Laminar plane jet from a line force ($f(\eta )=A\tanh (A\eta /6),A=(9/2{)}^{1/3}$)

A steady two-dimensional laminar jet driven by force $F$ per span has conserved momentum flux $\rho \int u^{2}dy=F$. 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 $(F\nu x/\rho {)}^{1/3}f(\eta )$, $\eta =y[F/(\rho {\nu }^2{x}^2){]}^{1/3}$, obeys $3f^{\prime \prime \prime }+(f^{\prime }{)}^2+f{f}^{\prime \prime }=0$. The displayed profile has $\int (f^{\prime }{)}^{2}d\eta =2A^3/9$, 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.