Poiseuille's equation

Poiseuille analysed the steady flow of a liquid through a narrow capillary tube and worked out an expression for the volume of liquid flowing through it per second, under these conditions: the flow through the tube is streamlined; the tube is horizontal, so gravity plays no part in the flow; the layer of liquid in contact with the tube's wall is at rest (a standard no-slip boundary condition); and the pressure is uniform over any given cross section of the tube. DIMENSIONAL DERIVATION. Let $v=V/t$ be the volume of liquid flowing out per second through the capillary tube; this depends on the coefficient of viscosity $\eta$ of the liquid, the radius r of the tube, and the pressure gradient $P/l$ along the tube's length. Writing $v=k\,\eta^a r^b (P/l)^c$ and matching dimensions of volume-per-time, viscosity, length and pressure-per-length on both sides gives three equations in a, b and c, which solve to $a=-1$, $b=4$, $c=1$; so $v=k\,\eta^{-1}r^4(P/l)$. The value of k, found experimentally (dimensional analysis alone cannot fix a pure number), is $\pi/8$, giving POISEUILLE'S EQUATION: $v=\dfrac{\pi r^4 P}{8\eta l}$. This relation holds only for fluid velocities below the critical velocity $v_c$ (i.e., for streamlined flow, as assumed i …