A plate is drawn vertically out of a bath and the resultant liquid drains off the plate as a thin film. Using lubrication theory, show that the governing equation for the thickness of the film, $h(x, t)$ is

$$\frac{\partial h}{\partial t}+\left(\frac{g h^{2}}{\nu}\right) \frac{\partial h}{\partial x}=0$$

where $t$ is time and $x$ is the distance down the plate measured from the top.

Verify that

$$h(x, t)=F\left(x-\frac{g h^{2}}{\nu} t\right)$$

satisfies $(*)$ and identify the function $F(x)$. Using this relationship or otherwise, determine an explicit expression for the thickness of the film assuming that it was initially of uniform thickness $h_{0}$.