Hydrostatic equilibrium represents that state of a fluid for which all of it sparts are macroscopically at rest relative to an externally applied, invariant field of force. In such a state, the absolute rate of deformation is zero, and its velocity relative to the field of force is zero. The sea at rest in the presence of the field of gravity would represent a state of geohydrostatic equilibrium.
In a fluid which is in a state of hydrostatic equilibrium the pressure is isotropic, that is, the normal stress acting on any prescribed surface within the fluid is independent of the orientation of that surface for a given point in the fluid. Another way of saying the same thing is that the hydrostatic pressure is a scalar quantity.
This property of isotropy of the hydrostatic pressure can be proven from the basin definition of a fluid. Proof: Consider a small tetrahedron of fluid of arbitrary dimensions which is contained within a fluid region (Figure 3.02-1).
The forces to which the tetrahedron of fluid is subjected under hydrostatic conditions are the surface forces and pA, and the volume force components where p is pressure, is force per unit volume, A is area, and V is volume. No shearing stresses can exist at the surfaces or A if the fluid is in a state of hydrostatic equilibrium. The balance of forces under static conditions is therefore
where the angles are connected by the relation
and, from geometrical considerations,
where d is the length of the line which passes through the point 0 and is normal to the surface ABC. Furthermore, the surface areas of the right tetrahedron are connected by the relation
It follows from relations (2) and (3) that
but since , and it follows that
using relation (4). Therefore
The volume V is , consequently Eqs. (9) can be expressed as
If we now take the limit as approach zero, considering that are finite, then we find that
at the point 0. That is, the hydrostatic pressure at a given point is independent of direction, and therefore isotropic.