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In a thin boundary layer, take streamwise velocity and distance to be order one, set $z=\delta \zeta$, and use incompressibility to set $w=\delta W$. The streamwise convective acceleration is order one, whereas its dominant viscous contribution after division by the Reynolds number is ${\mathrm{Re}}^{-1}{\delta }^{-2}{u}_{\zeta \zeta }$. Their dominant balance selectsWith this choice, the exact rescaled Navier-Stokes equations areThus the leading pressure is independent of $\zeta$. Matching it to the outer solution from part (a), evaluated near $z=0$, gives $p=p_{\ast }-x^2/2$, hence $-p_x=x$. The leading boundary-layer equation is thereforeThe exact surface shear stress condition rescales to $u_{\zeta }+{\delta }^2{W}_x=\delta \beta x=\gamma x$. Since $W(x,0)=0$ for all $x$, its tangential derivative also vanishes there. The leading surface boundary conditions areMatching to the outer solution requiresThe second matching condition specifies the leading growth of $W$, allowing a constant displacement correction. Requiring $W+\zeta \to 0$ would impose an extra condition not supplied by the leading outer solution. This is the stress-driven stagnation-point boundary layer.

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.