4.1 Fluid Mechanics
Key Takeaways
- Hydrostatic pressure is p = γh; for fresh water γ = 9.81 kN/m³ (62.4 lb/ft³), so 5 m of water gives ≈ 49.05 kPa.
- Continuity for incompressible flow is Q = A₁V₁ = A₂V₂; halving the diameter quarters the area and quadruples the velocity.
- The energy (Bernoulli) equation per unit weight is p/γ + V²/2g + z + h_pump = (downstream terms) + h_L.
- Darcy-Weisbach head loss is h_f = f(L/D)(V²/2g); Hazen-Williams uses an empirical coefficient C for water in pressurized pipes.
- Reynolds number Re = ρVD/μ = VD/ν; pipe flow is laminar below Re ≈ 2,100 and turbulent above Re ≈ 4,000.
Why Fluid Mechanics Matters
Fluid Mechanics contributes roughly 5–8 questions on the FE Civil exam and is the foundation for the larger Hydraulics & Hydrologic Systems area. Every relationship here lives in the NCEES FE Reference Handbook (Fluid Mechanics chapter), so the exam tests whether you can pick the right equation and substitute consistent units under time pressure — not whether you memorized formulas.
Fluid Properties
Know these core properties and their handbook symbols:
| Property | Symbol | Definition | Water (≈20°C) |
|---|---|---|---|
| Density | ρ (rho) | mass per volume | 998 kg/m³ |
| Specific weight | γ (gamma) | ρg | 9.79–9.81 kN/m³ |
| Dynamic viscosity | μ (mu) | shear resistance | 1.00×10⁻³ Pa·s |
| Kinematic viscosity | ν (nu) | μ/ρ | 1.00×10⁻⁶ m²/s |
Kinematic viscosity ν equals dynamic viscosity μ divided by density ρ. The specific gravity (SG) of a fluid is its density relative to water; mercury has SG ≈ 13.6.
Hydrostatics
Pressure increases linearly with depth in a static fluid:
p = γh (gauge pressure below a free surface)
For fresh water, 5 m of depth gives p = 9.81 kN/m³ × 5 m = 49.05 kPa. The hydrostatic force on a submerged plane surface acts at the centroid of the pressure prism, located at depth h_cp = h̄ + I_c/(h̄·A) below the surface.
Continuity and the Energy Equation
For steady incompressible flow, mass conservation gives continuity:
Q = A₁V₁ = A₂V₂
If pipe diameter halves, area drops to one-quarter and velocity quadruples. The energy (Bernoulli) equation between two points, per unit weight, is:
p₁/γ + V₁²/2g + z₁ + h_pump = p₂/γ + V₂²/2g + z₂ + h_turbine + h_L
Bernoulli's classic form assumes steady, incompressible, frictionless flow along a streamline. Each term has units of length (head): pressure head p/γ, velocity head V²/2g, and elevation head z.
Pipe Flow and Head Loss
Friction (major) loss in a full pipe uses Darcy-Weisbach:
h_f = f · (L/D) · (V²/2g)
The friction factor f comes from the Moody diagram (a function of Reynolds number and relative roughness ε/D) in the handbook. For laminar flow, f = 64/Re. Minor losses from fittings use h_m = K(V²/2g).
The empirical Hazen-Williams equation (water only, turbulent, near 60°F) uses a roughness coefficient C (e.g., C ≈ 130–140 for new ductile iron). It is dimensional — use the handbook's stated coefficient (0.849 SI or 1.318 US) exactly.
Pumps and Power
The water (hydraulic) power added by a pump is:
P = ρgQh_pump = γQh_pump
Brake (shaft) power = water power / pump efficiency η. A pump delivering 0.05 m³/s of water against 20 m of head needs P = 9.81 × 0.05 × 20 = 9.81 kW of water power.
Dimensional Analysis and Reynolds Number
Reynolds number Re = ρVD/μ = VD/ν is the ratio of inertial to viscous forces. In pipes: laminar below ≈ 2,100, turbulent above ≈ 4,000. The Buckingham Pi theorem predicts the number of independent dimensionless groups; the Froude number governs open-channel and free-surface flow.
Water (γ = 9.81 kN/m³) flows in a horizontal pipe that contracts from a 200 mm diameter to a 100 mm diameter. If the velocity in the 200 mm section is 1.5 m/s, what is the velocity in the 100 mm section?
A pump delivers 0.04 m³/s of water (γ = 9.81 kN/m³) against a total dynamic head of 25 m. If the pump efficiency is 80%, what is the required brake (shaft) power input?
Water (ν = 1.0×10⁻⁶ m²/s) flows at 0.5 m/s in a 50 mm diameter pipe. Approximately what is the Reynolds number and the expected flow regime?