Viscous fluid occupying $z>0$ is bounded by a rigid plane at $z=0$ and is extracted through a small hole at the origin at a constant flow rate $Q=2 \pi A$. Assume that for sufficiently small values of $R=|\mathbf{x}|$ the velocity $\mathbf{u}(\mathbf{x})$ is well-approximated by

$$\mathbf{u}=-\frac{A \mathbf{x}}{R^{3}}$$

except within a thin axisymmetric boundary layer near $z=0$.

(a) Estimate the Reynolds number of the flow as a function of $R$, and thus give an estimate for how small $R$ needs to be for such a solution to be applicable. Show that the radial pressure gradient is proportional to $R^{-5}$.

(b) In cylindrical polar coordinates $(r, \theta, z)$, the steady axisymmetric boundary-layer equations for the velocity components $(u, 0, w)$ can be written as

$$u \frac{\partial u}{\partial r}+w \frac{\partial u}{\partial z}=-\frac{1}{\rho} \frac{d P}{d r}+\nu \frac{\partial^{2} u}{\partial z^{2}}, \quad \text { where } \quad u=-\frac{1}{r} \frac{\partial \Psi}{\partial z}, \quad w=\frac{1}{r} \frac{\partial \Psi}{\partial r}$$

and $\Psi(r, z)$ is the Stokes streamfunction. Verify that the condition of incompressibility is satisfied by the use of $\Psi$.

Use scaling arguments to estimate the thickness $\delta(r)$ of the boundary layer near $z=0$ and then to motivate seeking a similarity solution of the form

$$\Psi=(A \nu r)^{1 / 2} F(\eta), \quad \text { where } \quad \eta=z / \delta(r)$$

(c) Obtain the differential equation satisfied by $F$, and state the conditions that would determine its solution. [You are not required to find this solution.]

By considering the flux in the boundary layer, explain why there should be a correction to the approximation $(*)$ of relative magnitude $(\nu R / A)^{1 / 2} \ll 1$.