Viscous fluid is extracted through a small hole in the tip of the cone given by $\theta=\alpha$ in spherical polar coordinates $(R, \theta, \phi)$. The total volume flux through the hole takes the constant value $Q$. It is given that there is a steady solution of the Navier-Stokes equations for the fluid velocity $\mathbf{u}$. For small enough $R$, the velocity $\mathbf{u}$ is well approximated by $\mathbf{u} \sim\left(-A / R^{2}, 0,0\right)$, where $A=Q /[2 \pi(1-\cos \alpha)]$ except in thin boundary layers near $\theta=\alpha$.

(i) Verify that the volume flux through the hole is approximately $Q$.

(ii) Construct a Reynolds number (depending on $R$ ) in terms of $Q$ and the kinematic viscosity $\nu$, and thus give an estimate of the value of $R$ below which solutions of this type will appear.

(iii) Assuming that there is a boundary layer near $\theta=\alpha$, write down the boundary layer equations in the usual form, using local Cartesian coordinates $x$ and $y$ parallel and perpendicular to the boundary. Show that the boundary layer thickness $\delta(x)$ is proportional to $x^{\frac{3}{2}}$, and show that the $x$ component of the velocity $u_{x}$ may be written in the form

$$u_{x}=-\frac{A}{x^{2}} F^{\prime}(\eta), \quad \text { where } \quad \eta=\frac{y}{\delta(x)}$$

Derive the equation and boundary conditions satisfied by $F$. Give an expression, in terms of $F$, for the volume flux through the boundary layer, and use this to derive the $R$ dependence of the first correction to the flow outside the boundary layer.