 Similarity profile of an axisymmetric radial viscous free jet

In a symmetric thin radial free jet, $F=\int r{u}_r^{2}dz$ is conserved by the conservative momentum equation. Balancing advection and viscosity gives thickness $\delta =Cr$, with $C=({\nu }^2/F{)}^{1/3}$. The normalized similarity streamfunction uses $A=(F^2/\nu {)}^{1/3}$ and $f=2k\tanh (k\eta )$, $k=(3/16{)}^{1/3}$, so $\int f^{\prime 2}d\eta =1$. The displayed velocity follows, and the thin-layer requirement is $F\gg {\nu }^2$.