Categories: Fluid mechanics, Fluid statics, Physics, Surface tension.

Jurin’s law

A well-known example of capillary action is when a liquid climbs up a narrow vertical tube with radius RR, apparently defying gravity. Indeed, this occurs when the liquid’s surface tension can overpower gravity; specifically, when the capillary length Lc>RL_c > R.

Let us assume that the liquid-air interface has a spherical shape, which may point up or down depending on the liquid. This interface then has a constant curvature radius rr determined by the contact angle θ\theta of the liquid to the tube: r=R/cosθr = R / \cos{\theta}. We know that the liquid is at rest when the hydrostatic pressure equals the resulting Laplace pressure:

ρgh=α2r=2αcosθR\begin{aligned} \rho g h = \alpha \frac{2}{r} = 2 \alpha \frac{\cos{\theta}}{R} \end{aligned}

Note that hh is the height of interface’s highest/lowest point; we neglect the meniscus. By isolating the above equation for hh, we arrive at Jurin’s law:

h=2αcosθρgR=2Lc2Rcosθ\begin{aligned} \boxed{ h = \frac{2 \alpha \cos{\theta}}{\rho g R} = 2 \frac{L_c^2}{R} \cos{\theta} } \end{aligned}

Where Lcα/ρgL_c \equiv \sqrt{\alpha / \rho g}. This predicts the height climbed by a liquid in a narrow tube. If θ>90°\theta > 90\degree, then hh is negative, i.e. the liquid descends below the ambient level.

An alternative derivation of Jurin’s law balances the forces instead of the pressures. On the right, we have the gravitational force (i.e. the energy-per-distance to lift the liquid), and on the left, the surface tension force (i.e. the energy-per-distance of the liquid-tube interface):

πR2ρgh2πR(αsgαsl)\begin{aligned} \pi R^2 \rho g h \approx 2 \pi R (\alpha_{sg} - \alpha_{sl}) \end{aligned}

Where αsg\alpha_{sg} and αsl\alpha_{sl} are the energy costs of the solid-gas and solid-liquid interfaces. Thanks to the Young-Dupré relation, we can rewrite this as follows:

Rρgh=2αcosθ\begin{aligned} R \rho g h = 2 \alpha \cos\theta \end{aligned}

Isolating this for hh simply yields Jurin’s law again, as expected.


  1. B. Lautrup, Physics of continuous matter: exotic and everyday phenomena in the macroscopic world, 2nd edition, CRC Press.