Fluid Mechanics

§1Properties of fluids

Two properties run this page: density ρ, which prices pressure and power, and viscosity μ, which prices friction.

Working values at ~20 °C (indicative)
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.

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§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.

p = ρ g h absolute = gauge + atmospheric (≈ 101.3 kPa)

Example 1 — pressure at a tank drain

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.

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§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.

p equal ⇒ F₂ = F₁ × A₂A₁ (and the small piston travels A₂/A₁ times as far — no free work)

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.

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§4Buoyancy

A submerged body is pushed up by the weight of fluid it displaces — Archimedes, still on duty in every tank and sump.

upthrust = ρ_fluid × g × V_displaced

Example 2 — lifting a casting out of the quench

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.

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§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.

Q = A₁v₁ = A₂v₂ p + ½ρv² + ρgz = constant (ideal, along a streamline)

Example 3 — a pipe necking from Ø80 to Ø50

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.

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§6Laminar, turbulent, and Reynolds number

Slow, viscous flow slides in ordered layers; fast flow churns. One dimensionless number predicts which you have.

Re = ρ v dμ pipes: laminar below ≈ 2300, fully turbulent above ≈ 4000

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.

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§7Pipe friction

Every metre of pipe converts a little pressure into heat. The Darcy–Weisbach equation prices it as lost head.

h_f = f Ld v²2g f ≈ 0.02 – 0.03 for turbulent flow in commercial steel pipe (charts refine it)

Example 4 — a 60 m delivery run

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.

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§8Pump power

A pump buys head for a flow. Its power bill is the product, marked up by inefficiency.

P = ρ g Q Hη H = static lift + friction head (§7)

Example 5 — sizing the motor

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.

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§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

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Handbook application: from concept to controlled practice

Purpose. This expanded section turns the original page into a practical handbook. It preserves the supplied material and adds a repeatable way to apply, check and review Fluid Mechanics. It does not replace a contract, legislation, a controlled standard, competent engineering judgement or specialist advice.

The operating aim is to carry the subject from function and assumptions through design evidence, verification and controlled release. Read the original explanation first, then use the workflow and checks below to convert knowledge into evidence.

Apply Fluid Mechanics by beginning with the duty, not the component or software command. Convert the key ideas—fluid, properties, pressure, head, principle—into measurable requirements and interfaces. Record operating and non-operating environments, duty cycle, expected life, loads, energy sources, human interaction and reasonably foreseeable abnormal conditions. When a value is not a project requirement or verified supplier datum, identify it as an assumption or illustrative value.

Create a calculation and evidence trail that another competent person can audit. Every input should carry a source, unit, revision and uncertainty or tolerance where relevant. Every model should state its boundary conditions and limitations. Keep nominal capacity separate from design capacity, and keep verification margin separate from an arbitrary safety factor. If a code or standard governs the work, confirm the applicable edition and contractual status rather than copying a number from a secondary summary.

Design for manufacture, assembly, inspection, operation and maintenance at the same time. A technically valid geometry can still fail because it cannot be fixtured, measured, cleaned, guarded, reached or replaced. Review process capability, datum or reference strategy, tolerance accumulation, access, error-proofing and changeover. Where people interact with plant, apply the hierarchy of controls and consult those who will operate, clean, maintain and recover the equipment.

Plan verification before release. Define the characteristic, method, equipment, sample or test condition, acceptance criterion, record and responsible person. Validation then asks a different question: whether the resulting system is effective and suitable in the intended use context. A passed drawing check or analysis does not by itself validate usability, maintainability or production performance.

Step-by-step operating method

Define the duty. Capture the required function, interfaces, operating environment, life, loads and unacceptable outcomes.
Establish the model. Identify governing principles, units, material or process data, assumptions and uncertainty.
Develop alternatives. Compare feasible concepts against performance, manufacturability, safety, maintainability and cost.
Verify the design. Use analysis, test, inspection or demonstration with acceptance criteria defined before execution.
Release and learn. Baseline the design, control changes, retain evidence and feed operating results into the next revision.

Illustrative design review record

Illustrative values only. Build a one-page record with the required function, input sources, assumptions, governing load or process condition, failure consequences, selected concept, verification method and acceptance criterion. Mark every numerical input as project requirement, verified supplier data, measured value, calculation output or assumption. Review the weakest evidence first. If an assumption can change safety, compliance, interchangeability or capacity, it must be resolved before release rather than buried in a calculation note.

Evidence class	Question	Release expectation
Requirement	What must the design do and under which conditions?	Approved and traceable
Input	Where did the load, property, tolerance or process limit come from?	Source, unit and revision recorded
Analysis	Which model and assumptions connect input to result?	Checkable calculation or simulation
Verification	How will conformity be demonstrated?	Method and acceptance criterion agreed
Validation	Will the solution work for intended users and conditions?	Representative use evidence

Common failure modes and recovery actions

1. Watch for

Starting detailed design before interfaces and operating limits are agreed.

Recovery: Return to the governing definition or requirement and restate the decision in one sentence.

2. Watch for

Using catalogue or typical values as though they were certified project inputs.

Recovery: Separate evidence from assumption, assign an owner and set a date for validation.

3. Watch for

Checking nominal performance while ignoring tolerances, degradation and foreseeable misuse.

Recovery: Run a small counterexample, boundary test, pilot or independent check before proceeding.

4. Watch for

Confusing verification of requirements with validation of user need.

Recovery: Record the consequence, decision and rationale, then update the controlled baseline.

5. Watch for

Releasing drawings or procedures without configuration, inspection and change controls.

Recovery: Escalate when the issue affects safety, compliance, acceptance, material value or an agreed tolerance.