does not, the same physics, but the inverse relationship makes the effect dramatic for narrow tubes. The underlying reason is geometric: surface tension acts along the tube's circumference (∝ r), but the weight of the lifted column is proportional to the cross-sectional area (∝ r²). The ratio is ∝ 1/r.

This is also why trees can move water from roots to leaves. Xylem vessels are typically 20 to 200 µm in radius, and the resulting Jurin height from capillary action alone is several metres. Combined with the negative pressure generated by evaporation at the leaves (the cohesion-tension theory), the same physics allows water transport in the tallest trees (over 100 m, far beyond what pure capillary action could achieve).

Real-world applications

Plant biology and transpiration. Xylem vessels are effectively narrow capillary tubes. The driving force for water transport is a combination of capillary action (Jurin) and the negative pressure generated by evaporation at the leaves (the cohesion-tension theory). Without capillary action, the initial meniscus formation in the xylem would be impossible, and the whole transport chain would fail.

Paper chromatography and wicking tests. A paper strip dipped in solvent draws the liquid upward by capillary action. The rate of climb is a quality-control metric for filter papers, textiles, and absorbent hygiene products. The same Jurin-related physics is used in lateral-flow assays (home pregnancy tests, COVID antigen tests) where the wicking rate determines the test's time-to-result.

Mercury barometers and thermometers. Mercury barometers and mercury-in-glass thermometers rely on the depression of mercury in glass (θ ≈ 140°, h < 0). The non-wetting behaviour keeps the mercury column stable and free of air back-contamination. Mercury's high surface tension (0.485 N/m) and high density (13,534 kg/m³) are essential to making the depression visible in a practical tube size.

Oil recovery and porous media. Enhanced oil recovery depends on understanding capillary pressures in reservoir rock. The same Jurin equation, extended to networks of pores of varying radii, predicts how water displaces oil (or vice versa) through the pore space. Reservoir engineers use capillary pressure curves to design waterfloods and EOR strategies.

Building materials and rising damp. Rising damp in masonry walls is a capillary phenomenon. Engineers use Jurin's-law-derived calculations to choose damp-proof courses and to design pore-size distributions in rendering materials that minimise capillary rise. Conversely, archaeologists use the capillary rise of soluble salts in historic stone to predict the rate of weathering.

Inkjet printing and microfluidics. The stability of an inkjet droplet and the flow of liquid in a microfluidic channel depend on the same surface-tension and contact-angle physics that Jurin's law describes. Lab-on-a-chip devices are designed with characteristic channel widths chosen so that capillary forces drive the flow without pumps.

Why g appears in the denominator

Gravity is the restoring force that limits the capillary rise. The cap of the lifted column is in equilibrium when the upward force of surface tension exactly balances the downward weight of the column. On the Moon (g ≈ 1.62 m/s²), the same tube would draw water about 6× higher than on Earth. On Jupiter (g ≈ 24.79 m/s²), the rise would be only about 40% of the Earth value. This is why proposed lunar greenhouses can rely partly on capillary-driven irrigation in low gravity, a bigger height of water per unit surface area.

Edge cases and pitfalls

θ = 90°. The liquid neither rises nor falls. This is the precise condition for a perfectly neutral surface, drops neither spread nor bead. Such surfaces are rare in everyday life but achievable with specially prepared polymer coatings.

θ > 90°. The liquid is depressed. Mercury on glass (θ ≈ 140°) and water on PTFE / Teflon (θ ≈ 110°) are the everyday examples. The meniscus is convex (curves downward) and the liquid level inside the tube is below the surrounding flat surface.

Very thin tubes. For r smaller than about 1 µm, the linear Jurin formula begins to underestimate the true rise. The full Young-Laplace equation must be used, which accounts for the changing curvature of the meniscus as the tube narrows. This is the regime of nanofluidics and the physics of biological nanopores.

Non-Newtonian or viscous liquids. Jurin's law assumes equilibrium. The time to reach equilibrium scales with viscosity. For glycerol (about 1000× the viscosity of water), the rise may take several minutes rather than the fraction of a second for water. For very viscous liquids the equilibrium may be impractical to observe.

Dynamic contact angle. On real surfaces, the contact angle during advance (the liquid is climbing) is greater than the equilibrium angle, and during recede (the liquid is falling) it is smaller. The difference is contact-angle hysteresis. Jurin's law uses the equilibrium value, which gives the maximum height; the actual rise is usually slightly less.

Worked example: water in a 0.5 mm glass tube

The most cited capillary-rise demonstration. A clean glass tube of inner radius 0.5 mm is dipped in water at 20 °C. The relevant parameters are:

• γ = 0.0728 N/m
• θ ≈ 0° (water on clean glass is essentially complete wetting)
• ρ = 998 kg/m³
• g = 9.80665 m/s²
• r = 0.0005 m

Plugging in:

h = (2 × 0.0728 × cos(0°)) / (998 × 9.80665 × 0.0005) h = 0.1456 / 4.8949 h ≈ 0.02974 m ≈ 2.97 cm

This is the textbook value. Cutting the radius in half (to 0.25 mm) doubles the height to about 5.95 cm. Cutting the radius by a factor of 10 (to 0.05 mm) increases the height by a factor of 10 to about 29.7 cm, a dramatic demonstration of the 1/r scaling.

