= Bickley jet
{c}
A steady planar viscous jet whose conserved momentum flux and convection-diffusion balance give the displayed scaling. Choosing $U\delta^2=\nu x$ and $U$ as centre-line speed gives streamfunction profile $f''' +(ff''+(f')^2)/3=0$, with $f(0)=f''(0)=0$, $f'(0)=1$ and $f'(\pm\infty)=0$. Integration yields $f=\sqrt6\tanh(\eta/\sqrt6)$ and velocity profile $f'=\operatorname{sech}^2(\eta/\sqrt6)$. Its momentum integral is $4\sqrt6/3$, fixing the dimensional prefactors. Transverse entrainment does not require $f(\infty)=0$.
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