Fluid Properties and Statics

FE Exam Weight: Fluid Mechanics is the single highest-weighted topic on the FE Other Disciplines exam — 12–18 of 110 questions (~14%). The exam is open to the searchable NCEES FE Reference Handbook, so master which formula applies and keep units consistent.

Fluid Properties

Water at 20°C (memorize)

• ρ ≈ 998 kg/m³ ≈ 1,000 kg/m³
• γ ≈ 9,790 N/m³ ≈ 9,810 N/m³ (use 9.81 kN/m³)
• μ ≈ 1.0 × 10⁻³ Pa·s; ν ≈ 1.0 × 10⁻⁶ m²/s

Specific gravity is the fast bridge: SG = 0.8 means ρ = 800 kg/m³ and γ = 7,848 N/m³.

Viscosity and Newton's Law

Newton's law of viscosity relates shear stress to the velocity gradient:

τ = μ (du/dy)

Worked example — viscous shear stress

Oil with μ = 0.1 Pa·s fills a 2 mm gap; the top plate moves at 0.5 m/s while the bottom is fixed. The velocity gradient is linear: du/dy = 0.5/0.002 = 250 s⁻¹. So τ = μ(du/dy) = 0.1 × 250 = 25 Pa. This is the drag stress on the moving plate.

Hydrostatic Pressure

In a fluid at rest, pressure increases linearly with depth:

p = p₀ + ρgh = p₀ + γh

Key facts the FE tests:

• Pressure acts equally in all directions at a point (Pascal's law).
• Pressure is the same at every point on a horizontal plane in a connected fluid.
• Gauge pressure = absolute pressure − atmospheric (p_atm ≈ 101.3 kPa).

Worked example — gauge pressure with depth

Gauge pressure 10 m down in water: p = ρgh = 1,000 × 9.81 × 10 = 98,100 Pa = 98.1 kPa — almost exactly 1 atmosphere. So every ~10 m of water adds one atmosphere of gauge pressure.

Manometers

For a U-tube manometer, walk from one known end to the other, adding γh going down and subtracting γh going up:

p_A + γ₁h₁ − γ₂h₂ − γ₃h₃ = p_B

The manometer fluid (often mercury, SG = 13.6) amplifies small pressure differences into readable height changes.

Hydrostatic Force on a Plane Surface

The resultant force on a submerged flat plate is:

F = γ h̄ A

where h̄ is the vertical depth of the centroid of the surface and A is its area. The force acts at the center of pressure, located below the centroid:

y_cp = ȳ + I_x̄ / (ȳ A)

The extra term I_x̄/(ȳA) pushes the resultant deeper than the centroid because pressure grows with depth — the lower part of the plate feels more pressure.

Worked example — force on a gate

A rectangular gate 2 m wide × 3 m tall has its top edge 1 m below the surface. Centroid depth h̄ = 1 + 3/2 = 2.5 m; area A = 2 × 3 = 6 m².

F = γh̄A = 9,810 × 2.5 × 6 = 147,150 N ≈ 147 kN.

Curved Surfaces

Resolve into components: the horizontal force equals γh̄A on the surface's vertical projection; the vertical force equals the weight of fluid (real or virtual) directly above the curved surface.

Buoyancy — Archimedes' Principle

A fully or partly submerged body feels an upward buoyant force equal to the weight of fluid it displaces:

F_B = ρ_fluid · g · V_displaced = γ_fluid · V_displaced

Floating bodies

A floating object displaces exactly its own weight of fluid, so:

fraction submerged = ρ_object / ρ_fluid = SG (in water)

An iceberg (SG ≈ 0.92) floats with ~92% of its volume underwater; an SG = 0.8 block floats 80% submerged. If ρ_object > ρ_fluid the body sinks; if equal it is neutrally buoyant.

Worked example — apparent weight

A 0.01 m³ steel block (ρ = 7,850 kg/m³) is fully submerged in water. Its weight in air = ρgV = 7,850 × 9.81 × 0.01 = 770 N. Buoyant force = 1,000 × 9.81 × 0.01 = 98.1 N. Apparent (submerged) weight = 770 − 98.1 = 672 N.

Stability of floating bodies

A floating body is stable when its metacenter M lies above the center of gravity G. When it tilts, the center of buoyancy shifts, creating a righting moment if M is above G. A low, wide hull (ballast low) is stable; a top-heavy one capsizes.

Common Traps

• γ vs. ρ. Use ρgh when you have density; γh when you have specific weight. Don't multiply by g twice.
• h̄ is the centroid depth, not the depth of the top edge.
• SI consistency. kPa = kN/m²; keep pressures, areas, and forces in compatible units.
• Gauge vs. absolute. Manometers and gate problems use gauge pressure unless atmospheric acts on only one side.

Quick reference

• γ = ρg; SG = ρ/ρ_water; ν = μ/ρ.
• p = p₀ + γh; F = γh̄A on a plane surface.
• F_B = γ_fluid·V_displaced; floating fraction submerged = SG.
• Water: ρ ≈ 1,000 kg/m³, γ ≈ 9,810 N/m³.