Viscosity: A Fluid's Resistance to Flow
Viscosity is the internal friction of a fluid — the property that makes honey pour slowly and water splash freely. Formally, dynamic viscosity μ is the ratio of shear stress to strain rate: thick fluids resist deformation. Water has a viscosity of about 0.001 Pa·s, while honey is roughly 10,000 times more viscous. Jean Léonard Marie Poiseuille, a physician studying blood flow, quantified how viscosity governs flow through pipes in the 1840s.
Poiseuille's Law and the Power of r⁴
Poiseuille's law states Q = πr⁴ΔP/(8μL) — flow rate is proportional to the fourth power of radius. This r⁴ dependence is extraordinary: doubling the pipe radius increases flow by a factor of 16. Halving the radius reduces it to just 6.25% of the original. This is why even a small narrowing of blood vessels (atherosclerosis) can dramatically reduce blood flow, and why a slight widening (vasodilation) can provide enormous relief.
The Parabolic Velocity Profile
In this simulation, look at the velocity arrows across the pipe diameter. The no-slip boundary condition forces fluid velocity to zero at the pipe walls. Viscous friction between fluid layers creates a parabolic velocity profile: v(r) = v_max(1 - r²/R²). The centerline velocity is exactly twice the mean velocity. This profile is beautifully visible when comparing low-viscosity fluids (sharp, narrow parabola) with high-viscosity fluids (broad, flat-topped flow).
From IV Drips to Volcanic Eruptions
Poiseuille flow appears everywhere: IV drip rates in hospitals depend on needle gauge (radius), oil pipeline economics depend on pipe diameter, and volcanic eruption styles depend on magma viscosity. Low-viscosity basaltic lava (like in Hawaii) flows gently; high-viscosity rhyolitic magma builds pressure until explosive eruptions occur. In the body, blood viscosity increases with hematocrit (red blood cell concentration), which is why polycythemia can cause dangerous circulatory strain.