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Small-scale experiments have been carried out for determining the minimum vertical thermal gradient which is required to cause convection in liquids entrapped in porous media. Observations relative to the onset of convective flow in unconsolidated sands indicate that the present theories predict minimum gradients which are excessive by considerable amounts, possibly because they neglect the temperature-dependence of viscosity. The ratio of theoretical to observed gradients is found to be roughly R=(Kh2μAV/kρα)0.51,where h2 is the thermal diffusivity, μAv is average viscosity, k is flow-permeability, ρ is density of liquid, α is the coefficient of cubical expansion of the liquid, and where K = 10−3 sec.2 per cm2 °C for c.g.s. units. By extrapolation, it is possible to strengthen the earlier conclusion that convection occurs in the Woodbine sand near the Mexia fault zone.

In 1964, Spencer proposed a model for plane deformation of soils based upon the concept that the strain at any point may be considered as the resultant of simple shears on the two surface elements where Coulomb’s yield condition is met. Gravitation and acceleration terms were neglected in his full field equations. These terms are included in the present treatment, however, since they play an essential role in granular materials flow problems. It is shown that the field equations remain hyperbolic, and the characteristic equations are derived. In addition, a streamline equation, similar to Bernoulli’s classical equation for fluid flow, is derived and used together with the continuity equation to obtain one-dimensional approximate solutions to some typical hopper and chute problems. A solution is obtained for the nonsteady flow from a wedge-shaped hopper when the gate is suddenly opened, including an equation for the minimum slope necessary to prevent arching. Another solution is obtained for the profile of a hopper which has constant wall pressure. Still another solution is obtained for the relationship between the height of a chute and its exit velocity, and it suggests that the maximum trajectory usually is obtained with a horizontal exit since an upward-sloping exit requires a velocity jump at the minimum point in the chute, similar to a hydraulic jump in fluid flow. All of these solutions are ideal in the sense that they include no wall resistance to the flow, and therefore represent maximum flow rates.

The theory of the critical gradient for onset of thermal convection in a fluid entrapped in a porous medium is extended to allow for the exponential dependence of viscosity upon temperature and for non-linear vertical temperature-distributions which characterize transients in heat flow. Although there are approximations in the mathematical treatment, the extended theory agrees rather well with experimental data, as the simple theory does not in certain instances. New experimental data are reported for the critical gradient, obtained largely with silicone fluids in unconsolidated sands.