(a) Briefly outline the derivation of the boundary layer equation

$$u u_{x}+v u_{y}=U d U / d x+\nu u_{y y}$$

explaining the significance of the symbols used and what sets the $x$-direction.

(b) Viscous fluid occupies the sector $0<\theta<\alpha$ in cylindrical coordinates which is bounded by rigid walls and there is a line sink at the origin of strength $\alpha Q$ with $Q / \nu \gg 1$. Assume that vorticity is confined to boundary layers along the rigid walls $\theta=0$ $(x>0, y=0)$ and $\theta=\alpha$.

(i) Find the flow outside the boundary layers and clarify why boundary layers exist at all.

(ii) Show that the boundary layer thickness along the wall $y=0$ is proportional to

$$\delta:=\left(\frac{\nu}{Q}\right)^{1 / 2} x$$

(iii) Show that the boundary layer equation admits a similarity solution for the streamfunction $\psi(x, y)$ of the form

$$\psi=(\nu Q)^{1 / 2} f(\eta)$$

where $\eta=y / \delta$. You should find the equation and boundary conditions satisfied by $f$.

(iv) Verify that

$$\frac{d f}{d \eta}=\frac{5-\cosh (\sqrt{2} \eta+c)}{1+\cosh (\sqrt{2} \eta+c)}$$

yields a solution provided the constant $c$ has one of two possible values. Which is the likely physical choice?