= Similarity profile of an axisymmetric radial viscous free jet
{title2=$u_r=\frac{(9F^2/(32\nu))^{1/3}}r\operatorname{sech}^2\!\left[(3F/(16\nu^2))^{1/3}z/r\right]$}
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 $\int f'^2d\eta=1$. The displayed velocity follows, and the thin-layer requirement is $F\gg\nu^2$.
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