Air is blown over the surface of a large, deep reservoir of water in such a way as to exert a tangential stress in the $x$-direction of magnitude $K x^{2}$ for $x>0$, with $K>0$. The water is otherwise at rest and occupies the region $y>0$. The surface $y=0$ remains flat.

Find order-of-magnitude estimates for the boundary-layer thickness $\delta(x)$ and tangential surface velocity $U(x)$ in terms of the relevant physical parameters.

Using the boundary-layer equations, find the ordinary differential equation governing the dimensionless function $f$ defined in the streamfunction

$$\psi(x, y)=U(x) \delta(x) f(\eta), \quad \text { where } \eta=y / \delta(x)$$

What are the boundary conditions on $f$ ?

Does $f \rightarrow 0$ as $\eta \rightarrow \infty$ ? Why, or why not?

The total horizontal momentum flux $P(X)$ across the vertical line $x=X$ is proportional to $X^{a}$ for $X>0$. Find the exponent $a$. By considering the steadiness of the momentum balance in the region $0<x<X$, explain why the value of $a$ is consistent with the form of the stress exerted on the boundary.