= Laminar plane jet from a line force
{title2=$f(\eta)=A\tanh(A\eta/6),\quad A=(9/2)^{1/3}$}
A steady two-dimensional laminar jet driven by force $F$ per span has conserved momentum flux $\rho\int u^2dy=F$. The boundary-layer pressure gradient is zero. Balancing inertia and viscosity with this flux gives width proportional to $x^{2/3}$ and centre velocity proportional to $x^{-1/3}$. The streamfunction $(F\nu x/\rho)^{1/3}f(\eta)$, $\eta=y[F/(\rho\nu^2x^2)]^{1/3}$, obeys $3f'''+(f')^2+ff''=0$. The displayed profile has $\int(f')^2d\eta=2A^3/9$, so momentum equality fixes $A$. Merely requiring independence of $x$ does not fix the amplitude. Entrainment makes volume flux $2A(F\nu x/\rho)^{1/3}$ grow downstream.
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