The steady two-dimensional boundary-layer equations for flow primarily in the $x$ direction are

$$\begin{gathered} \rho\left(u \frac{\partial u}{\partial x}+v \frac{\partial u}{\partial y}\right)=-\frac{d P}{d x}+\mu \frac{\partial^{2} u}{\partial y^{2}} \\ \frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0 \end{gathered}$$

A thin, steady, two-dimensional jet emerges from a point at the origin and flows along the $x$-axis in a fluid at rest far from the $x$-axis. Show that the momentum flux

$$F=\int_{-\infty}^{\infty} \rho u^{2} d y$$

is independent of position $x$ along the jet. Deduce that the thickness $\delta(x)$ of the jet increases along the jet as $x^{2 / 3}$, while the centre-line velocity $U(x)$ decreases as $x^{-1 / 3}$.

A similarity solution for the jet is sought with a streamfunction $\psi$ of the form

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

Derive the nonlinear third-order non-dimensional differential equation governing $f$, and write down the boundary and normalisation conditions which must be applied.