4.I $. 7 \mathrm{C} \quad$

If $\mathbf{u}$ is given in Cartesian co-ordinates as $\mathbf{u}=(-\Omega y, \Omega x, 0)$, with $\Omega$ a constant, verify that

$\mathbf{u} \cdot \nabla \mathbf{u}=\nabla\left(-\frac{1}{2} \mathbf{u}^{2}\right)$

When incompressible fluid is placed in a stationary cylindrical container of radius $a$ with its axis vertical, the depth of the fluid is $h$. Assuming that the free surface does not reach the bottom of the container, use cylindrical polar co-ordinates to find the equation of the free surface when the fluid and the container rotate steadily about this axis with angular velocity $\Omega$.

Deduce the angular velocity at which the free surface first touches the bottom of the container.

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