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

Fluid Dynamics | Part IB, 2002

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

uu=(12u2)\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 aa with its axis vertical, the depth of the fluid is hh. 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 Ω\Omega.

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

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