Last updated: 2026-07-13

Capillary rise

Capillary rise is the climb of a liquid up a narrow tube against gravity, driven by surface tension rather than any applied pressure. Dip a thin glass tube into water and the water rises inside it, forming a curved surface — the meniscus — above the surrounding level. The same effect wicks water into a paper towel, draws ink to a pen nib, and pulls moisture up through soil and porous brick. Jurin's law turns this everyday observation into a simple, quantitative prediction.

The equation

For a thin, circular tube dipped in a wetting liquid, the equilibrium rise is:

$h = \frac{2\gamma \cos\theta}{\rho\, g\, r}$

where $h$ is the height the liquid rises, $\gamma$ the surface tension of the liquid, $\theta$ the contact angle between the liquid and the tube wall, $\rho$ the liquid density, $g$ the acceleration due to gravity, and $r$ the internal radius of the tube. The rise grows with surface tension and shrinks with density and radius: the narrower the tube, the higher the climb.

Symbol table

The value of $g$ used is the standard gravitational acceleration, $9.80665\ \text{m/s}^2$.

Where the formula comes from

At equilibrium, the upward pull of surface tension around the circle where the liquid meets the wall balances the weight of the raised column of liquid. The surface tension acts along the contact line of circumference $2\pi r$, and its vertical component is $\gamma\cos\theta$, giving an upward force of $2\pi r\,\gamma\cos\theta$. The weight of the column of height $h$ and cross-section $\pi r^2$ is $\rho g\,\pi r^2 h$. Setting the two equal,

$2\pi r\,\gamma\cos\theta = \rho g\,\pi r^2 h,$

and solving for $h$ gives Jurin's law. The factor $\cos\theta$ is what makes a wetting liquid rise and a non-wetting one fall.

Worked example

The role of the contact angle

The contact angle measures how well the liquid wets the tube. When adhesion to the wall beats cohesion within the liquid, the angle is small: water on clean glass sits near 0°, so $\cos\theta \approx 1$ and the rise is at its maximum. As the surface becomes less wettable the angle grows, $\cos\theta$ shrinks, and the rise falls. At exactly 90° the cosine is zero and there is no rise at all.

Beyond 90° the cosine turns negative and the formula predicts a negative height — the liquid is pushed down below the outside level. This is capillary depression, and mercury in a glass tube is the classic case: its contact angle is around 140°, so the mercury dips rather than climbs.

Why narrower tubes lift liquid higher

Because $h$ is inversely proportional to $r$, halving the tube radius doubles the rise. The upward force scales with the circumference ($2\pi r$), while the weight it must support scales with the cross-sectional area ($\pi r^2$) times the height. The ratio of force to supported weight therefore improves as the tube narrows, letting thin tubes lift liquid much higher than wide ones. This is why fine pores in soil, fabric, and blotting paper wick liquid so effectively, while the same liquid barely rises in a wide vessel.

Wicking, plants, and porous materials

Capillary action moves liquid through any network of fine channels — the fibres of a paper towel, the pores of a sponge, the gaps between soil grains. In plants, capillarity helps draw water into the narrow xylem vessels, though on its own it lifts water only a fraction of a metre. Reaching the top of a tall tree relies mainly on transpiration pull, where evaporation at the leaves places the water column under tension and cohesion keeps it intact. Capillarity is one contributor among several, most important at the small scales where surface tension dominates over gravity.

Where the simple formula breaks down

Jurin's law assumes a thin, uniform, circular tube in which the meniscus is nearly spherical and gravity across the meniscus is negligible. It also assumes a clean surface and a well-defined, stable contact angle. In wide tubes the meniscus flattens, the spherical approximation fails, and the formula overestimates the rise. Contamination, surfactants, temperature changes, and rough or non-uniform walls all shift the contact angle and surface tension, so real measurements can differ from the ideal prediction. Non-circular or interconnected pores, as in real soils and textiles, need more elaborate models.

Related tools

To relate the height of a liquid column to the pressure at its base, use the Hydrostatic Pressure Calculator. For the viscous drag on small particles settling in a fluid, see the Stokes' Law Calculator, and for steady flow through a narrow tube, the Hagen-Poiseuille Flow Calculator applies the same thin-tube geometry to volumetric flow rate.