Fluid dynamics
The flow that enters is the flow that leaves, and where the fluid speeds up the pressure drops. With those two ideas you can explain the wing of an airplane, the flowmeter and the fan of a heat sink.
01Flow rate and continuity
The flow rate is the volume that passes through a cross-section per unit time, Q = A·v, in m³/s. If the fluid is incompressible and there are no leaks, what enters at one end leaves at the other:
The continuity equation is the conservation of mass, and its electrical twin is Kirchhoff’s current law: the current entering a node is the current leaving it. Flow rate and current play the same role; so do pressure and voltage.
02Bernoulli’s equation
For an ideal fluid —incompressible, inviscid and in steady flow—, the energy per unit volume is conserved along a streamline:
Where the fluid moves faster, the pressure is lower. It is not magic: the fluid sped up because a pressure difference pushed it, and that energy came out of the pressure term. This is where the lift of a wing, the atomizer, the shower curtain that clings to you, and the fact that two ships sailing side by side attract each other come from.
The flow rate is the same in both sections, but the velocity and the pressure are not. The columns show the pressure head.
03What is measured with Bernoulli
| Instrument | What it does | Used in |
|---|---|---|
| Venturi tube | Measures flow rate from the pressure drop at a throat | Industrial plants, carburetors |
| Pitot tube | Measures velocity by comparing total and static pressure | Aircraft, wind tunnels |
| Orifice plate | Simpler and more dissipative version of the Venturi | Flow measurement in piping |
| Torricelli’s theorem | \( v = \sqrt{2 \cdot g \cdot h} \) at an orifice | Draining tanks |
The flowmeter of an industrial process is usually a Venturi with two pressure sensors and a microcontroller that calculates Q = A₂·√(2Δp/(ρ(1−(A₂/A₁)²))). That is where this subject and instrumentation electronics come together.
04Viscosity, Reynolds and head loss
No real fluid is ideal: viscosity dissipates energy and makes the pressure drop along a straight pipe. The Reynolds number compares inertial forces with viscous ones and tells you what kind of flow there is:
| Regime | Reynolds | How it behaves |
|---|---|---|
| Laminar | < 2300 | Orderly layers; the loss grows with v |
| Transition | 2300 to 4000 | Unstable |
| Turbulent | > 4000 | Chaotic mixing; the loss grows with v² |
The head loss in a straight section is calculated with the Darcy–Weisbach equation, Δp = f·(L/D)·½·ρ·v², where the factor f comes from the Moody chart as a function of Reynolds and of the roughness of the pipe. That is the calculation that decides the size of a pump.
If at some point the absolute pressure falls below the vapor pressure of the liquid, bubbles form and collapse violently downstream. It erodes pump impellers and propellers, and sounds as if stones were flowing through. In a heavily constricted Venturi it is easy to reach that condition.
05What it is good for in electronics
- Air cooling. Choosing a fan requires knowing the necessary flow rate and the back pressure of the enclosure: the fan curve crosses the system resistance curve.
- Liquid cooling. In high-power equipment, the pump is sized from the flow rate and head loss of the circuit.
- Flowmeters. Turbine, Venturi, ultrasonic (transit-time or Doppler), and electromagnetic based on Faraday’s law: all of them are electronic instrumentation.
- Hot-wire anemometers. They measure air speed from the power needed to keep a wire hot: a forced-convection problem solved with a Wheatstone bridge.
- Enclosure testing. The protection rating against water jets is specified in flow rate and pressure.
06In the lab
With a hose and two nozzles of different diameter, measure the flow rate by volume and time, and calculate the exit velocity in each case. Check that the product A·v is maintained.
Drill holes in a bottle at different heights and measure the range of the jet. Verify that the exit velocity follows √(2gh) and that the maximum range occurs with the hole at mid-height.
Measure the flow rate of an enclosure fan against different restrictions and plot its curve of flow rate against back pressure. Cross it with the system curve to find the actual operating point.
07Common mistakes
- Applying Bernoulli between points that are not on the same streamline, or with a pump or turbine in between without adding its term.
- Using Bernoulli in highly viscous flow and ignoring the head loss.
- Believing that more velocity means more pressure. It is exactly the opposite.
- Confusing volumetric flow rate with mass flow rate. In gases the difference matters, because the density changes.
- Computing Reynolds with mixed units. It is dimensionless: if the number does not come out unitless, there is a mistake.
- Forgetting that air is a fluid and treating it differently from water.
08Self-assessment
A 50 mm pipe carries 2 L/s. What is the velocity of the water?
A = π·0.025² = 1.96·10⁻³ m², so v = 0.002/1.96·10⁻³ ≈ 1.02 m/s.
If the diameter is reduced to half, by how much does the velocity increase?
The area drops to a quarter, so the velocity quadruples.
Is that 50 mm pipe with water at 1 m/s laminar or turbulent?
Re = 1000·1·0.05/10⁻³ = 50,000: turbulent, by a wide margin. With water, laminar flow in pipes is rare.
At what velocity does water leave an orifice 2 m below the surface?
v = √(2·9.8·2) ≈ 6.3 m/s, the same as a body would have after falling that height.
Why can the roof of a house be blown upward in a strong wind?
Because the air accelerates as it passes over it and the pressure above drops below that inside. The difference, multiplied by the area, is an enormous force.
What does a Pitot tube measure?
The difference between total and static pressure, from which the velocity follows: v = √(2Δp/ρ). It is the airspeed instrument of an aircraft.
09Further reading
- Yunus A. Çengel and John M. Cimbala. Fluid Mechanics: Fundamentals and Applications. 4th ed., McGraw-Hill, 2018 (Spanish edition, “Mecánica de fluidos: fundamentos y aplicaciones”). The reference text: Bernoulli, Reynolds, Moody and head losses with an engineering approach.
- Raymond A. Serway and John W. Jewett. Physics for Scientists and Engineers with Modern Physics, Volume 1. 9th ed. (Spanish edition, “Física para ciencias e ingeniería”), Cengage Learning, 2014. Chapter 14 presents continuity and Bernoulli at the level of the course.
- Robert L. Mott. Applied Fluid Mechanics. 6th ed., Pearson, 2006 (Spanish edition, “Mecánica de fluidos”). Strongly oriented toward the practical calculation of installations and pump selection.