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Poiseuille's equation describes steady laminar flow of a viscous fluid through a cylindrical pipe: $Q = \pi\Delta Pr^4/(8\eta L)$. The flow rate depends on the fourth power of the radius — this is the most critical dependence.

Practical implications of the $r^4$ law: a 10% reduction in artery radius causes a 34% reduction in blood flow. A 50% blockage reduces flow by 93.75%. This is why atherosclerosis (artery narrowing) is so dangerous — even modest narrowing dramatically reduces flow.

The velocity profile is parabolic: $v(r) = (\Delta P/4\eta L)(R^2 - r^2)$, with maximum velocity at the center equal to twice the average velocity. The no-slip condition forces zero velocity at the walls.

Electrical analogy: $\Delta P \leftrightarrow V$, $Q \leftrightarrow I$, $R_{\text{flow}} = 8\eta L/(\pi r^4) \leftrightarrow R$. Pipes in series add resistances; pipes in parallel add conductances (1/R). This makes complex pipe network problems tractable.