Archimedes force is a particular case of mass or volume forces. By mass (volume) forces we mean forces acting upon each element of a volume or mass of a body. In mechanics of continua the mass is introduced proceeding from the volumetric density ρ(xi, t), determined for each time instant t and any space point xi The mass in a volume v at time t has the integral ρ(xi, t)dV.
Similarly, the mechanics of continua is concerned not with forces themselves, but with densities of forces, and their distributions in space and in time.
Thus, we may interpret the distribution density of volume forces ∫v (xi, t) at the given point xi of a medium as the ratio of the basic vector of forces applied to a small volume including point xi, to the mass of the volume, when the latter tends to zero.
As an example of volume forces, we may consider gravity force ρ dV, where ρ is the average value of density in volume dV, is the vector of the acceleration due to gravity. Other examples include centrifugal forces or electromagnetic forces on a fluid carrying an electrical charge.
The pressure gradient is the analog of the volume force (taken with a reverse sign) affecting the liquid element if the pressure itself varies from point to point. If we separate a certain volume in the fluid, then the force, acting upon this volume, is equal to the integral −∫pdA, where dA is the surface element, where the integral is taken over the surface surrounding this volume. Transforming the surface integral into a volume one, we find
The last integral is the volume force acting on the whole volume.
The fact that the equation of motion includes not the pressure itself, but only its gradient, shows that the pressure value in the liquid is determined only with respect to an arbitrary constant.
If the external volume force is = dV, where is force referred to unit mass, and FA is the force affecting volume V from the surrounding medium through the boundary surface A, then the sum of the forces acting upon the separated volume will be as follows
Force is assumed to be a known function of time and space. If field is related to a potential Φ which is independent of time, then Φ = −grad Φ.
In hydromechanics, in many practically interesting cases the main driving force is caused by the presence of a temperature or concentration field. Variation of temperature or concentration leads to a change in density; this leads to the development of a buoyant force, which is formed because of the presence of the volume force field. The buoyant force is also called the Archimedes or the hydrostatic lift force. At small temperature drops the dependence of the fluid density on temperature may be taken to be linear, i.e., ρ=ρ0[1 + β(T0 - T)], where ρ0 is the fluid density at temperature T0; β is the coefficient of cubic expansion. The product βg is called also the buoyancy parameter.
In the case of the gravitation field a hot substance moves under the action of the Archimedes force F = (ρ − ρ0)gV with reference to a cooler substance in the direction opposite to the gravity force direction. The convection intensity depends on the temperature difference between layers, thermal conductivity and on the medium viscosity.