Continuity Equation (A₁v₁ = A₂v₂)

Water does not pile up inside a pipe and it does not vanish, so whatever volume enters one end each second must leave the other. That is the whole argument. If the pipe is narrower at the second point, the same volume has to get through a smaller opening in the same second, and the only way it can is by moving faster: in exact inverse proportion to the area. $A_1v_1 = A_2v_2$ is conservation of mass for a fluid whose density does not change, dressed in the units a pipe fitter would use.

Work a reducer. A 100 mm main carrying 1.5 m/s steps down to 50 mm. The diameter halves, so the area falls by a factor of four, and the velocity must rise by four: $v_2 = 6$ m/s. Now look at what that does to the energy carried as motion. Velocity head goes as $v^2$, so it climbs from $1.5^2/19.61 = 0.115$ m to $6^2/19.61 = 1.84$ m: a sixteenfold increase, and that energy has to come from somewhere. It comes out of the pressure, which is why a gauge downstream of a sudden contraction reads lower than one upstream even before any friction is counted.

That trade is Bernoulli's equation, and continuity is the half of it that most people find intuitive. Together they explain the venturi meter, where a deliberate constriction converts a measurable pressure drop into a flow reading; the carburettor, where the same drop pulls fuel into an air stream; and the bruit a stethoscope picks up over a narrowed artery, which is the sound of blood forced through a reduced bore at a speed that has tipped it into turbulence. Rivers do it too, running slow and broad across a plain and fast through a gorge with the same discharge in both places.

The dominant error is working in diameters instead of areas. Halving the diameter does not double the velocity, it quadruples it, because area carries the square. Halving the area doubles the velocity. The two sentences sound alike and differ by a factor of two, and the mistake usually survives because the answer still looks plausible. Convert to areas first, every time.

Three conditions on when the equation applies. It is written for incompressible flow: fine for liquids and for gases well below about Mach 0.3, but a compressible flow obeys $\rho_1A_1v_1 = \rho_2A_2v_2$ instead, and above Mach 1 the behaviour inverts so that a converging duct slows the flow rather than speeding it. It is a two-point statement along a single stream: at a tee, the inlet flow equals the sum of the branches, and applying the two-term form across a branch fitting quietly loses whatever went the other way. And unlike Bernoulli, continuity does not care about friction at all. It is pure kinematics, so it holds through a filthy pipe, across a valve, and through a pump: friction and pumps change the pressure, never the volume balance.