Categories: Fluid mechanics, Fluid statics, Physics.

Newton’s bucket

Newton’s bucket is a cylindrical bucket that rotates at angular velocity ω\omega. Due to viscosity, any liquid in the bucket is affected by the rotation, eventually achieving the exact same ω\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 F⃗f=mf⃗\va{F}_\mathrm{f} = m \va{f} on a molecule with mass mm:

f⃗=ω2r⃗\begin{aligned} \va{f} = \omega^2 \va{r} \end{aligned}

Where r⃗\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 Φf\Phi_\mathrm{f}, such that f⃗=−∇Φf\va{f} = - \nabla \Phi_\mathrm{f}:

Φf=−ω22r2=−ω22(x2+y2)\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 F⃗g=mg⃗\va{F}_\mathrm{g} = m \va{g}, where g⃗=−∇Φg\va{g} = - \nabla \Phi_\mathrm{g}:

Φg=gz\begin{aligned} \Phi_\mathrm{g} = \mathrm{g} z \end{aligned}

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

Φ=Φg+Φf=gz−ω22(x2+y2)\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 pp in the liquid is the one that satisfies:

∇pρ=−∇Φ\begin{aligned} \frac{\nabla p}{\rho} = - \nabla \Phi \end{aligned}

Removing the gradients gives integration constants p0p_0 and Φ0\Phi_0, so the equilibrium equation is:

p−p0=−ρ(Φ−Φ0)\begin{aligned} p - p_0 = - \rho (\Phi - \Phi_0) \end{aligned}

We isolate this for pp and rewrite Φ0=gz0\Phi_0 = \mathrm{g} z_0, where z0z_0 is the liquid height at the center:

p=p0−ρg(z−z0)+ω22ρ(x2+y2)\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=p0p = p_0, where p0p_0 is the air pressure. The zz-coordinate at which this is satisfied is as follows, telling us that the surface is parabolic:

z=z0+ω22g(x2+y2)\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.