Viscous incompressible fluid of uniform density is extruded axisymmetrically from a thin circular slit of small radius centred at the origin and lying in the plane $z=0$ in cylindrical polar coordinates $r, \theta, z$. There is no external radial pressure gradient. It is assumed that the fluid forms a thin boundary layer, close to and symmetric about the plane $z=0$. The layer has thickness $\delta(r) \ll r$. The $r$-component of the steady Navier-Stokes equations may be approximated by

$$u_{r} \frac{\partial u_{r}}{\partial r}+u_{z} \frac{\partial u_{r}}{\partial z}=\nu \frac{\partial^{2} u_{r}}{\partial z^{2}}$$

(i) Prove that the quantity (proportional to the flux of radial momentum)

$$\mathcal{F}=\int_{-\infty}^{\infty} u_{r}^{2} r d z$$

is independent of $r$.

(ii) Show, by balancing terms in the momentum equation and assuming constancy of $\mathcal{F}$, that there is a similarity solution of the form

$$u_{r}=-\frac{1}{r} \frac{\partial \Psi}{\partial z}, \quad u_{z}=\frac{1}{r} \frac{\partial \Psi}{\partial r}, \quad \Psi=-A \delta(r) f(\eta), \quad \eta=\frac{z}{\delta(r)}, \quad \delta(r)=C r$$

where $A, C$ are constants. Show that for suitable choices of $A$ and $C$ the equation for $f$ takes the form

$$\begin{gathered} -f^{\prime 2}-f f^{\prime \prime}=f^{\prime \prime \prime} ; \\ f=f^{\prime \prime}=0 \text { at } \eta=0 ; \quad f^{\prime} \rightarrow 0 \text { as } \eta \rightarrow \infty ; \\ \int_{-\infty}^{\infty} f_{\eta}^{2} d \eta=1 . \end{gathered}$$

(iii) Give an inequality connecting $\mathcal{F}$ and $\nu$ that ensures that the boundary layer approximation $(\delta \ll r)$ is valid. Solve the equation to give a complete solution to the problem for $u_{r}$ when this inequality holds.

[Hint: $\left.\int_{-\infty}^{\infty} \operatorname{sech}^{4} x d x=4 / 3 .\right]$