Bickley jet

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 $f^{\prime \prime \prime }+(f{f}^{\prime \prime }+(f^{\prime }{)}^2)/3=0$, with $f(0)=f^{\prime \prime }(0)=0$, $f^{\prime }(0)=1$ and $f^{\prime }(\pm \infty )=0$. Integration yields $f=\sqrt{6}\tanh (\eta /\sqrt{6})$ and velocity profile $f^{\prime }={\mathrm{sech}}^2(\eta /\sqrt{6})$. Its momentum integral is $4\sqrt{6}/3$, fixing the dimensional prefactors. Transverse entrainment does not require $f(\infty )=0$.