### Stress in fluids

The stress in a point in a material is given by a stress tensor; three components (σxx, σyy and σzz) denote the stress along the three normals (speaking in a Cartesian coordinate system) and the remaining six components (σxy = σyx, σxz = σzx and σyz = σzy) denote the shear stress in the point.

In fluids that are at rest the components of the shear stress are always zero, which results in the components of the normal stress being equal to one another (σxx = σyy = σzz; the stress in the fluid is isotropic, this is Pascal's Law).

In fluids that are in motion the components of the shear stress are not zero and generally the stress in the fluid is also not isotropic, so it is not possible to speak of "the" normal stress in a point. However, it can be shown that the average of the three components of the normal stress taken in three perpendicular faces (in relation to each other) is independent of the orientation of these faces. This average equates to the isotropic part of the stress. The deviation of the components of the normal stress from this average is called the deviator stress. The shear stress belongs, in its entirety, to the deviator stress. The average of the components of the normal stress is given as follows:
σ0 = 1/3 (σxx + σyy + σzz)

In mechanics the usual way to indicate the difference between tensile and compressive stress in mathematical notations is to denote tensile stress (pulling) with a positive value and compressive stress with a negative value. So, if one comes across a line that reads:
σcable = 600 [N]
this means that the tension in the (imaginary) cable is equivalent to the force needed to keep about 60 [kg] from falling to the earth.

In fluids and gasses, however, tensile stress is a very rare occurrence, and therefore the definition pressure (p) is introduced, which is equivalent to the isotropic part of the compressive stress in a fluid or gas:
p = - σ0 = - 1/3 (σxx + σyy + σzz)
Variations in the isotropic part of the stress (tension or pressure) result in changes in the volume of the gas or fluid, while variations in the deviator stress result in changes in the shape of the gas or fluid (usually resulting in the fluid or gas being in motion).

Support write-up for Fluid mechanics

Sources: