• id: JME-09-N12
• title: Poiseuille's Law for Pipe Flow
• tags: poiseuille, pipe-flow, viscous

For steady laminar flow of a viscous fluid through a cylindrical pipe: $Q = \frac{\pi \Delta P r^4}{8\eta L}$, where $Q$ is volume flow rate, $\Delta P$ is pressure difference, $r$ is pipe radius, $\eta$ is viscosity, and $L$ is pipe length. Key dependence: $Q \propto r^4$ — doubling the pipe radius increases flow rate by 16 times. This is why even small arterial blockages drastically reduce blood flow. The velocity profile is parabolic: $v(r) = \frac{\Delta P}{4\eta L}(R^2 - r^2)$, with maximum velocity at the center (twice the average velocity).