= Explicit opposing-flow Falkner-Skan branches
{title2=$g(\eta)=4-6\tanh^2(\eta\pm\eta_0)$}
For outer velocity $U_s=-2/x$, the <Falkner-Skan equation> reduces for the scaled velocity $g=f'$ to $g''+g^2-4=0$. Multiplication by $g'$ gives $\tfrac12(g')^2+g^3/3-4g=16/3$ for profiles tending to $-2$ at infinity. No slip gives $g(0)=0$, so $g'(0)=\pm4\sqrt{2/3}$. The two smooth solutions are the displayed translates with $\eta_0=\operatorname{arctanh}\sqrt{2/3}$. 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.
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