= Boundary layer with exponentially varying outer velocity
{title2=$\Psi=e^{\alpha x/2}f(e^{\alpha x/2}Y)$}
For outer tangential speed $U_s=Ae^{\alpha x}$ in steady <Prandtl boundary-layer equations>, the displayed <similarity solution> makes the streamwise convection, <pressure gradient> and transverse viscous diffusion have the same exponential factor. Its profile equation is $f'''+(\alpha/2)ff''+\alpha(A^2-(f')^2)=0$, with $f(0)=f'(0)=0$ and $f'(\infty)=A$. Attached far-field matching requires $\alpha A>0$: the outer speed increases along the direction of motion. Linearizing $g=f'-A$ at large $\eta$ gives $g''+(\alpha A/2)\eta g'-2\alpha A g\simeq0$. For positive $\alpha A$ there is a Gaussian-decaying mode, while negative $\alpha A$ leaves only growing nontrivial asymptotic modes. Local thinness additionally requires a large <Reynolds number> based on the speed and its streamwise variation length.
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