Explicit opposing-flow Falkner-Skan branches ($g(\eta )=4-6{\tanh }^2(\eta \pm {\eta }_0)$)

For outer velocity $U_s=-2/x$, the Falkner-Skan equation reduces for the scaled velocity $g=f^{\prime }$ to $g^{\prime \prime }+g^2-4=0$. Multiplication by $g^{\prime }$ gives $\frac{1}{2}(g^{\prime }{)}^2+g^3/3-4g=16/3$ for profiles tending to $-2$ at infinity. No slip gives $g(0)=0$, so $g^{\prime }(0)=\pm 4\sqrt{2/3}$. The two smooth solutions are the displayed translates with ${\eta }_0=\mathrm{arctanh}\sqrt{2/3}$. One decreases monotonically to the outer velocity; the other rises to $4$ before descending, producing an internal reversed-flow region. The latter has inflection points, motivating concern about shear instability, but the steady profile equation alone does not determine temporal stability.