Royal Society Publishing

The Stability of a Viscous Liquid in a Vertical Tube Containing Porous Material

R. A. Wooding


If a long vertical tube filled with porous material contains a viscous solution, the density of which increases with height as a result of the presence of the dissolved substance, the equilibrium of the liquid is stable provided that the density gradient does not anywhere exceed the value $\frac{\text{d}\rho}{\text{d}Z}=\frac{3\cdot 390\mu \kappa}{gkb^{2}}$. Here $\kappa $, the diffusivity of the solute through the saturated porous medium, is defined to be the quantity of solute diffusing across unit area within the porous medium per unit time under unit density gradient. The above expression for the density gradient at neutral stability has been compared experimentally with Taylor's value for the corresponding density gradient in a vertical capillary tube. For a porous medium consisting of randomly packed glass spheres of mean diameter about 0$\cdot $2 mm and porosity $\epsilon $ = 0$\cdot $365, it has been found that the two results are consistent provided that the ratio $\kappa $/D$\epsilon $ = 0$\cdot $633, where D is the molecular diffusivity of the solute when the porous medium is absent. As this dimensionless ratio is a property of the porous material alone, it can be determined directly by diffusion measurements. An alternative method of measuring $\kappa $/D$\epsilon $, based upon an electrical analogue, has led to a value of 0$\cdot $641 for the same porous material, which is in good numerical agreement.