= Weak surface-shear perturbation of stagnation flow
{title2=$u(x,0)=x[1+\sqrt{\pi/8}\,\gamma+O(\gamma^2)]$}
For the <stress-driven stagnation-point boundary layer> on $\zeta<0$, a <regular perturbation> $f=\zeta+\gamma f_1+O(\gamma^2)$ gives $g=f_1'$ satisfying $g''+\zeta g'-2g=0$, $g'(0)=1$ and $g(-\infty)=0$. <Reduction of order> with the known solution $1+\zeta^2$ gives
$$
g(\zeta)=(1+\zeta^2)\int_{-\infty}^{\zeta}
\frac{e^{-s^2/2}}{(1+s^2)^2}\,ds.
$$
Indeed, writing $g=(1+\zeta^2)v$ gives $v'=C e^{-\zeta^2/2}/(1+\zeta^2)^2$; decay at $-\infty$ fixes the lower endpoint and $g'(0)=1$ fixes $C=1$. The <integral> at zero is $\sqrt{2\pi}/4=\sqrt{\pi/8}$, as follows by differentiating the antiderivative
$$
\frac14\left[\frac{2s}{1+s^2}e^{-s^2/2}
+\sqrt{2\pi}\operatorname{erf}(s/\sqrt2)\right].
$$
Here $\operatorname{erf}$ is the <error function>. Thus positive imposed outward <shear stress> increases the outward surface <velocity> by the displayed first-order factor. Replacing decay at $-\infty$ by decay at $+\infty$ while leaving the equation and wall <derivative> unchanged reverses this coefficient and describes a different problem.
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