Let us now consider a real cylindrical tube of radius $R$ and length $L$, through which a fluid with viscosity $\eta$ is flowing.
A pressure difference $\Delta p = p_2 - p_1$ is applied across the ends of the cylinder. The force resulting from this pressure difference during steady flow must be equal to the frictional force along the cylinder wall with area $A = 2\pi r L$. Thus, we have:
$$
-\eta 2\pi r L \frac{\mathrm{d}v}{\mathrm{d}r} = \pi r^2 \Delta p
$$
Due to symmetry, only the distance $r$ from the cylinder axis is relevant.
To determine the velocity $v$ as a function of $r$, we rearrange and integrate the equation:
$$
v(r) = \int_{r}^{R} \frac{\Delta p}{2\eta L} \tilde{r} \, \mathrm{d}\tilde{r}
$$
which leads to the parabolic velocity profile:
$$
v(r) = \frac{\Delta p}{4\eta L}\left(R^2 - r^2\right)
$$
This gives rise to the parabolic flow profile found in cylindrical pipes.
To determine the amount of fluid flowing per unit time through a small annular shell of thickness $\mathrm{d}r$, we write:
$$
\frac{\mathrm{d}V(r)}{\mathrm{d}t}\, \mathrm{d}r = 2\pi r v(r)\, \mathrm{d}r
$$
Inserting $v(r)$ yields:
$$
\frac{\mathrm{d}V(r)}{\mathrm{d}t}\, \mathrm{d}r = \frac{2\pi r (R^2 - r^2)}{4\eta L} \Delta p \, \mathrm{d}r
$$
Integrating over $r$ finally results in the Hagen–Poiseuille law:
$$
\boxed{\frac{\mathrm{d}V}{\mathrm{d}t} = \frac{\pi R^4}{8\eta L} \Delta p}
$$
This law states that the volume flow rate in a cylindrical pipe is proportional to the pressure difference between the ends and increases with the fourth power of the radius.
This strong dependence on the radius explains why the narrowing of blood vessels in smokers can be dangerous. It also shows how the human body can regulate blood flow by narrowing or widening blood vessels.
Note: The Hagen–Poiseuille law applies strictly to Newtonian fluids. Blood is not a perfect Newtonian fluid—its viscosity depends slightly on pressure—but the law still provides a good approximation in most practical cases.