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# FLOW FIELDS

When we discussed sound, we mostly used the example of air, because air happens to be easily compressible. Therefore it was easy to talk about the relations between the pressure in a bicycle tire and the amount of air we had forced into the confined space. Similarly, the motions that make up sound waves in air are fairly gross, because the compressibility allows them to be.

Sound exists in water as well, though. The important difference is that water, unlike air, is very difficult to compress. Another way to say this is that a very great pressure is required to decrease the volume of liquid water even a little, or correspondingly, that a slight decrease in the volume of some small amount of water results in a very large increase in pressure. Thus, sound waves in water, even when they are loud enough to be fatal, involve very small actual changes in the volume of any little amount of the water, and correspondingly small motions. These characteristics made water an intuitively poor candidate for introducing properties of sound, because most of them are almost impossible to observe with simple means. The same things, though, make it a very good candidate to introduce our next topic, which is the properties of flow fields.

For this case, we recognize that sound is not the only motion possible for water, and that an even more familiar one is ordinary fluid flow. This motion comes from the simple sliding of the molecules past each other, and does not require compression or any change in pressure at all. Making use of the fact that water is so hard to compress, we will will choose to ignore compression entirely, and think of water in the examples that follow as if it were an idealized, completely incompressible fluid, at which point the pressure becomes an irrelevant field, and the motion of each little amount of the water is the only field left to consider.

The important thing to recognize about flow is that it transports water from one place to another. In some cases, this can be transport on a large scale, the way a river takes water from a lake to an ocean. In others, all the transport can take place in some small region, like a whirling eddy, where water is transported around and around in circles, but on large scale doesn't go anywhere.

The transport-function of flows is encoded in the notion of a current. To define it, we consider the simplest case, of water in two reservoirs connected by a channel. These can be large containers connected by a pipe, or lakes connected by a river, or whatever we like. The only important requirement is that we be able to keep track of all the water. Anything that flows out of a reservoir must do so by means of the channel, and the channel can only gain or lose water from one of the reservoirs.

Because the water does not compress, it is easy to tell how much water is in one of the reservoirs at any given time by measuring its volume. For this example the water flowing through the channel is the current that will concern us. Clearly, some currents are `bigger' than others. We will make this intuition concrete, in terms of experiments, by saying that the size of a current is measured by how much water it transports out of the reservoir in a given time. This is an important choice to make, because a current can have several properties, most of which we are choosing to ignore in assigning it a size. We say that two currents are the same size if they transport the same amount of water from a reservoir. One of them may be a slow flow through a large pipe, and another may be a raging torrent in a very small tube. How the current is composed, we do not care. This choice is made because it turns out to be the useful notion of size for what we will say about electricity and magnetism. It is also the one that is most directly related to some very useful conservation laws.

Currents can be specified even when flow is not restricted to be through pipes. The notion of a current makes sense any time the amount of flow across some area can be measured. It is as if a pipe could have been there, to direct that same amount of fluid across the same area. Of course, once the area and the current are both specified, on a fine enough scale, all the properties of the flow are specified, because if we know both how big the area is and how much is being transported through it, we know how fast the fluid must be moving.

While we now have a language capable of describing them in principle, flow fields can be complicated, and under many circumstances it is not the flow fields themselves, but particular properties that can be extracted from them, that are simple enough to be useful. We now describe two such properties.

Next: CONSERVATION AND DIVERGENCE Up: Classical Fields and Previous: HOW MANY MODES

desmith@
Thu Aug 31 12:01:42 CDT 1995