### Stress in fluids

The stress in a point in a

material is given by a stress

tensor; three components (σ

_{xx}, σ

_{yy} and σ

_{zz}) denote the stress along the three

normals (speaking in a

Cartesian coordinate system) and the remaining six components (σ

_{xy} = σ

_{yx}, σ

_{xz} = σ

_{zx} and σ

_{yz} = σ

_{zy}) 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 (σ

_{xx} = σ

_{yy} = σ

_{zz}; 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:

σ_{0} = 1/3 (σ_{xx} + σ_{yy} + σ_{zz})

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:

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* = - σ_{0} = - 1/3 (σ_{xx} + σ_{yy} + σ_{zz})

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

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*July 8, 2001*