Two-dimensional point sink ($W(z)=-\frac{Q}{2\pi }\log (z-z_0)$)

A two-dimensional point sink of strength $Q>0$ removes volume flux $Q$ per unit depth at $z_0$. Its complex potential of a two-dimensional potential flow is the displayed expression, so $u-iv=-Q/[2\pi (z-z_0)]$. The radial velocity is $u_r=-Q/(2\pi r)$: integrating over a circle gives outward flux $2\pi r{u}_r=-Q$. This is the planar cross-section of a line sink. A passive particle satisfies $d(r^2)/dt=-Q/\pi$ and reaches the ideal point singularity in finite time. This is a flow singularity, distinct from a stable sink equilibrium of a regular dynamical system.