Using surface integrals for checking Archimedes' law of buoyancy

A mathematical derivation of the force exerted by an inhomogeneous (i.e. compressible) fluid on the surface of an arbitrarily shaped body immersed in it is not found in the literature, which may be attributed to our trust in 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 with the container walls, as is more evident in the case of a block immersed in a liquid and in contact with the bottom, in which a downward force that 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 with the force predicted by AP, as long as the fluid density is a 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 with experiments. It also yields a formula for this force which shows that it increases with the area of contact.