In which flow regime is friction head loss proportional to velocity rather than velocity squared?

In which flow regime is friction head loss proportional to velocity rather than velocity squared?

• A. In laminar flow
• B. In turbulent flow
• C. In non-Newtonian fluids
• D. In smooth pipes

Answer: (A)

In laminar pipe flow, friction head loss grows linearly with velocity. This happens because the Darcy friction factor is f = 64/Re, and Re = ρ v D / μ. Substituting into h_f = f L v^2 /(2 g D) yields h_f ∝ v, since the 1/v from f cancels one of the v's in v^2. In turbulent flow, the friction factor is roughly constant for a given roughness, so head loss scales with v^2. Non-Newtonian fluids can change the relationship depending on shear, and pipe smoothness mainly affects pressure loss in the turbulent regime. So the linear relationship is a hallmark of laminar flow.

In laminar pipe flow, friction head loss grows linearly with velocity. This happens because the Darcy friction factor is f = 64/Re, and Re = ρ v D / μ. Substituting into h_f = f L v^2 /(2 g D) yields h_f ∝ v, since the 1/v from f cancels one of the v's in v^2. In turbulent flow, the friction factor is roughly constant for a given roughness, so head loss scales with v^2. Non-Newtonian fluids can change the relationship depending on shear, and pipe smoothness mainly affects pressure loss in the turbulent regime. So the linear relationship is a hallmark of laminar flow.