the continuity equation
The continuity equation is the simple bookkeeping rule that a fluid speeds up when it is forced through a narrower space. Put your thumb over the end of a garden hose and the water shoots out faster; watch a river quicken where its banks pinch together. It answers a very intuitive question: if the same amount of fluid has to keep passing through a pipe, what happens to its speed when the pipe gets thinner?
Precisely, for an incompressible fluid in steady flow, the volume passing any cross-section each second must be the same everywhere, because fluid is neither created nor destroyed and cannot pile up. This gives A_1 v_1 = A_2 v_2, where A is the cross-sectional area of the pipe and v is the fluid's speed there. The product A v, the volume flow rate, stays constant along the pipe. So where the area is small the speed must be large, and where the area is wide the speed drops.
This is really the law of conservation of mass wearing a fluid disguise, and it underlies everything from blood flow through arteries to air through a ventilation duct to water through a river delta. One honest condition: the clean form A_1 v_1 = A_2 v_2 assumes the fluid is incompressible and nothing leaks in or out. For a gas being strongly compressed the more general statement keeps the mass flow, density times area times speed, constant instead.
Water flows at 1 m/s in a pipe of cross-section 0.01 m^2. Where the pipe narrows to 0.0025 m^2, the continuity equation gives v_2 = A_1 v_1 / A_2 = (0.01 × 1) / 0.0025 = 4 m/s, four times faster in a quarter the area.
Narrow the channel and the fluid speeds up in exact proportion, keeping A v constant.
The tidy form A_1 v_1 = A_2 v_2 assumes an incompressible fluid with no leaks. It is conservation of mass applied to a flow.