Weak surface-shear perturbation of stagnation flow ($u(x,0)=x[1+\sqrt{\pi /8}\gamma +O({\gamma }^2)]$)

For the stress-driven stagnation-point boundary layer on $\zeta <0$, a regular perturbation $f=\zeta +\gamma f_1+O({\gamma }^2)$ gives $g=f_1^{\prime }$ satisfying $g^{\prime \prime }+\zeta g^{\prime }-2g=0$, $g^{\prime }(0)=1$ and $g(-\infty )=0$. Reduction of order with the known solution $1+{\zeta }^2$ givesIndeed, writing $g=(1+{\zeta }^2)v$ gives $v^{\prime }=C{e}^{-{\zeta }^2/2}/(1+{\zeta }^2{)}^2$; decay at $-\infty$ fixes the lower endpoint and $g^{\prime }(0)=1$ fixes $C=1$. The integral at zero is $\sqrt{2\pi }/4=\sqrt{\pi /8}$, as follows by differentiating the antiderivativeHere $\mathrm{erf}$ 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.