A steady, two-dimensional flow in the region $y>0$ takes the form $(u, v)=$ $(E x,-E y)$ at large $y$, where $E$ is a positive constant. The boundary at $y=0$ is rigid and no-slip. Consider the velocity field $u=\partial \psi / \partial y, v=-\partial \psi / \partial x$ with stream function $\psi=\operatorname{Ex} \delta f(\eta)$, where $\eta=y / \delta$ and $\delta=(\nu / E)^{1 / 2}$ and $\nu$ is the kinematic viscosity. Show that this velocity field satisfies the Navier-Stokes equations provided that $f(\eta)$ satisfies

$$f^{\prime \prime \prime}+f f^{\prime \prime}-\left(f^{\prime}\right)^{2}=-1$$

What are the conditions on $f$ at $\eta=0$ and as $\eta \rightarrow \infty$ ?