Hagen-Poisseille Equation

The force due to the pressure difference $\Delta p = p_2 - p_1$ on both ends of a cylinder is equal to the force along the cylindrical surface $A = 2\pi r L$ due to internal friction of the fluid: $$-\eta 2\pi r L \frac{\mathrm{d} v}{\mathrm{d} r} = \pi r^2 \Delta p$$ In order to determine the speed $v$ as a function of the radius $r$, this equation can be rearranged and integrated according to: $$v(r) = \int_{r}^{R} \frac{\Delta p}{2\eta L}r'\,\mathrm{d} r'$$ This leads to the following equation: $$v(r) = \frac{\Delta p}{4\pi L}\left(R^2 - r^2\right)$$ The result is a parabolic shape of the fluid inside the cylinder. The fluid volume in a small layer can be written as $$ \frac{\mathrm{d} V(r)}{\mathrm{d} t}\,\mathrm{d} r = 2\pi v(r)\,\mathrm{d} r$$ Inserting $v(r)$ results in $$\frac{\mathrm{d} V(r)}{\mathrm{d} t}\,\mathrm{d} r = \frac{2\pi r \left(R^2 - r^2\right)}{2\eta L}\Delta p\,\mathrm{d} r$$ Computing the integral with $\mathrm{d}r$ finally leads to the Hagen-Poisseille equation: $$\boxed{\frac{\mathrm{d} V}{\mathrm{d}t} = \frac{\pi R^4}{8\eta L}\Delta p}$$ This equation states that the volume flow is proportional to the pressure difference in a cylindrical pipe and that it increases with the fourth power of the radius of the pipe.