Atherosclerotic plaque reduces an artery's internal radius from R to R/2. Assuming the same pressure gradient, by what factor does blood flow through the artery decrease?
ABy a factor of 2 (proportional to the radius reduction)
BBy a factor of 4 (proportional to radius squared)
CBy a factor of 8 (proportional to radius cubed)
DBy a factor of 16 (proportional to radius to the fourth power)
The Hagen-Poiseuille law Q = πR⁴ΔP/(8μL) shows flow scales with R⁴. Halving the radius gives (R/2)⁴ = R⁴/16, so flow decreases to 1/16 of original. This dramatic R⁴ dependence explains why arterial stenosis is so dangerous: a 50% reduction in radius cuts blood flow to 1/16 of normal. Options A, B, and C represent common underestimates of how sensitively flow responds to radius changes.
Question 2 Multiple Choice
In fully developed laminar pipe flow, what is the centerline velocity relative to the cross-sectional average velocity?
AEqual to the average velocity (uniform flow profile)
B1.5 times the average velocity
C2 times the average velocity
DDependent on the Reynolds number and pipe roughness
The parabolic velocity profile u(r) = (R² − r²)/(4μ)·(−dP/dx) reaches its maximum at the centerline (r = 0). Integrating this profile over the cross-section and dividing by area gives an average velocity equal to exactly half the centerline velocity. This is a consequence of the parabola's shape. Option D is incorrect because in laminar flow, the profile is determined entirely by the governing physics and does not depend on roughness.
Question 3 True / False
In laminar pipe flow, a smooth-walled pipe and a rough-walled pipe of the same diameter will have the same friction factor if both operate at the same Reynolds number.
TTrue
FFalse
Answer: True
The laminar friction factor f = 64/Re depends only on the Reynolds number, not on pipe roughness. Laminar flow consists of orderly, parallel layers — the viscous sublayer extends to the wall and completely suppresses the effect of surface imperfections. Fluid never 'feels' the roughness because the flow is not turbulent enough to throw fluid into contact with wall features. In turbulent flow, roughness becomes the dominant factor controlling friction.
Question 4 True / False
According to the Hagen-Poiseuille law, doubling the pipe length has a larger effect on flow rate than halving the pipe radius.
TTrue
FFalse
Answer: False
Q = πR⁴ΔP/(8μL). Doubling pipe length (L → 2L) reduces flow by a factor of 2. Halving the pipe radius (R → R/2) reduces flow by a factor of (1/2)⁴ = 16. The R⁴ dependence on radius far outweighs the linear dependence on length. This asymmetry is practically important: a modest reduction in pipe diameter creates a far larger pressure drop than a large increase in pipe length.
Question 5 Short Answer
Why does the Hagen-Poiseuille law's R⁴ dependence make the internal radius of a blood vessel so critical to blood flow? Use the law to explain what happens during arterial narrowing.
Think about your answer, then reveal below.
Model answer: Q = πR⁴ΔP/(8μL) shows that flow rate is proportional to the fourth power of radius — flow is extraordinarily sensitive to radius changes. A 50% reduction in radius (R → R/2) reduces flow to (1/2)⁴ = 1/16 of its original value at the same driving pressure. During arterial narrowing (stenosis), even modest reductions in internal radius from plaque buildup produce catastrophic drops in blood flow to downstream tissue. The heart must increase pressure significantly to maintain flow, raising blood pressure and cardiac workload.
The R⁴ relationship is the key result of laminar pipe flow analysis and is routinely underestimated because people expect a linear or quadratic relationship. The fourth-power sensitivity arises because radius appears both in the cross-sectional area (πR²) and in the parabolic velocity profile (which steepens as the pipe widens, giving higher average velocity). These effects compound, yielding the fourth-power scaling.