= Solution
For the steady, fully developed ansatz $\mathbf u=w(r)\mathbf e_z$, both $\partial_t\mathbf u$ and $(\mathbf u\cdot\nabla)\mathbf u$ vanish. The axial <Navier-Stokes equation> reduces to
$$
0=-\frac{\partial p}{\partial z}
+\mu\frac1r\frac d{dr}\left(r\frac{dw}{dr}\right).
$$
Two integrations give
$$
w(r)=\frac1{4\mu}\frac{\partial p}{\partial z}r^2
+C_1\log r+C_2.
$$
Regularity at the axis forces $C_1=0$, and the <no-slip boundary condition> $w(R)=0$ gives the <Hagen-Poiseuille flow>
$$
\boxed{
w(r)=-\frac1{4\mu}\frac{\partial p}{\partial z}
(R^2-r^2)}.
$$
For flow in the positive $z$ direction, $\partial p/\partial z<0$.
Back to article page
