A pipe carrying water in laminar flow has its radius halved by mineral scale buildup, while the pressure drop across the pipe is held constant. By what factor does the volumetric flow rate change?
AIt decreases by a factor of 16, because Q ∝ R⁴ and (1/2)⁴ = 1/16
BIt decreases by a factor of 2, because flow rate scales linearly with the pipe cross-section
CIt decreases by a factor of 4, because Q scales with the cross-sectional area R²
DIt decreases by a factor of 8, because Q scales with volume (R³)
The Hagen-Poiseuille equation states Q = πR⁴ΔP/(8μL). Flow rate depends on the FOURTH power of radius. Halving R gives Q_new = Q_old × (R/2)⁴/R⁴ = Q_old/16. This extreme sensitivity is why even modest deposits dramatically reduce flow. Options 1–3 reflect intuitions from 2D area or 3D volume scaling — neither applies here. The R⁴ dependence is the single most important practical implication of Hagen-Poiseuille.
Question 2 Multiple Choice
An engineer compares two laminar flow systems: one with smooth glass tubing, one with rough steel pipes where roughness elements protrude about 5% of the pipe radius. Both have the same diameter, length, fluid, and pressure drop. Which system delivers greater flow rate?
ABoth deliver the same flow rate — roughness has no effect on laminar flow because viscous forces dominate and the flow never interacts with wall features
BThe smooth glass system delivers more, because roughness increases friction and pressure losses
CThe rough steel system delivers more, because roughness disrupts the boundary layer and promotes mixing that reduces viscous losses
DThe smooth glass system delivers more, but only for Reynolds numbers above 1,000; below that, roughness is irrelevant
In laminar flow, f = 64/Re — roughness does not appear in this expression at all. The viscous sublayer in laminar flow is so thick that it completely engulfs wall roughness features; the orderly, layer-by-layer flow never 'sees' the wall texture. Roughness only matters in turbulent flow, where high-momentum fluid reaches the wall. This is why the Moody chart's laminar region (f = 64/Re) is a single line with no roughness parameter.
Question 3 True / False
In fully developed laminar pipe flow, the maximum fluid velocity occurs at the pipe wall.
TTrue
FFalse
Answer: False
Maximum velocity occurs at the CENTERLINE (r = 0), where V = Vmax. The no-slip condition requires the fluid velocity to be exactly zero at the wall (r = R). The parabolic profile V(r) = Vmax(1 − r²/R²) decreases monotonically from center to wall. This is the opposite of what might be intuitively expected if one imagines 'friction at the wall slowing everything equally' — in fact, the wall completely arrests flow locally, and the centerline fluid is least affected by wall friction.
Question 4 True / False
The average velocity in fully developed laminar pipe flow equals exactly half the centerline (maximum) velocity.
TTrue
FFalse
Answer: True
Integrating the parabolic profile V(r) = Vmax(1 − r²/R²) over the circular cross-section yields Vavg = Vmax/2. This factor of two has practical importance: a velocity probe placed at the centerline (common in measurement) reads twice the average velocity. An engineer who misidentifies centerline velocity as average velocity will overestimate volumetric flow rate by exactly 2×.
Question 5 Short Answer
Why does the Hagen-Poiseuille equation predict that surface roughness has no effect on laminar flow resistance, and what happens to this immunity as Reynolds number increases toward transition?
Think about your answer, then reveal below.
Model answer: In laminar flow, the viscous sublayer is thick relative to wall roughness elements — the orderly, layered flow is entirely within this viscous region and never contacts rough wall features with appreciable momentum. Friction depends only on the velocity gradient at the wall (a viscous effect), not on surface texture. As Re increases toward ~2,300, the viscous sublayer thins and eventually turbulent fluctuations begin to sweep fluid directly against the wall. Once fully turbulent, roughness elements protrude through the viscous sublayer and dramatically increase friction — which is why the Moody chart's turbulent region splits into many roughness-dependent curves.
The immunity to roughness in laminar flow is not approximate — it is exact. The laminar f = 64/Re formula applies identically to commercial steel, glass, and corroded pipes, as long as flow remains laminar. This changes categorically at the laminar-turbulent transition, one of the most consequential regime shifts in fluid mechanics.