Free convection of fluid in a vertical tube filled with porous material

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Abstract
The problem of an unstable fluid overturning in a vertical tube filled with porous material is treated by an approximation of boundary-layer type. It is shown that the fluid can experience a pseudo-inertial effect, in which variations in density across the tube exhibit properties analogous to variations of momentum in an inertial flow. The mean fluid density and mean-square vertical velocity over a horizontal cross-section of the tube are related by a pair of hyperbolic equations, for which there exist two systems of characteristics. It is shown that changes in the mean density of the fluid can be propagated as discontinuities. For discontinuities of finite amplitude, two jump conditions are derived, one of which is found to involve an undetermined parameter λ. The theory is applied to the case of a vertical tube containing porous material saturated with water, which is attached at the top to a reservoir filled with an aqueous solution (an analogue of Taylor's (1954) experiment). The motion of a finite discontinuity which arises at the initial unstable interface is calculated by two approximate methods. These results compare satisfactorily with the data from three experiments, using tubes of circular cross-section, provided that the value of λ is about 0·75. If the theoretical interpretation is correct, it appears that convective flow ceases when the vertical density gradient is slightly less than the neutral value.

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