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

# Capillary length

Capillary action refers to the movement of liquid through narrow spaces due to surface tension, often against gravity. It occurs when the Laplace pressure from surface tension is much larger in magnitude than the 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}
1. B. Lautrup, Physics of continuous matter: exotic and everyday phenomena in the macroscopic world, 2nd edition, CRC Press.