Past exam of the physics B course of the University of Cambridge / 2018 / Paper 1 / Section A / A2

Let the signed axial velocity on the axis be $w(z)$. Smooth axisymmetric flow has $u_z(r,z)=w(z)+O(r^2)$ and regular radial velocity at $r=0$. The incompressible flow condition in cylindrical coordinates isTo leading order near the axis, integrate ${\partial }_r(r{u}_r)=-r{w}^{\prime }(z)$ from zero to $r$. The integration constant vanishes by regularity, giving the near-axis radial velocity in incompressible axisymmetric flow,With the usual signed interpretation $w(z)=A/z^3$,For $A>0$ this is outward: the outward axial flow slows as its distance from the origin grows, and incompressibility requires radial spreading. Here $A$ has units ${\mathrm{m}}^4{\mathrm{s}}^{-1}$, so the answer is a speed.

The source specifies a magnitude rather than an unambiguous signed component. On $z>0$, reversing the axial flow while keeping its magnitude would reverse the radial answer. The general relation $u_r=-r{w}^{\prime }/2$ makes that dependence explicit; the boxed expression adopts $u_z=A/z^3$, outward on both halves of the axis.