In fluid mechanics it is assumed, that the normal components of the stress tensor are all the same and identical to the pressure p:
$\sigma_{xx}= \sigma_{yy}=\sigma_{zz} = p$
Where does this come from? In solid materials it is not the case in general, but why in fluids? In all my texts on fluid mechanics this assumption is not justified. What particular property of the state "fluid" is responsible for the assertion
$\sigma_{xy}= p \delta_{i,j}+\tau_{i,j}$
where $\tau$ is the viscous stress tensor.
Best Answer
The formula $\sigma_{xx}=\sigma_{yy}=\sigma_{zz}=-p$ is true only for a stationary fluid. For a fluid in motion the three normal stresses will be different from each other, and then pressure is defined to be their average: $-p\equiv(\sigma_{xx}+\sigma_{yy}+\sigma_{zz})/3$.
For a stationary fluid, $\sigma_{xx}=\sigma_{yy}=\sigma_{zz}=-p$ is a consequence of the fact that a fluid cannot remain at rest when acted upon by shear stress, no matter how small its magnitude (in fact this is taken to be the definition of simple fluids like air and water; solids on the other hand deform to finite extent and come to rest). This definition implies that in a stationary fluid there can be no shear stresses; every area element within the fluid must experience a purely normal stress. Then it is simple enough to prove that this normal stress must be independent of the orientation of the area element at a given point.