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 RR. The hydrostatic pressure difference between the top and bottom of the drop is much smaller than the Laplace pressure:

2Rρg≪2αR\begin{aligned} 2 R \rho g \ll 2 \frac{\alpha}{R} \end{aligned}

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

R2≪αρg\begin{aligned} R^2 \ll \frac{\alpha}{\rho g} \end{aligned}

From this, we define the capillary length LcL_c such that gravity is negligible if R≪LcR \ll L_c:

Lc≡αρg\begin{aligned} \boxed{ L_c \equiv \sqrt{\frac{\alpha}{\rho g}} } \end{aligned}

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

Bo≡Eo≡L2Lc2\begin{aligned} \boxed{ \mathrm{Bo} \equiv \mathrm{Eo} \equiv \frac{L^2}{L_c^2} } \end{aligned}

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

For an alternative interpretation of Bo\mathrm{Bo}, let m≡ρL3m \equiv \rho L^3 be the mass of a cube with side LL such that its weight is mgm g. The tension force on its face is αL\alpha L, so Bo\mathrm{Bo} is simply the force ratio:

Bo=mgαL\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.