Flow and continuity
The continuity equation is not mainly a formula about pipes. It is a statement that mass is accounted for. Once that idea is clear, contractions, branches, nozzles and even compressible-flow problems become variations of the same conservation argument.
Flow rate and mass flow
Take a cross-section of area A. In a short time Δt, fluid with average normal speed v travels a distance vΔt, so the volume crossing the section is approximately ΔV = AvΔt. Therefore:
Q = dV/dt = Av
Q is the volume flow rate, in m³ s⁻¹.
If density is ρ, the mass crossing per second is ṁ = ρAv. This is the more general quantity because mass remains conserved even when density changes.
Continuity as conservation of mass
For steady flow through one inlet and one outlet, mass cannot accumulate inside the chosen region. The mass flow rates must match:
ρ₁A₁v₁ = ρ₂A₂v₂
For an incompressible fluid, ρ₁ = ρ₂, so:
A₁v₁ = A₂v₂ = Q
If area falls to one quarter, average speed rises by a factor of four. The geometry matters because area, not diameter, appears in the equation.
Worked example: a nozzle
Water moves through a hose of diameter 3.0 cm at 1.5 m s⁻¹ and enters a nozzle of diameter 1.0 cm.
Because A ∝ d², A₁/A₂ = (3.0/1.0)² = 9.
Then v₂ = (A₁/A₂)v₁ = 9 × 1.5 = 13.5 m s⁻¹.
The speed rises by nine, not three, because reducing diameter by three reduces area by nine.
Branches and storage
Steady incompressible junction
Qin = Q₁ + Q₂ + …
The total volume flow leaving equals the total volume flow entering.
General mass balance
ṁin − ṁout = dMinside/dt
If a tank is filling or emptying, inflow and outflow do not have to be equal at that instant.
This control-volume view prevents a common mistake: continuity does not mean every branch must carry the same flow.
Local form of the equation
The pipe formula is one special case of a field equation:
∂ρ/∂t + ∇·(ρu) = 0
The first term measures local density change; the divergence term measures whether mass flux spreads out from or converges into a small region. For incompressible flow, a common consequence is ∇·u = 0.
“Incompressible” means a moving fluid element keeps essentially the same density. It does not require every point in every problem to have the same numerical density under all circumstances.
Continuity is not Bernoulli
Continuity
Comes from mass conservation.
It connects ρ, A and v.
Bernoulli
Comes from an energy balance under additional assumptions.
It connects pressure, speed and elevation along a streamline.
A contraction can force speed to rise through continuity, but continuity alone does not determine the pressure. Pressure requires a momentum or energy relation and the assumptions that go with it.
Experiment and model limits
A simple test is to collect the outlet fluid for a measured time. The collected volume divided by time gives Q; measuring the internal diameter gives A and hence an average speed v = Q/A. Repeating with a different nozzle directly tests the area–speed relation.
The simple form A₁v₁ = A₂v₂ assumes steady, incompressible flow with no leakage or storage between the sections. For a gas that changes density strongly, use ρAv. For a filling tank, include accumulation. For a leaking line, include the leak as an additional outlet.
Exercises
Pipe contraction
An incompressible liquid moves at 2.0 m s⁻¹ through 6.0 cm². The pipe narrows to 2.0 cm². Find the new average speed.
Solution
A₁v₁ = A₂v₂, so v₂ = (6.0/2.0) × 2.0 = 6.0 m s⁻¹.
Junction
A steady incompressible flow of 12 L s⁻¹ reaches a junction. One branch carries 4 L s⁻¹. Find the other branch flow if there is no storage.
Solution
Q₂ = 12 − 4 = 8 L s⁻¹.
Compressible flow
Why is A₁v₁ = A₂v₂ insufficient when gas density changes substantially?
Solution
Because volume flow is not conserved when density changes. For steady one-inlet/one-outlet flow, use ρ₁A₁v₁ = ρ₂A₂v₂.