continuity equation fluid flow
Send water through a pipe that was thick and then grows thin, and the current grows strong at the narrow throat. The same water has to leave through a narrower door, so it has no choice but to go faster there. Squeeze the throat yourself and the reason comes into the hand. The one fact that the water which entered does not vanish ties every cross section of the pipe.
Drag the probe along the pipe and read the area A and the speed v, and the narrower it is the faster it is. The wide side is easy and the throat is fierce. Yet the product A·v is the same everywhere. Even when the place changes, that product staying one value is the whole of this picture.
Now squeeze the throat yourself and the relation grows clear. Cut the area in half and the speed doubles, so area and speed lock in inverse proportion. The more you squeeze, the stronger the current, and when you open it again the current grows easy again. The speed fills the place by as much as the face shrank.
A and v move opposite each other, but multiply them and they are the same at the two sections. This product A·v is the volume that passes in a unit of time, the flow rate. The volume that left one side has to equal the volume that entered the other, so as long as the pipe does not leak the flow rate is kept as one line.
In symbols, A1 v1 = A2 v2, that is Q = A·v stays constant. Drag the speed v1 of the front section and the speed at the throat follows as v2 = v1·A1A2. Water is hardly compressed, so it passes with its volume intact, and when the density changes, as in a gas, ρ·A·v including that density stays constant.
Narrow the end of a hose and the same water crowds into the thin hole and shoots far. A nozzle, a spray, and a syringe are all this principle. Narrowing the hole is the same act as strengthening the current, a picture of the same amount leaving through a narrower door.