Why might the plot of flow versus radius not be linear in the activity?

Why might the plot of flow versus radius not be linear in the activity?

• A. There is an exponential relationship between radius and flow rate
• B. Radius has no effect on flow
• C. Flow is random
• D. Viscosity changes linearly

Answer: (A)

The key idea is that flow through a vessel changes very nonlinearly with radius. Poiseuille’s law shows that flow is proportional to the radius to the fourth power (Q ∝ r^4) when pressure drop, viscosity, and length are held constant. That means small increases in radius produce disproportionately large increases in flow—for example, doubling the radius would raise flow by about 16 times. Because of this r^4 relationship, the plot of flow versus radius is curved rather than a straight line, which is why it isn’t linear. The idea that radius has a strong, non-proportional effect on flow fits the concept of an exponential-type increase with radius. The other ideas—radius having no effect, flow being random, or viscosity changing in a simple linear way—do not capture why the relationship would curve as radius changes.

The key idea is that flow through a vessel changes very nonlinearly with radius. Poiseuille’s law shows that flow is proportional to the radius to the fourth power (Q ∝ r^4) when pressure drop, viscosity, and length are held constant. That means small increases in radius produce disproportionately large increases in flow—for example, doubling the radius would raise flow by about 16 times. Because of this r^4 relationship, the plot of flow versus radius is curved rather than a straight line, which is why it isn’t linear. The idea that radius has a strong, non-proportional effect on flow fits the concept of an exponential-type increase with radius. The other ideas—radius having no effect, flow being random, or viscosity changing in a simple linear way—do not capture why the relationship would curve as radius changes.