The equation for the vorticity $\omega(x, y)$ in two-dimensional incompressible flow takes the form

$$\frac{\partial \omega}{\partial t}+u \frac{\partial \omega}{\partial x}+v \frac{\partial \omega}{\partial y}=\nu\left(\frac{\partial^{2} \omega}{\partial x^{2}}+\frac{\partial^{2} \omega}{\partial y^{2}}\right)$$

where

$$u=\frac{\partial \psi}{\partial y}, \quad v=-\frac{\partial \psi}{\partial x} \quad \text { and } \quad \omega=-\left(\frac{\partial^{2} \psi}{\partial x^{2}}+\frac{\partial^{2} \psi}{\partial y^{2}}\right)$$

and $\psi(x, y)$ is the stream function.

Show that this equation has a time-dependent similarity solution of the form

$$\psi=C x H(t)^{-1} \phi(\eta), \quad \omega=-C x H(t)^{-3} \phi_{\eta \eta}(\eta) \quad \text { for } \quad \eta=y H(t)^{-1}$$

if $H(t)=\sqrt{2 C t}$ and $\phi$ satisfies the equation

$$3 \phi_{\eta \eta}+\eta \phi_{\eta \eta \eta}-\phi_{\eta} \phi_{\eta \eta}+\phi \phi_{\eta \eta \eta}+\frac{1}{R} \phi_{\eta \eta \eta \eta}=0$$

and $R=C / \nu$ is the effective Reynolds number.

Show that this solution is appropriate for the problem of two-dimensional flow between the rigid planes $y=\pm H(t)$, and determine the boundary conditions on $\phi$ in that case.

Verify that $(*)$ has exact solutions, satisfying the boundary conditions, of the form

$$\phi=\frac{(-1)^{k}}{k \pi} \sin (k \pi \eta)-\eta, \quad k=1,2, \ldots$$

when $R=k^{2} \pi^{2} / 4$. Sketch this solution when $k$ is large, and discuss whether such solutions are likely to be realised in practice.