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 $R$, apparently defying gravity. Indeed, this occurs when the liquid’s surface tension can overpower gravity; specifically, when the capillary length $L_c > R$.

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 $r$ determined by the contact angle $\theta$ of the liquid to the tube: $r = R / \cos{\theta}$. We know that the liquid is at rest when the hydrostatic pressure equals the resulting Laplace pressure:

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

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

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

Where $L_c \equiv \sqrt{\alpha / \rho g}$. This predicts the height climbed by a liquid in a narrow tube. If $\theta > 90\degree$, then $h$ 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):

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

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

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

Isolating this for $h$ 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.