Solution
= Solution
The steady axial <Navier-Stokes equation> for the straight-pipe approximation is $0=G/\rho+\nu w''(y)$. No-slip boundary conditions are $w(\pm a)=0$. Integrating twice gives
$$
\boxed{\mathbf u=(0,0,w(y)),\qquad w(y)=\frac{G}{2\rho\nu}(a^2-y^2).}
$$
Ignoring the radial variation is justified away from the side walls by the given ordering $a\ll b$.