10 / CHAPTER 07
Boundary layers and separation
See how wall friction, adverse pressure gradients, and detachment arise.
ANCHOR EQUATION
Boundary-layer scale
In symbols: δ/L ∼ Re⁻¹ᐟ²
Viscous effects organize into a thin boundary layer at high Reynolds number.
Poiseuille flow comes from a pressure–viscous balance
In a long, straight, circular pipe with steady laminar flow, pressure gradient drives the fluid while viscosity resists layer shear. The parabolic profile and strong radius dependence follow from integrating this balance.
Integrate the cylindrical momentum equation
For axial speed u(r), steady momentum gives 0 = −dp/dx + μ(1/r)d/dr(r du/dr). Integrate once: du/dr = (dp/dx)r/(2μ) after removing the singular center term. Integrate again and impose no-slip u(R)=0: u(r) = −(dp/dx)(R² − r²)/(4μ). Integrating Q = ∫₀^R 2πr u(r)dr yields Q = πR⁴Δp/(8μL).
For R = 1 mm, L = 1 m, Δp = 10 kPa, μ = 0.001 Pa·s, Q = π(10⁻³)⁴(10⁴)/(8×10⁻³) = 3.93×10⁻⁶ m³/s. Doubling R multiplies Q by 16.
Laminar is not automatic
The derivation assumes fully developed, steady, incompressible, Newtonian flow with no-slip and a circular tube. At high Re, turbulence changes the profile and pressure-drop law.
Work it: Use the formula to compare flow through 2 mm and 1 mm radius tubes at the same Δp and L. Explain why a partial blockage can have a dramatic effect.