Continuity: Why the Jet Narrows

Water leaving a tap narrows as it falls, and it is not evaporating on the way down. The same mass passes every level of the stream per second. Gravity makes the lower fluid faster. Same mass per second, moving faster, must occupy a smaller cross-section.

That sentence is the entire equation. Everything else is bookkeeping notation for it.

Draw an imaginary boundary anywhere in the flow. Mass entering minus mass leaving equals the rate of accumulation inside. In steady flow nothing accumulates, so inflow equals outflow exactly.

"Steady" means the pattern does not change with time — not that nothing moves. A river in flood at constant discharge is steady; a tap being opened is not.

The general statement is ρ₁A₁v₁ = ρ₂A₂v₂ — the mass flow rate ṁ in kg/s is the same at both sections. The familiar A₁v₁ = A₂v₂ drops density and is only valid when density is constant.

Area scales as the square of diameter. Halve the diameter and area falls to a quarter, so velocity must quadruple. A 100 mm main narrowing to 50 mm turns 1 m/s into 4 m/s.

Friction loss goes roughly as v², so that section dissipates sixteen times the head per metre. This is why a single small restriction can dominate an entire pipe run, and why pipe sizing is dominated by the narrowest section.

At a branch, total inflow equals total outflow, because the junction has no capacity to store. A 60 L/s main splitting into 25 and 35 L/s branches closes exactly.

This is structurally identical to Kirchhoff's current law at a node — and that is not a coincidence worth ignoring, since pipe-network solvers borrow their iterative methods directly from circuit analysis.

Continuity is a kinematic constraint. It relates velocities using geometry only — no forces, no energy, no fluid properties. That makes it the free half of most problems, and the reason Bernoulli becomes solvable: two unknowns, and continuity supplies the second equation.

A venturi meter is exactly this pairing. The area ratio fixes the velocity ratio through continuity; the measured pressure drop then converts to a flow rate through Bernoulli.

• It says nothing about why the velocity changed — that is the momentum equation's job.
• Av gives only a mean velocity. The real profile is zero at the wall and peaks at the centre, which matters when computing kinetic energy.
• The simple form assumes steady flow. A filling tank or a slammed valve brings the storage term back and it cannot be dropped.