 Solution

Fix the scaling constants as $U=A/z$, $\delta =Bz$, with $A=F/(\rho \nu )$ and $B=\nu \sqrt{\rho /F}$, so $A{B}^2=\nu$. The Stokes streamfunction is then $\psi =\nu zf(\eta )$. Write $g=f^{\prime }/\eta$. Differentiation givesSubstitution in the boundary-layer equation, cancelling $A^2/z^3$, gives $-(f/\eta )g^{\prime }-g^2=g^{\prime \prime }+g^{\prime }/\eta$. Multiplication by ${\eta }^3$ therefore gives the required similarity equationAxis regularity requires zero radial velocity and a finite smooth axial velocity: choose the streamfunction gauge $f(0)=0$ and require $f(\eta )=c{\eta }^2+O({\eta }^4)$. Equivalently $f^{\prime }(0)=0$, $f^{\prime }/\eta$ has a finite limit, and $(f^{\prime }/\eta {)}^{\prime }\to 0$ at the axis. The constant $c$ is not specified in advance. In the quiescent ambient fluid require $f^{\prime }/\eta \to 0$ and $f^{\prime }-f/\eta \to 0$ as $\eta \to \infty$, so both velocity components vanish. The localized physical jet has finite axial volume flux, corresponding to finite $f(\infty )$.

Finally the prescribed force fixes the amplitude through the finite momentum-flux normalizationThe axis conditions express regularity rather than independently prescribed arbitrary derivatives; the far-field conditions express matching to the resting fluid, and the integral condition supplies the jet strength.