Continuity: what goes in must come out
A liquid hardly compresses, so in a pipe with no leaks the same volume has to pass every section each second. The volume flow rate is the cross-section times the mean speed, so
Q = A₁ v₁ = A₂ v₂, with A = π D² ÷ 4
Where the pipe narrows the fluid speeds up, by the ratio of the areas, which is the square of the ratio of the diameters. Halve the diameter and the speed goes up four times. This is why putting your thumb over a garden hose makes the water shoot further.
Bernoulli's equation
Speeding the fluid up takes energy, and in a steady flow without friction it comes from the pressure. Daniel Bernoulli showed in 1738 that along a streamline
P + ½ ρ v² + ρ g z = constant
- P is the pressure, the work done by the fluid behind pushing the fluid in front.
- ½ρv² is the kinetic energy per unit volume, sometimes called the dynamic pressure.
- ρgz is the potential energy per unit volume at height z.
So where the fluid goes faster, or higher, its pressure is lower. It is conservation of energy written per cubic meter of fluid.
A worked example
Water (ρ = 998 kg/m³) flows at 10 L/s through a 10 cm pipe that narrows to a 5 cm throat, then rises 1 m. The pressure at the inlet is 150 kPa above the atmosphere.
- Areas: A₁ = π × 0.1² ÷ 4 = 0.00785 m², A₂ = 0.00196 m².
- Speeds: v₁ = 0.010 ÷ 0.00785 = 1.27 m/s, v₂ = 0.010 ÷ 0.00196 = 5.09 m/s.
- Pressure in the throat: ½ × 998 × (5.09² − 1.27²) = 12.1 kPa lower, so 137.9 kPa.
- After the rise, back in the 10 cm pipe: the speed is 1.27 m/s again, so the only change from the inlet is ρgh = 998 × 9.81 × 1 = 9.8 kPa, leaving 140.2 kPa.
The pressure recovers after the throat: nothing is lost (without friction), it was only turned into speed and back.
The Venturi meter
Turning the example around: measure the pressure drop between the pipe and the throat, and continuity and Bernoulli together give the flow,
Q = A₂ √(2 ΔP ÷ (ρ (1 − (A₂ ÷ A₁)²)))
With ΔP = 12.1 kPa this returns 10 L/s. Venturi meters measure flow in water mains and industrial pipes with no moving parts; real ones multiply by a discharge coefficient of about 0.98 to allow for friction. The same principle runs carburetors, paint sprayers and the Venturi masks that mix oxygen with air in hospitals.
Viscosity and the Reynolds number
Real fluids are viscous: they stick to the wall, so the speed is zero there and greatest in the middle. Which way the flow behaves depends on the Reynolds number,
Re = ρ v D ÷ μ
where μ is the viscosity. Below about 2,300 the flow in a pipe is laminar: smooth layers sliding past each other, with a parabolic speed profile whose center moves at twice the mean. Above about 4,000 it is turbulent: chaotic eddies mix the fluid, flattening the profile. In between it flips back and forth. Water at 1.27 m/s in a 10 cm pipe has Re = 127,000, firmly turbulent; honey (μ about 10 Pa·s) at 0.3 m/s in a 2 cm pipe has Re of about 1.
Friction losses
Viscosity turns some of the flow's energy into heat, so P + ½ρv² + ρgz falls along the pipe. The Darcy–Weisbach equation gives the loss over a length L:
ΔP = f (L ÷ D) ½ ρ v²
- Laminar: f = 64 ÷ Re, which is the Hagen–Poiseuille law, ΔP = 8 μ L Q ÷ (π R⁴). The loss rises with the fourth power of the narrowness: halving the radius needs sixteen times the pressure.
- Turbulent, smooth pipe: f ≈ 0.316 Re−1/4 (Blasius). For the 10 cm water pipe f = 0.017 and the loss is about 0.14 kPa per meter; in the 5 cm throat, about 3.6 kPa per meter.
Honey at 0.1 L/s through a 2 cm pipe loses about 255 kPa every meter, which is why thick liquids need big pipes and strong pumps.
Using the simulation
In the Fluid Flow and Bernoulli's Principle Simulator, set the flow rate, diameters, height and inlet pressure, and read the columns over sections A, B and C. Things to try:
- Halve the throat diameter from 10 cm to 5 cm: the speed in it quadruples and the drop grows sixteen-fold.
- Set the height negative: the outlet is lower, and its pressure is higher than the inlet's.
- Narrow the throat to 2 cm at 10 L/s: the pressure there would have to fall below the vapor pressure of water, and the simulation warns of cavitation.
- Tick friction and switch to honey: the profile becomes a parabola, Re is about 1, and the dashed line of total energy falls steeply.
What the model assumes
- Steady, incompressible flow. The flow rate does not change with time and the density is constant, even for air.
- One speed per section. Bernoulli's equation is applied to the mean speed; the speed profile only changes how the dots move.
- Friction from straight-pipe formulas, integrated along the pipe: 64 ÷ Re for laminar flow, Blasius for turbulent flow in a smooth pipe, and a straight line between them from Re 2,300 to 4,000.
- No minor losses. The contraction and expansion themselves lose nothing extra.
- Fixed properties at about 20 °C: water 998 kg/m³ and 1.0 mPa·s, olive oil 911 kg/m³ and 81 mPa·s, glycerol 1,260 kg/m³ and 1.41 Pa·s, honey 1,420 kg/m³ and about 10 Pa·s, air 1.2 kg/m³ and 18 µPa·s.
- Columns that exaggerate. The heights show the differences between the gauges stretched to fit, so they are not to scale with each other; the labels give the real pressures.
Edge cases
- Throat as wide as the pipe: no speed-up, no drop, and the Venturi formula has nothing to measure.
- Throat wider than the pipe: the fluid slows and the pressure rises, the principle of a diffuser.
- Very low pressure: if the absolute pressure would fall below the vapor pressure (2.3 kPa for water at 20 °C), the liquid boils into bubbles that collapse violently downstream. This cavitation erodes ship propellers and pump impellers.
- Negative absolute pressure is impossible for a liquid in a pipe; the flow you asked for cannot happen with that inlet pressure.
Where the model stops being right
- Sudden expansions lose a lot of energy to eddies: the Borda–Carnot loss, ½ρ(v₁ − v₂)², is about 7.3 kPa for the throat in the example opening abruptly into the pipe. Gentle cones, like the one drawn, recover most of it, which is why Venturi meters have long tapered outlets.
- Compressible flow. Above about a third of the speed of sound (roughly 100 m/s in air) gases compress noticeably and Bernoulli's equation in this form underestimates the pressure changes; near Mach 1 the flow chokes.
- Rough pipes and very high Reynolds numbers. Blasius only holds up to about Re 100,000 in smooth pipes; real pipes need the Colebrook equation or a Moody chart with the roughness of the wall.
- Unsteady flow. Closing a valve quickly sends a pressure wave back up the pipe (water hammer) far larger than any steady value.
- Non-Newtonian fluids like blood, ketchup and paint change viscosity with how fast they are sheared, and honey's viscosity depends strongly on its temperature and water content.
- Lift on wings. Bernoulli's equation describes the lower pressure over a wing, but it does not on its own explain why the air goes faster there; the "equal transit time" explanation is false.
Calculators
Work out flow rates, speeds and pressure drops in real pipes with the Pipe Flow Calculator, check whether a flow is laminar with the Reynolds Number Calculator, and convert between liters, gallons and cubic meters with the Volume Converter. How to Calculate Pipe Flow Rate and Pressure Drop covers the sums by hand.