Viscous Flow and Flow Through Pipes

Real fluids lose energy to friction. In pipes, this loss decides pump sizes and flow rates. The first question is always: is the flow laminar or turbulent?

Reynolds number

Re = ρVD/μ = VD/ν (inertia forces / viscous forces)

• In a pipe: laminar below about 2000, turbulent above about 4000, transitional in between.

Laminar flow in a pipe (Hagen–Poiseuille)

• Velocity profile is a parabola: u = u_max (1 − r²/R²).
• u_max = 2 V_avg.
• Pressure drop: Δp = 32μVL/D².
• Friction factor f = 64/Re.
• Shear stress varies linearly from zero at the centre to its maximum at the wall.

Between fixed parallel plates: u_max = 1.5 V_avg and Δp = 12μVL/h².

Head loss (Darcy–Weisbach)

h_f = f L V²/(2 g D)

• Laminar: f = 64/Re.
• Turbulent: f depends on Re and relative roughness ε/D (Moody chart / Colebrook equation). For smooth pipes, Blasius: f = 0.316/Re^0.25 (Re < 10⁵).
• In terms of flow: h_f = f L Q²/(12.1 D⁵), so h_f ∝ 1/D⁵ at fixed Q.

Minor losses

h = K V²/(2g):

• Sudden expansion: h = (V₁ − V₂)²/(2g).
• Pipe entrance: K ≈ 0.5. Exit into a tank: K = 1.
• Bends, valves and fittings each have their own K.

Pipes in series and parallel

• Series: same Q; head losses add.
• Parallel: same head loss across each branch; flows add.
• Equivalent pipe: a single pipe with the same loss as a series combination: L/D⁵ = Σ Lᵢ/Dᵢ⁵.

Worked example

Oil (μ = 0.1 Pa·s, ρ = 900 kg/m³) flows at 1 m/s in a 50 mm pipe 100 m long. Find Re and the head loss.

• Re = 900 × 1 × 0.05/0.1 = 450 → laminar.
• f = 64/450 = 0.1422; h_f = 0.1422 × 100 × 1/(2 × 9.81 × 0.05) = 14.5 m of oil.

Practice