Using surface integrals for checking the Archimedes' law of buoyancy

TL;DR

This paper uses surface integrals and the divergence theorem to analyze the forces on bodies in fluids, confirming the validity of Archimedes' principle for fully immersed objects and explaining the observed downward force when in contact with container bottoms.

Contribution

It provides a mathematical derivation of buoyant forces for inhomogeneous fluids and arbitrarily shaped bodies, extending the understanding of forces in contact scenarios beyond classical Archimedes' law.

Findings

Archimedes' principle holds for fully immersed bodies with continuous density functions.

The approach explains the downward force observed when bodies contact container bottoms.

The derived formula shows the contact force increases with contact area.

Abstract

A mathematical derivation of the force exerted by an \emph{inhomogeneous} (i.e., compressible) fluid on the surface of an \emph{arbitrarily-shaped} body immersed in it is not found in literature, which may be attributed to our trust on Archimedes' law of buoyancy. However, this law, also known as Archimedes' principle (AP), does not yield the force observed when the body is in contact to the container walls, as is more evident in the case of a block immersed in a liquid and in contact to the bottom, in which a \emph{downward} force that \emph{increases with depth} is observed. In this work, by taking into account the surface integral of the pressure force exerted by a fluid over the surface of a body, the general validity of AP is checked. For a body fully surrounded by a fluid, homogeneous or not, a gradient version of the divergence theorem applies, yielding a volume integral that simplifies to an upward force which agrees to the force predicted by AP, as long as the fluid density is a \emph{continuous function of depth}. For the bottom case, this approach yields a downward force that increases with depth, which contrasts to AP but is in agreement to experiments. It also yields a formula for this force which shows that it increases with the area of contact.