Poiseuille's law indicates that flow through a vessel is proportional to which factors?

• A. ΔP r^4 /(η L)
• B. ΔP /(η L)
• C. η L /(ΔP r^4)
• D. ΔP r^2 /(η L)

Answer: (A)

Poiseuille's law shows how laminar flow through a long, straight cylindrical vessel depends on the driving pressure, the radius of the vessel, the fluid's viscosity, and the vessel’s length. The flow rate is proportional to the pressure difference and to the radius to the fourth power, while being inversely proportional to viscosity and length. In other words, flow ∝ ΔP × r^4 ÷ (η × L). The constants π and 1/8 are fixed, so they’re not needed when considering proportional relationships.

This makes the option that includes ΔP times r^4 divided by η times L the best representation. The radius to the fourth power means even small changes in radius dramatically change flow, while increasing viscosity or length slows flow, and increasing the pressure difference speeds it up. The other forms either drop the r^4 dependence, use the wrong radius exponent, or place ηL in the numerator, which would suggest the opposite effect.