The Dynamics of Capillary Flow

TLDR

The rate of liquid penetration into a small cylindrical capillary of radius r is given by dl/dt = P(r²+4εr)/(8ηl), where P is driving pressure, ε the slip coefficient, and η viscosity. The study aims to show that, for a porous body modeled as many small cylindrical capillaries, the penetrated volume after time t scales with the square root of γt/η. By integrating the capillary flow equation, the distance penetrated by a liquid under capillary pressure alone in a horizontal capillary is √(γ r t cosθ / 2η), with γ surface tension and θ contact angle. Experiments with mercury, water and other liquids confirm the theoretical predictions, and the dynamic capillary method for measuring surface tension is shown to have advantages over the static method. The coefficient of penetrance (γ cosθ / 2η) defines liquid penetrativity.

Abstract

Penetration of Liquids into Cylindrical Capillaries.---The rate of penetration into a small capillary of radius $r$ is shown to be: $\frac{\mathrm{dl}}{\mathrm{dt}}=\frac{P({r}^{2}+4\ensuremath{\epsilon}r)}{8\ensuremath{\eta}l}$, where $P$ is the driving pressure, $\ensuremath{\epsilon}$ the coefficient of slip and $\ensuremath{\eta}$ 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 ($\frac{\ensuremath{\gamma}\mathrm{rt}\ifmmode\cdot\else\textperiodcentered\fi{}cos\ensuremath{\theta}}{2\ensuremath{\eta}}$), where $\ensuremath{\gamma}$ is the surface tension and $\ensuremath{\theta}$ the angle of contact. The quantity ($\frac{\ensuremath{\gamma}cos\ensuremath{\theta}}{2\ensuremath{\eta}}$) is called the coefficient of penetrance or the penetrativity of the liquid.Penetration of Liquids into a Porous Body.---(1) Theory. If a porous body behaves as an assemblage of very small cylindrical capillaries, the volume which penetrates in a time $t$ would be proportional to the square root of ($\frac{\ensuremath{\gamma}t}{\ensuremath{\eta}}$). (2) Experiments with mercury, water and other liquids completely verify the theoretical deductions.Dynamic capillary method of measuring surface tension is described. It possesses certain advantages on the static method of capillary rise.