Radial velocity in cylindrical coordinates

Away from the $z$ -axis, the radial velocity is $u_r=u\cdot e_r$ , where $e_r$ is the outward radial unit vector in cylindrical coordinates. Along a particle trajectory,

\frac{dr}{dt}=u_r(t,X(t)).

Thus $u_r<0$ describes radial inflow toward the axis, and $u_r>0$ describes radial outflow.

Cylindrical versus spherical radius

Here $r=\sqrt{x^2+y^2}$ is distance from the axis. It is not distance from the origin in $\mathbb R^3$ . A flow can move toward the axis while moving away from the origin in the axial direction.