Radial source flow ($u_r=Q/(2\pi r)$)

A two-dimensional incompressible line source with area flux $Q$ has radial velocity $Q/(2\pi r)$, obtained from flux conservation around a circle. It is irrotational away from the source, with potential $(Q/(2\pi ))\ln r$. For a circular expanding interface, $Q=2\pi a{a}_t$ and $u_r=a{a}_t/r$, whose radial strain is $-a{a}_t/r^2$. That strain supplies a normal viscous stress even though the velocity is a potential flow.