= Solution
Let $\delta=Re^{-1/2}$, stretch $Y=y/\delta$ and write the original transverse <velocity> as $v=\delta V$. Continuity then gives $u_x+V_Y=0$. In the streamwise momentum equation $u_{yy}/Re=u_{YY}$ is order one, while $u_{xx}/Re$ is negligible. The transverse equation gives $p_Y=0$ to leading order, so <pressure> is supplied by the outer Euler flow. Its streamwise balance is $p_x=-U U'$. Consequently
$$
\boxed{u u_x+V u_Y=U U'+u_{YY},\qquad u_x+V_Y=0.}
$$
For a stationary impermeable plate, $\boxed{u(x,0)=V(x,0)=0}$, with matching $\boxed{u(x,Y)\to U(x)}$ as $Y\to\infty$. The transverse <velocity> must match the outer displacement flow; it is generally wrong to impose $V\to0$ independently at infinity.
At the leading edge the wall no-slip data meet the external stream abruptly, and the <boundary layer>[boundary-layer approximation] is nonuniform. For a constant external speed, the thickness is $O(\sqrt{x/Re})$, whose ratio to $x$ ceases to be small when the local <Reynolds number> is order one. A leading-edge region retaining neglected derivatives is needed there. An <adverse pressure gradient> can also cause separation farther downstream; a large global <Reynolds number> alone does not prevent it.
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