Fluid mechanics: Fluid statics

If the sum of the external forces acting on a fluid element is zero, the fluid will be at rest or moving as a solid body - in either case, the fluid element is in equilibrium. This section considers fluids in such an equilibrium state. For fluids in equilibrium, the only internal stresses acting will be normal forces because the shear stresses depend on velocity gradients, and all such gradients, by the definition of equilibrium, are zero. If one then carries out a balance between the body forces, assumed proportional to volume or mass - such as gravity - and the normal surface stresses acting on an elementary prismatic fluid volume, the resulting equilibrium equations, after shrinking the volume to zero, show that the normal stresses at a point are the same in all directions; because they are known to be negative, this common value, called the pressure, is denoted by -p.