Poiseuille's law
Summary
Poiseuille's law gives the volume flow rate of a viscous fluid in steady laminar flow through a long cylindrical tube: Q = πr⁴ΔP / (8ηL). Most testable point: flow depends on the fourth power of the radius, so a small change in radius changes flow or resistance a great deal.
Detail
Meaning: Poiseuille's law describes steady laminar flow of a viscous fluid through a long, rigid, cylindrical tube. A pressure difference drives the flow, while viscosity and friction at the wall oppose it. Flow rate rises with pressure difference and radius, and falls with viscosity and length.
Equation: Q = πr⁴ΔP / (8ηL), where Q is volume flow rate (m³/s), r is tube radius (m), ΔP is the pressure difference between the ends (Pa), η is viscosity (Pa·s), and L is tube length (m). Resistance is R = 8ηL/(πr⁴), so ΔP = QR.
On the MCAT: Blood-vessel passages ask how flow or resistance changes when radius, length or viscosity changes. Halving the radius cuts flow 16-fold at fixed ΔP, so the same flow would need 16 times the pressure difference. Doubling the length or viscosity halves the flow. Resistances add in series; in parallel their reciprocals add.
Common confusion: Students apply Bernoulli's equation, which neglects viscosity, to a real pipe. With viscosity, pressure falls along a tube even when its radius is constant. Another slip is using r² because Q = Av: at fixed ΔP the speed v also grows with r², giving r⁴ overall. The law does not hold for turbulent flow.
Sources
- Fundamentals of Physics
- University Physics
- Physics
Written for the MCAT and checked before it was published. For study reference only. Spotted an error? Let us know.