Categories: Fluid mechanics, Fluid statics, Physics.

**Newton’s bucket** is a cylindrical bucket that rotates at angular velocity \(\omega\). Due to viscosity, any liquid in the bucket is affected by the rotation, eventually achieving the exact same \(\omega\).

However, once in equilibrium, the liquid’s surface is not flat, but curved upwards from the center. This is due to the centrifugal force \(\va{F}_\mathrm{f} = m \va{f}\) on a molecule with mass \(m\):

\[\begin{aligned} \va{f} = \omega^2 \va{r} \end{aligned}\]

Where \(\va{r}\) is the molecule’s position relative to the axis of rotation. This (fictitious) force can be written as the gradient of a potential \(\Phi_\mathrm{f}\), such that \(\va{f} = - \nabla \Phi_\mathrm{f}\):

\[\begin{aligned} \Phi_\mathrm{f} = - \frac{\omega^2}{2} r^2 = - \frac{\omega^2}{2} (x^2 + y^2) \end{aligned}\]

In addition, each molecule feels a gravitational force \(\va{F}_\mathrm{g} = m \va{g}\), where \(\va{g} = - \nabla \Phi_\mathrm{g}\):

\[\begin{aligned} \Phi_\mathrm{g} = \mathrm{g} z \end{aligned}\]

Overall, the molecule therefore feels an “effective” force with a potential \(\Phi\) given by:

\[\begin{aligned} \Phi = \Phi_\mathrm{g} + \Phi_\mathrm{f} = \mathrm{g} z - \frac{\omega^2}{2} (x^2 + y^2) \end{aligned}\]

At equilibrium, the hydrostatic pressure \(p\) in the liquid is the one that satisfies:

\[\begin{aligned} \frac{\nabla p}{\rho} = - \nabla \Phi \end{aligned}\]

Removing the gradients gives integration constants \(p_0\) and \(\Phi_0\), so the equilibrium equation is:

\[\begin{aligned} p - p_0 = - \rho (\Phi - \Phi_0) \end{aligned}\]

We isolate this for \(p\) and rewrite \(\Phi_0 = \mathrm{g} z_0\), where \(z_0\) is the liquid height at the center:

\[\begin{aligned} p = p_0 - \rho \mathrm{g} (z - z_0) + \frac{\omega^2}{2} \rho (x^2 + y^2) \end{aligned}\]

At the surface, we demand that \(p = p_0\), where \(p_0\) is the air pressure. The \(z\)-coordinate at which this is satisfied is as follows, telling us that the surface is parabolic:

\[\begin{aligned} z = z_0 + \frac{\omega^2}{2 \mathrm{g}} (x^2 + y^2) \end{aligned}\]

- B. Lautrup,
*Physics of continuous matter: exotic and everyday phenomena in the macroscopic world*, 2nd edition, CRC Press.

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