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}

## References

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