§1Properties of fluids
Two properties run this page: density ρ, which prices pressure and power, and viscosity μ, which prices friction.
| Fluid | ρ (kg/m³) | μ (Pa·s) |
|---|---|---|
| Water | 1000 | 1.0 × 10⁻³ |
| Air (sea level) | 1.20 | 1.8 × 10⁻⁵ |
| Hydraulic / machine oils | ≈ 850 – 900 | strongly temperature-dependent |
| Specific gravity SG = ρ/1000. Oil viscosity can change an order of magnitude between a cold start and running temperature — the reason hydraulic systems warm up before they behave. | ||
§2Pressure and head
Still fluid presses equally in all directions, and harder with depth. Depth and pressure are interchangeable currencies — “head” is pressure quoted in metres.
3.5 m of water above a fitting: p = 1000 × 9.81 × 3.5 = 34.3 kPa gauge (135.7 kPa absolute). Handy rule hiding in the formula: every metre of water is 9.81 kPa, and 10.2 m of water is one atmosphere.
§3Pascal’s principle — hydraulics
Pressure applied to a confined fluid arrives undiminished everywhere. Two pistons of different size turn that fact into a force multiplier.
200 N on a Ø20 mm master piston pressurises the line to 0.64 MPa and delivers 200 × (50/20)² = 1250 N at a Ø50 mm slave — the jack, the brake and the press in one line of arithmetic. The distance penalty is exactly the force gain: the simple-machines ledger of the Mechanics page, kept in fluid.
Contents§4Buoyancy
A submerged body is pushed up by the weight of fluid it displaces — Archimedes, still on duty in every tank and sump.
A 0.02 m³ steel fixture weighs 0.02 × 7850 × 9.81 = 1540 N in air. Submerged in water it displaces 196 N of water, so the crane sees 1344 N — until the moment it breaks the surface, when the full 1540 N returns at a jerk. Rig for the dry weight.
§5Continuity and Bernoulli
What flows in must flow out; and along a streamline, pressure, velocity and height trade against each other at fixed total. Together they solve the venturi on the hero sheet.
Water at 1.8 m/s in the Ø80: Q = A₁v₁ = 9.05 L/s, and continuity forces v₂ = 1.8 × (80/50)² = 4.61 m/s in the Ø50. Bernoulli prices the speed-up: Δp = ½ρ(v₂² − v₁²) = 9.0 kPa drop at the throat. Measure that drop and you have built a flowmeter; recover it badly in a sudden enlargement and you have built a loss.
§6Laminar, turbulent, and Reynolds number
Slow, viscous flow slides in ordered layers; fast flow churns. One dimensionless number predicts which you have.
Water at 2 m/s in a 50 mm pipe: Re = 1000 × 2 × 0.05/10⁻³ = 100 000 — deeply turbulent, like almost every water and air system in industry. Laminar flow belongs to oils, small clearances and instrument lines; hydraulic leakage paths and journal-bearing films live there, which is why viscosity dominates those calculations and barely features in plant pipework.
Contents§7Pipe friction
Every metre of pipe converts a little pressure into heat. The Darcy–Weisbach equation prices it as lost head.
Water at 2 m/s through 60 m of 50 mm pipe, f = 0.02: h_f = 0.02 × (60/0.05) × (2²/19.62) = 4.9 m of head — 48 kPa spent before the water arrives. The v² is the lever: halving velocity (doubling pipe area) cuts friction to a quarter, the standing argument for one size larger pipe. Fittings add their own tolls, usually reckoned as equivalent lengths.
§8Pump power
A pump buys head for a flow. Its power bill is the product, marked up by inefficiency.
12 L/s raised through a total head of 25 m at η = 0.65: P = 1000 × 9.81 × 0.012 × 25 / 0.65 = 4.53 kW — call it a 5.5 kW motor for margin. Note where the head came from: if 5 of those 25 metres are friction, one pipe size up would shrink the motor too — pipes and pumps are sized together or badly.
§9Quick reference
The working core of the page on one card rack.
Statics
p = ρgh
1 m water ≈ 9.81 kPa
Hydraulics
F₂ = F₁ A₂/A₁
upthrust = ρgV
Flow
Q = Av
p + ½ρv² + ρgz = const
Regime
Re = ρvd/μ
laminar < 2300
Friction
h_f = f(L/d)(v²/2g)
halve v → quarter h_f
Pump
P = ρgQH/η