Boundary layer with exponentially varying outer velocity ($\Psi =e^{\alpha x/2}f(e^{\alpha x/2}Y)$)

For outer tangential speed $U_s=A{e}^{\alpha x}$ 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 $f^{\prime \prime \prime }+(\alpha /2)f{f}^{\prime \prime }+\alpha (A^2-(f^{\prime }{)}^2)=0$, with $f(0)=f^{\prime }(0)=0$ and $f^{\prime }(\infty )=A$. Attached far-field matching requires $\alpha A>0$: the outer speed increases along the direction of motion. Linearizing $g=f^{\prime }-A$ at large $\eta$ gives $g^{\prime \prime }+(\alpha A/2)\eta g^{\prime }-2\alpha Ag\simeq 0$. 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.