Viscous Flow Through A Pipe

Steady laminar flow of a viscous liquid through a narrow tube is governed by Poiseuille's equation, $Q=\frac{\pi P{r}^4}{8\eta L}$, where $P$ is the pressure difference, $r$ the radius, $L$ the length and $\eta$ the viscosity. The striking feature is the fourth-power dependence on radius, which arises because both the cross-sectional area and the velocity profile favour wider tubes. With pressure, length and viscosity held constant, $Q\propto r^4$. Replacing the tube with one of half the radius gives $Q_2/Q_1=(1/2{)}^4=1/16$. The option 1/2 assumes flow scales with radius, ignoring area and velocity effects. The option 1/4 treats $Q$ as proportional to area alone. The option 1/8 corresponds to an $r^3$ law and is still too generous. This dramatic $r^4$ sensitivity is exactly the result emphasised in the NCERT discussion of viscous flow and is why narrowed arteries so severely restrict blood flow. The fourth power emerges because the cross-sectional area itself contributes an $r^2$ factor while the parabolic velocity profile contributes a further $r^2$, since fluid near the centre of a wider tube faces far less wall drag. As a plausibility check, halving the radius slashes the flow far more than halving it, and the factor 1/16 reflects this extreme sensitivity correctly.