Velocity as dependent variable in the Falkner-Skan equation ($w=f^{\prime },w^2{w}_{ff}+w(w_f{)}^2+\beta fw{w}_f+m(1-w^2)=0$)

For the Falkner-Skan equation $f^{(3)}+\beta f{f}^{\prime \prime }+m(1-(f^{\prime }{)}^2)=0$, an attached branch has $f^{\prime }>0$ away from the wall. It is then possible to replace the similarity coordinate by $f$ itself. Define $w(f)=f^{\prime }$. The chain rule gives $f^{\prime \prime }=w{w}_f$ and $f^{(3)}=w(w_f^2+w{w}_{ff})$, proving the displayed second-order equation. The wall and outer conditions become $w(0)=0$ and $w(\infty )=1$ when the stream function is set to zero at the wall.

The reduction is singular at the wall: if $s=f^{\prime \prime }(0)>0$, then $f\sim s{\eta }^2/2$ and $w\sim \sqrt{2sf}$. Hence $w_f$ diverges while $w{w}_f\to s$ remains finite. A finite-slope condition on $w$ at zero would incorrectly exclude the ordinary positive-wall-shear branch. At flow reversal or separation, the invertibility of the coordinate change must be checked again.