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:
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:
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
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July 8, 2001