Starting from the steady planar vorticity equation

$$\mathbf{u} \cdot \nabla \omega=\nu \nabla^{2} \omega,$$

outline briefly 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.

Viscous fluid occupies the region $y>0$ with rigid stationary walls along $y=0$ for $x>0$ and $x<0$. There is a line sink at the origin of strength $\pi Q, Q>0$, with $Q / \nu \gg 1$. Assuming that vorticity is confined to boundary layers along the rigid walls:

(a) Find the flow outside the boundary layers.

(b) Explain why the boundary layer thickness $\delta$ along the wall $x>0$ is proportional to $x$, and deduce that

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

(c) Show that the boundary layer equation admits a solution having stream function

$$\psi=(\nu Q)^{1 / 2} f(\eta) \quad \text { with } \quad \eta=y / \delta$$

Find the equation and boundary conditions satisfied by $f$.

(d) Verify that a solution is

$$f^{\prime}=\frac{6}{1+\cosh (\eta \sqrt{2}+c)}-1$$

provided that $c$ has one of two values to be determined. Should the positive or negative value be chosen?