Fluid Flow & Continuity

Following every parcel of water through a pipe is hopeless, so fluid mechanics watches fixed points instead and records the velocity there. If the velocity at every point stays the same from one moment to the next, the flow is steady. Steady does not mean that nothing accelerates: a parcel entering a nozzle speeds up because it moves to a place where the velocity is larger, while the pattern itself never changes. A streamline is a curve tangent to the velocity all along its length, and in steady flow it is also the path a parcel follows. Two streamlines can never cross, because fluid at the crossing point would need two velocities at once. Wrap a closed loop of streamlines around some fluid and you have a stream tube: the velocity is tangent to its walls, so nothing leaks through them. A stream tube is a pipe drawn by the flow itself, and everything below applies to it as well as to real pipes.

Take a flat cross-section of area A where the fluid moves at speed v perpendicular to it. In a short time dt every parcel advances v dt, so the fluid that crosses fills a slab of volume dV = A v dt. Divide by dt: Q = dV/dt = Av, in m³ s⁻¹. A tap that fills a 1.5 L jug in 12 s delivers Q = 1.5 × 10⁻³ m³ ÷ 12 s = 1.25 × 10⁻⁴ m³ s⁻¹; through a spout of area 0.80 cm² that requires v = Q/A = (1.25 × 10⁻⁴) ÷ (0.80 × 10⁻⁴) = 1.6 m s⁻¹. Multiply by the density for the mass flow rate: dm/dt = ρQ = 0.125 kg s⁻¹ here. Because Q is a derivative, the volume delivered is its integral. A valve opened gradually so that Q(t) = (2.0 × 10⁻⁵ m³ s⁻²) t delivers ∫₀¹⁰ Q dt = ½(2.0 × 10⁻⁵)(10)² = 1.0 × 10⁻³ m³ in its first 10 s: one litre, half of what the final flow rate would have delivered in the same time.

Mark two sections of a pipe or stream tube and treat the space between them as a control volume. Mass enters through section 1 at the rate ρ₁A₁v₁ and leaves through section 2 at the rate ρ₂A₂v₂; nothing crosses the side walls. In steady flow the density at each interior point is fixed, so the mass stored in the control volume cannot change, and the two rates must be equal: ρ₁A₁v₁ = ρ₂A₂v₂. This is conservation of mass written per second, and in this form it handles gases. Air of density 1.20 kg m⁻³ enters a heater duct of constant area at 3.0 m s⁻¹ and leaves hot at 1.00 kg m⁻³. The area cancels, so v₂ = (1.20/1.00)(3.0) = 3.6 m s⁻¹. The volume flow rate has risen by 20% while the mass flow rate has not changed at all, which is why mass, not volume, is the thing to balance.

Water shrinks by only about 0.005% for each extra atmosphere of pressure, and air moving slower than roughly 100 m s⁻¹ changes density by only a few per cent, so in most first-year problems ρ₁ = ρ₂ and continuity becomes A₁v₁ = A₂v₂: the volume flow rate Q is the same at every section. Speed is inversely proportional to area. For a round pipe A = πd²/4, so v₂ = v₁(d₁/d₂)². Water moving at 0.40 m s⁻¹ in a 25 mm pipe that necks down to 10 mm leaves at 0.40 × (2.5)² = 2.5 m s⁻¹, not the 1.0 m s⁻¹ obtained by scaling with the diameter. Streamline pictures carry the same information: if neighbouring streamlines bound tubes of equal flow, they must sit closer together where the fluid is faster, so crowded streamlines mean high speed.

A junction is a control volume with more than two openings. In steady incompressible flow the flow rates in must add up to the flow rates out: ΣQᵢₙ = ΣQₒᵤₜ. Flow rates add; speeds do not. A main carrying 0.90 L s⁻¹ that feeds two branches, one metered at 0.35 L s⁻¹, must send 0.55 L s⁻¹ down the other whatever the pipe sizes are, and only then do the areas turn those flow rates into speeds. If the totals do not balance, the difference is being stored: dV/dt = Qᵢₙ − Qₒᵤₜ. A tank with a free surface of 0.60 m² that receives 0.90 L s⁻¹ while 0.30 L s⁻¹ drains away gains 6.0 × 10⁻⁴ m³ every second, so its level rises at dh/dt = (6.0 × 10⁻⁴) ÷ 0.60 = 1.0 × 10⁻³ m s⁻¹, one millimetre per second. The pipe rule A₁v₁ = A₂v₂ is the special case in which the stored volume is constant.

Real fluid sticks to the wall, so the speed across a section is not uniform: zero at the wall, greatest on the axis. The flow rate is then the integral Q = ∫ v dA, which for a round pipe is ∫₀ᴿ v(r) 2πr dr, a sum over rings of area 2πr dr. Continuity survives untouched provided v in A₁v₁ = A₂v₂ is read as the mean speed Q/A. For the parabolic profile of slow, smooth pipe flow the mean is exactly half the centreline speed, so a probe on the axis overstates Q by a factor of two. Turbulent flow is unsteady in its detail, yet its time-averaged flow rates still balance. What does fail is the constant-density shortcut, for gas flows faster than a few tenths of the speed of sound or with strong heating, and the two-section balance whenever fluid accumulates in between. And continuity is silent about pressure: that takes the energy argument of Bernoulli's equation, which comes next.