Hagen–Poiseuille equation

The Hagen-Poiseuille equation describes steady, fully developed, laminar flow of an incompressible Newtonian fluid through a straight rigid circular pipe. With pipe radius $R$, axial coordinate $z$ and constant viscosity $\mu$, force balance reduces to $\mu r^{-1}\frac{d}{dr}(r\frac{dw}{dr})=dp/dz$.

Steady pressure-driven flow through a circular pipe has the parabolic profile

Regularity at the centre and the no-slip boundary condition at $r=R$ fix the displayed profile. Integrating over the cross-section gives $Q=-\pi R^4(dp/dz)/(8\mu )=\pi R^4\Delta p/(8\mu L)$ for length $L$ and positive pressure drop $\Delta p$. The fourth power makes volume flux strongly sensitive to radius; the assumptions exclude arbitrary turbulent or developing pipe flow. ^[ref]