For a fluid with kinematic viscosity $\nu$, the steady axisymmetric boundary-layer equations for flow primarily in the $z$-direction are

$$\begin{aligned} u \frac{\partial w}{\partial r}+w \frac{\partial w}{\partial z} &=\frac{\nu}{r} \frac{\partial}{\partial r}\left(r \frac{\partial w}{\partial r}\right) \\ \frac{1}{r} \frac{\partial(r u)}{\partial r}+\frac{\partial w}{\partial z} &=0 \end{aligned}$$

where $u$ is the fluid velocity in the $r$-direction and $w$ is the fluid velocity in the $z$-direction. A thin, steady, axisymmetric jet emerges from a point at the origin and flows along the $z$-axis in a fluid which is at rest far from the $z$-axis.

(a) Show that the momentum flux

$$M:=\int_{0}^{\infty} r w^{2} d r$$

is independent of the position $z$ along the jet. Deduce that the thickness $\delta(z)$ of the jet increases linearly with $z$. Determine the scaling dependence on $z$ of the centre-line velocity $W(z)$. Hence show that the jet entrains fluid.

(b) A similarity solution for the streamfunction,

$$\psi(x, y, z)=\nu z g(\eta) \quad \text { with } \quad \eta:=r / z$$

exists if $g$ satisfies the second order differential equation

$$\eta g^{\prime \prime}-g^{\prime}+g g^{\prime}=0$$

Using appropriate boundary and normalisation conditions (which you should state clearly) to solve this equation, show that

$$g(\eta)=\frac{12 M \eta^{2}}{32 \nu^{2}+3 M \eta^{2}}$$