Kinematic viscosity ν = μ/ρ = 0.001 / 1000 = 1 × 10⁻⁶ m²/s. This is the well-known value for water at about 20°C.

Because hydrostatic pressure increases with depth, the resultant force acts below the centroid. The centre of pressure is at y_cp = ȳ + I/(ȳ·A), always deeper than the centroid.

Continuity A₁V₁ = A₂V₂. Area scales with D², so halving the diameter quarters the area and velocity quadruples: V₂ = 2 × (100/50)² = 8 m/s. Bernoulli then converts this speed-up into the measured pressure drop.

The impact force equals the rate of change of momentum: F = ρQV = ρAV². The linear momentum equation (control-volume form) is the correct tool for jet-force problems.

In the turbulent regime the Colebrook equation (and the Moody chart) make f a function of both Reynolds number and relative roughness ε/D. In laminar flow f = 64/Re depends on Re alone.

For a laminar boundary layer, δ ≈ 5x/√(Re_x), and since Re_x ∝ x, this gives δ ∝ √x. The boundary layer thickens with the square root of distance.

Inviscid flow gives zero drag (d'Alembert's paradox) because the pressure is fore-aft symmetric. Lift, however, follows the Kutta-Joukowski theorem L' = ρVΓ, proportional to density, free-stream velocity, and circulation.

Speed of sound a = √(γRT) = √(1.4 × 287 × 288) ≈ 340 m/s. Mach number M = V/a = 170 / 340 = 0.50.

Choked flow means the throat has reached sonic conditions (M = 1) and the mass flow rate is maxed out for the given stagnation conditions. Lowering the downstream pressure further cannot increase it.

The convective (advective) acceleration term ρ(V·∇)V is nonlinear because velocity multiplies its own gradient. This nonlinearity gives rise to turbulence and is why most flows have no closed-form solution and require CFD.