Categories: Fluid mechanics, Fluid statics, Physics.

# Newton’s bucket

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