---
title: "Capillary length"
sort_title: "Capillary length"
date: 2021-03-29
categories:
- Physics
- Fluid mechanics
- Fluid statics
- Surface tension
layout: "concept"
---

**Capillary action** refers to the movement of liquid
through narrow spaces due to surface tension, often against gravity.
It occurs when the [Laplace pressure](/know/concept/young-laplace-law/)
from surface tension is much larger in magnitude than the
[hydrostatic pressure](/know/concept/hydrostatic-pressure/) from gravity.

Consider a spherical droplet of liquid with radius $$R$$.
The hydrostatic pressure difference
between the top and bottom of the drop
is much smaller than the Laplace pressure:

$$\begin{aligned}
    2 R \rho g \ll 2 \frac{\alpha}{R}
\end{aligned}$$

Where $$\rho$$ is the density of the liquid,
$$g$$ is the acceleration due to gravity,
and $$\alpha$$ is the energy cost per unit surface area.
Rearranging the inequality yields:

$$\begin{aligned}
    R^2 \ll \frac{\alpha}{\rho g}
\end{aligned}$$

From this, we define the **capillary length** $$L_c$$
such that gravity is negligible if $$R \ll L_c$$:

$$\begin{aligned}
    \boxed{
        L_c
        \equiv \sqrt{\frac{\alpha}{\rho g}}
    }
\end{aligned}$$

In general, for a system with characteristic length $$L$$,
the relative strength of gravity compared to surface tension
is described by the **Bond number** $$\mathrm{Bo}$$
or **Eötvös number** $$\mathrm{Eo}$$:

$$\begin{aligned}
    \boxed{
        \mathrm{Bo}
        \equiv \mathrm{Eo}
        \equiv \frac{L^2}{L_c^2}
    }
\end{aligned}$$

Capillary action is observed when $$\mathrm{Bo \ll 1}$$,
while for $$\mathrm{Bo} \gg 1$$ surface tension is negligible.

For an alternative interpretation of $$\mathrm{Bo}$$,
let $$m \equiv \rho L^3$$ be the mass of a cube with side $$L$$
such that its weight is $$m g$$.
The tension force on its face is $$\alpha L$$,
so $$\mathrm{Bo}$$ is simply the force ratio:

$$\begin{aligned}
    \mathrm{Bo}
    = \frac{m g}{\alpha L}
\end{aligned}$$



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