Solution
= Solution
Let $U$ be the streamwise speed and $\delta$ the layer width. Continuity gives $v\sim U\delta/x$, while inertia--viscosity balance gives
$$
\frac{U^2}{x}\sim\frac{\nu U}{\delta^2}.
$$
The imposed stress gives $\rho\nu U/\delta\sim Sx^{-1/2}$. Eliminating $U$ yields
$$
\delta\sim\left(\frac{\rho\nu^2}{S}\right)^{1/3}x^{1/2},
$$
and
$$
U\sim\left(\frac{S^2}{\rho^2\nu}\right)^{1/3},
$$
which is independent of $x$.