Boundary-layer equation ($u{u}_x+v{u}_y=-P_x/\rho +\nu u_{yy},u_x+v_y=0$)

The leading momentum and mass equations in a thin laminar shear region with flow predominantly along a surface or jet axis. Longitudinal viscous derivatives are smaller than transverse ones, and pressure is supplied by the outer flow. The transverse velocity remains necessary for mass conservation and entrainment even when it is small compared with the axial velocity.

For a fluid occupying $z<0$, match an outer stagnation point flow $u=x,w=-z$ to a flat impermeable interface with dimensionless shear stress $\beta x$. With boundary layer thickness $\delta ={\mathrm{Re}}^{-1/2}$, inner variable $\zeta =z/\delta$ and $\gamma =\delta \beta$, the leading boundary-layer equation admits $u=x{f}^{\prime }(\zeta )$ and $w=-\delta f(\zeta )$. Incompressibility is automatic, and tangential momentum conservation gives the displayed ordinary differential equation, with $f(0)=0$, $f^{\prime \prime }(0)=\gamma$ and $f^{\prime }\to 1$ at $\zeta \to -\infty$. The sign of this far-field endpoint is fixed by the fluid domain. This is a prescribed-traction interface; it does not impose a no-slip boundary condition.

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.