Rotating fluid under gravity, fluid dynamics question

Question

An incompressible inviscid fluid is rotating under gravity g with constant angular velocity $\Omega$ about the z-axis, which is vertical, so that $u = (−\Omega y, \Omega x, 0)$ relative to fixed Cartesian axes. We wish to find the surfaces of constant pressure, and hence surface of a uniformly rotating bucket of water (which will be at atmospheric pressure). Bernoulli's equation suggests that $$p/\rho+|u|^2/2+gz=\text{constant. So,}$$

$$z=\text{constant}-\frac{\Omega^2}{2g}(x^2+y^2)$$

But this suggests that the surface of a rotating bucket of water is at its highest in the middle, where is this going wrong?

Many thanks

Answer 1

In the Physics Forums you should find the answer to your exact problem 'Fluid dynamics - finding pressure for a rotating fluid'. Another answer is here (the point is that Bernoulli's law is applicable only along a streamline so that we must use a rotating frame and add the centripetal acceleration).

From another point of view see Newton's 'Bucket argument'.