Communicating Vessels

If we connect two containers of different shape holding the same liquid with a tube, the liquid arranges itself so that the free surface is at the same height in every container. The reason follows directly from Stevin’s law: at the base of the connecting tube the pressure must be the same on both sides (otherwise the liquid would flow), and since the surface pressure is atmospheric everywhere, the height of the free surfaces must also coincide.

Principle — Communicating vessels

Liquids of the same kind, in connected vessels open to the same atmosphere, settle at equilibrium with a horizontal free surface at the same height in every container, regardless of their shape.

Aqueducts

This is the principle behind ancient aqueducts: it was enough to build reservoirs and pipes with the right gradient, and the water found its own level by itself, with no need for pumps.

Two immiscible liquids

If we pour two liquids of different densities $\rho_1$ and $\rho_2$, immiscible with each other (such as water and oil, or mercury and water), into a U-tube, the heights of the two columns above the common interface arrange themselves so that the pressures at the interface height are equal. Setting $\rho_1\, g\, h_1 = \rho_2\, g\, h_2$ and cancelling $g$ gives the ratio between the heights:

Law — Ratio of the columns

$\ev{\frac{h_1}{h_2} = \frac{\rho_2}{\rho_1}}$

The column of the denser liquid is the shorter one: at equal pressure at the interface, less height of a heavy liquid is needed to balance a greater height of a light liquid.

Mercury and water in a U-tube

A U-tube is partially filled with mercury ($\rho_\text{Hg} = 13600\;\text{kg/m}^3$). In one arm we pour a column of $20$ cm of water ($\rho_w = 1000\;\text{kg/m}^3$). By how much does the mercury rise in the other arm relative to the bottom of the water?

Let $h_w = 20$ cm be the water column and $h_\text{Hg}$ the mercury level difference; at the height of the water-mercury interface it must hold that $\rho_w g h_w = \rho_\text{Hg} g h_\text{Hg}$, from which $h_\text{Hg} = (\rho_w/\rho_\text{Hg})\,h_w = (1000/13600)\cdot 20 \approx 1{,}47\;\text{cm}$.

The density of mercury is about $13{,}6$ times that of water, and indeed the level difference turns out to be about $13{,}6$ times smaller.

Links

Topics: Fluidostatica e fluidodinamica Concepts: Legge di Stevino Objects: Vasi comunicanti

Related exercises: Esercizio svolto — La colonna d’aria che schiaccia i nostri piedi · Problema — Derivazione di Stevino · Esercizio svolto — Pressa per la frutta