Fluid Mechanics and Hydraulics
Fluid mechanics tests conservation laws applied to real systems, from a tank draining through an orifice to a pump moving water through a pipe network.
Core rules
- Continuity for incompressible flow: Q = A₁V₁ = A₂V₂.
- Bernoulli along a streamline: p/γ + V²/2g + z = constant, with head loss h_L added for real fluids.
- Darcy–Weisbach: h_f = f(L/D)(V²/2g); minor losses h_m = K(V²/2g). Reynolds number Re = ρVD/μ; laminar below about 2000, turbulent above about 4000.
- Hydrostatics: p = γh, acting normal to surfaces; the resultant force on a submerged vertical gate acts at the center of pressure, below the centroid.
- Pumps: head added h_p; affinity laws give Q ∝ N, H ∝ N², and P ∝ N³ at constant impeller diameter. Available NPSH must exceed required NPSH.
- Open channels: Manning's equation Q = (1.486/n)A R^(2/3) S^(1/2) in US customary units.
Worked example
Water flows at 2 m/s through a 0.1 m pipe that is 100 m long, with f = 0.02. Then h_f = 0.02(100/0.1)(2²/19.62) = 0.02(1000)(0.2039) = 4.08 m. A pump must supply at least this head plus minor losses and any elevation change.
Traps
- Mixing gauge and absolute pressures within one equation.
- Treating the velocity head at a large tank surface as significant while ignoring it at the outlet, or the reverse.
- Applying affinity laws with equal exponents instead of Q ∝ N, H ∝ N², P ∝ N³.
- Confusing the hydraulic grade line (p/γ + z) with the energy grade line.