The Dynamics of Capillary Flow

Penetration of Liquids into Cylindrical Capillaries.—The rate of penetration into a small capillary of radius $r$ is shown to be: $\frac{\mathrm{dl}}{\mathrm{dt}}=\frac{P(r^2+4\mathrm{}\epsilon r)}{8\eta l}$, where $P$ is the driving pressure, $\epsilon$ the coefficient of slip and $\eta$ the viscosity. By integrating this expression, the distance penetrated by a liquid flowing under capillary pressure alone into a horizontal capillary or one with small internal surface is found to be the square root of ($\frac{\gamma \mathrm{rt}·cos\theta }{2\eta }$), where $\gamma$ is the surface tension and $\theta$ the angle of contact. The quantity ($\frac{\gamma cos\theta }{2\eta }$) is called the coefficient of penetrance or the penetrativity of the liquid.