# The relationship between height of water and the rotational speed inside a cup

When I stir the water inside a cup with a speed $v$ ( using a stick), I find that a vortex will be formed inside the cup, with the middle part of the water at the lowest point ( with the height $h_0$), and the water level gradually rises until it reaches the wall of the cup, with the height ($h_1$).

I also observe that the faster I stir the water, the steeper the difference ($h_1$-$h_0$) is.

What is the relationship between the height of the water and the rotational speed? Anyway to derive it?

• Can you derive the equation for the steady state height profile if the entire cup is rotating about its axis? – Chet Miller May 31 '16 at 4:24
• @ChesterMiller , not too sure. Can you elaborate in the answer? – Graviton May 31 '16 at 4:54
• See Floris' answer. You have centripetal acceleration of the fluid that requires a radial pressure gradient to sustain. The radial pressure gradient is provided hydrostatically by the radial profile of depth. – Chet Miller May 31 '16 at 10:27

$$y(r) = \frac{\omega^2r^2}{2g}$$