As in Chapter 2, so again here the intention is to review ideas that are already familiar, rather than to introduce the unfamiliar; to build a springboard, but not yet to leap off into space. The familiar idea is of flow down a gradient—water running downhill. Parallels are electric current in a wire, salt diffusing inland from the sea, heat flowing from the fevered brow into the cool windowpane, and helium diffusing through the membrane of a helium balloon. For any of these, we can imagine a linear relation: . . . Flow rate across a unit area = (conductivity) x (driving gradient) . . . where the conductivity retains a constant value, and if the other two quantities change, they do so in a strictly proportional way. Real life is not always so simple, but this relation serves to introduce the right quantities, some suitable units and some orders of magnitude. For present purposes, the second and fourth of the examples listed are the most relevant. To make comparison easier we imagine a barrier through which salt can diffuse and through which water can percolate, but we imagine circumstances such that only one process occurs at a time. Specifically, imagine a lagoon separated from the ocean by a manmade dike of gravel and sand 4 m thick, as in Figure 3.1. If the lagoon is full of seawater but the water levels on the two sides of the dike are unequal, water will percolate through the dike, whereas if the levels are the same and the dike is saturated but the lagoon is fresh water, salt will diffuse through but there will be no bulk flow of water. (More correctly, because seawater and fresh water have different densities, and because of other complications, the condition of no net water flow would be achieved in circumstances a little different from what was just stated. For present purposes all we need is the idea that conditions exist where water does not percolate but salt does diffuse.) For flow of water driven by a pressure gradient, suitable units are shown in the upper part of Table 3.1 and for diffusion of salt driven by a concentration gradient, suitable units are shown in the lower part.
Influence of slip velocity on the flow of viscous fluid through a porous medium in a permeable tube with a variable bulk flow rate