Worked example: mercury in a 0.5 mm glass tube

The classic depression example. The same glass tube, but now dipped in mercury at 20 °C:

• γ = 0.485 N/m (much higher than water)
• θ ≈ 140° (mercury does not wet glass)
• ρ = 13,534 kg/m³ (much higher than water)
• g = 9.81 m/s²
• r = 0.0005 m

Plugging in:

h = (2 × 0.485 × cos(140°)) / (13,534 × 9.81 × 0.0005) h = (2 × 0.485 × −0.7660) / (66.39) h = −0.7432 / 66.39 h ≈ −0.01119 m ≈ −1.12 cm

The mercury column inside the tube sits about 1.12 cm below the surrounding flat surface. The negative sign indicates depression. This is the physics that makes mercury barometers work: the mercury in the tube is held up by atmospheric pressure, but the level inside the tube is depressed below the level in the reservoir because of the non-wetting contact angle.

Worked example: water on the Moon

A hypothetical lunar greenhouse uses capillary irrigation to deliver water to plants. The same 0.5 mm tube as the first example, but on the Moon:

• γ = 0.0728 N/m (surface tension is a property of the liquid, not the gravity)
• θ = 0° (same as on Earth)
• ρ = 998 kg/m³ (same as on Earth)
• g = 1.62 m/s² (Moon's gravity)
• r = 0.0005 m

Plugging in:

h = (2 × 0.0728 × 1) / (998 × 1.62 × 0.0005) h = 0.1456 / 0.8084 h ≈ 0.180 m ≈ 18.0 cm

The same capillary tube would draw water about 18 cm on the Moon, roughly 6× the Earth height. This is consistent with the 1/g scaling implied by the formula.

Frequently Asked Questions

What is the difference between capillary rise and capillary depression? Capillary rise occurs when the liquid wets the tube (contact angle less than 90°). The liquid climbs the tube wall and forms a concave meniscus. Capillary depression occurs when the liquid does not wet the tube (contact angle greater than 90°); the liquid is pushed down and forms a convex meniscus. Mercury on glass is the classic example of depression.

Why does mercury behave differently from water in a glass tube? Mercury has a much higher surface tension than water (about 0.485 N/m vs 0.0728 N/m) and a much higher density (about 13,534 kg/m³ vs 998 kg/m³). More importantly, it does not wet glass, the contact angle is about 140° so cos θ is negative. The combination of high surface tension and a non-wetting contact angle produces a strong depression of several millimetres in a thin glass tube.

How do I measure the contact angle? The standard method is the sessile-drop technique: place a small drop of the liquid on a flat surface of the solid (cleaned and prepared to match the application), photograph the drop's profile from the side, and measure the angle between the drop's tangent at the contact line and the solid surface. Optical tensiometers automate this. For rough or reactive surfaces, the apparent contact angle varies with drop size and waiting time (hysteresis).

Does the formula work for non-circular tubes? The Jurin formula h = (2γ·cos θ) / (ρ·g·r) is exact for a cylindrical tube of radius r. For a tube with a non-circular cross-section, the equivalent radius is r = 2·A/P, where A is the cross-sectional area and P is the wetted perimeter. This is the hydraulic radius. The formula is a good approximation when the cross-section is roughly equidimensional and the tube is much longer than its width.

What gravity do I use on another planet? Use the local gravitational acceleration in m/s². Earth at sea level is 9.80665 m/s²; the Moon is 1.62 m/s²; Mars is 3.71 m/s²; Jupiter is 24.79 m/s²; the Sun is 274 m/s². The capillary rise is inversely proportional to g, so on the Moon it would be about 6× higher than on Earth for the same tube and liquid.

Why does the calculator give the same height in metres and millimetres? The metres value is the formula's raw output. The millimetres value is the same number scaled by 1000. Reading the rise in millimetres is convenient for typical laboratory glass tubes (radii 0.1 to 1 mm produce rises of 1 to 50 mm), whereas for very thin tubes or very large ones, the metres value is more natural.

What is the relationship between Jurin's law and the Young-Laplace equation? Jurin's law is a special case of the Young-Laplace equation applied to a cylindrical tube. The Young-Laplace equation gives the pressure difference across any curved interface as ΔP = γ (1/R₁ + 1/R₂), where R₁ and R₂ are the principal radii of curvature. For a meniscus in a cylindrical tube, one radius is the tube radius r and the other is infinite (the meniscus is a section of a cylinder), so ΔP = γ / r. Equating ΔP with ρ g h gives Jurin's law directly.

Can the calculator be used for oil recovery calculations? Yes, with care. The Jurin formula applies to a single capillary tube. For a network of pores in reservoir rock, the same pressure-balance equation is applied to each pore, and the capillary pressure curve (P_c vs saturation) is the integrated result. Reservoir engineers use this curve to design waterfloods and predict oil recovery. The calculator is a single-pore idealisation, useful for understanding the underlying physics but not a substitute for full reservoir simulation.

Why does temperature affect the result? Surface tension and density both depend on temperature. Surface tension of water decreases from about 0.074 N/m at 15 °C to about 0.069 N/m at 80 °C. Density decreases from 999.1 kg/m³ at 0 °C to 971.8 kg/m³ at 80 °C. The combined effect is a slightly lower capillary rise at higher temperature. The calculator's preset values are at 20 °C; for other temperatures, enter the appropriate γ and ρ.

Is the calculator suitable for measuring the surface tension of a liquid? Indirectly, yes. If you measure the capillary rise h for a liquid of known density ρ, known contact angle θ, and tube radius r, you can solve for γ: γ = ρ · g · r · h / (2 · cos θ). This is the capillary-rise method of surface-tension measurement, a classical technique in physical chemistry. The calculator reverses the usual flow: given γ, it computes h; given the experimental h, you can read off γ.

References

• Encyclopaedia Britannica, Capillary Action, the physics of surface tension and meniscus rise. https://www.britannica.com/science/capillary-action
• Batchelor, G. K. (1967), An Introduction to Fluid Dynamics, Cambridge University Press, for the derivation of the capillary rise equation.