Q.Derive Poiseuille's formula for the volume of a liquid flowing per second through a pipe under streamlined flow.

POISEUILLE'S EQUATION describes the steady, streamlined flow of a viscous liquid through a narrow (capillary) tube, under four conditions: the flow is streamlined; the tube is horizontal (so gravity plays no role); the layer of liquid touching the tube's wall is at rest (the standard no-slip condition); and the pressure is uniform across any one cross section of the tube. DIMENSIONAL DERIVATION: let $v=V/t$ be the volume of liquid flowing out per second; it depends on the liquid's coefficient of viscosity $\eta$, the tube's radius r, and the pressure gradient $P/l$ along the tube. Writing $v=k\,\eta^a r^b(P/l)^c$ and matching the dimensions of volume-per-time, viscosity, length, and pressure-per-length on both sides gives three simultaneous equations in the exponents a, b, c, solving to $a=-1$, $b=4$, $c=1$: $v=k\,\eta^{-1}r^4(P/l)$. The dimensionless constant k, found experimentally (dimensional analysis alone can never fix a pure number), is $\pi/8$, so POISEUILLE'S EQUATION is $v=\dfrac{\pi r^4 P}{8\eta l}$. This …