Hydrostatic stress

In the particular case of an incompressible fluid, the thermodynamic pressure coincides with the mechanical pressure (i.e. the opposite of the hydrostatic stress): ${\displaystyle p=-\sigma _{h}=-{\frac {1}{3}}\operatorname {tr} ({\boldsymbol {\sigma }})}$

In the general case of a compressible fluid, the thermodynamic pressure ${\displaystyle p}$ is no more proportional to the isotropic stress term (the mechanical pressure), since there is an additional term dependent on the trace of the strain rate tensor:

${\displaystyle p=-{\frac {1}{3}}\operatorname {tr} ({\boldsymbol {\sigma }})+\zeta \operatorname {tr} ({\boldsymbol {\epsilon }})}$

where the coefficient ${\displaystyle \zeta }$ is the bulk viscosity. The trace of the strain rate tensor corresponds to the flow compression (the divergence of the flow velocity):

${\displaystyle \operatorname {tr} ({\boldsymbol {\epsilon }})=\operatorname {tr} \left({\frac {1}{2}}(\nabla \mathbf {u} +(\nabla \mathbf {u} )^{T})\right)=\nabla \cdot \mathbf {u} }$

So the expression for the thermodynamic pressure is usually expressed as:

${\displaystyle p=-\sigma _{h}+\zeta \nabla \cdot \mathbf {u} ={\bar {p}}+\zeta \nabla \cdot \mathbf {u} }$

where the mechanical pressure has been denoted with ${\textstyle {\bar {p}}}$. In some cases, the second viscosity ${\textstyle \zeta }$ can be assumed to be constant in which case, the effect of the volume viscosity ${\textstyle \zeta }$ is that the mechanical pressure is not equivalent to the thermodynamic pressure^[3] as stated above. $${\displaystyle {\bar {p}}\equiv p-\zeta \,\nabla \cdot \mathbf {u} ,}$$ However, this difference is usually neglected most of the time (that is whenever we are not dealing with processes such as sound absorption and attenuation of shock waves,^[4] where second viscosity coefficient becomes important) by explicitly assuming ${\textstyle \zeta =0}$. The assumption of setting ${\textstyle \zeta =0}$ is called as the Stokes hypothesis.^[5] The validity of Stokes hypothesis can be demonstrated for monoatomic gas both experimentally and from the kinetic theory;^[6] for other gases and liquids, Stokes hypothesis is generally incorrect.

Its magnitude in a fluid, ${\displaystyle \sigma _{h}}$, can be given by Stevin's Law:

${\displaystyle \sigma _{h}=\displaystyle \sum _{i=1}^{n}\rho _{i}gh_{i}}$

where

• i is an index denoting each distinct layer of material above the point of interest;
• ${\displaystyle \rho _{i}}$ is the density of each layer;
• ${\displaystyle g}$ is the gravitational acceleration (assumed constant here; this can be substituted with any acceleration that is important in defining weight);
• ${\displaystyle h_{i}}$ is the height (or thickness) of each given layer of material.

For example, the magnitude of the hydrostatic stress felt at a point under ten meters of fresh water would be

${\displaystyle \sigma _{h}=\rho _{w}gh_{w}=1000\,{\text{kg m}}^{-3}\cdot 9.8\,{\text{m s}}^{-2}\cdot 10\,{\text{m}}=9.8\cdot {10^{4}}{\text{ kg m}}^{-1}{\text{s}}^{-2}=9.8\cdot 10^{4}{\text{ N m}}^{-2}}$

where the index w indicates "water".

Because the hydrostatic stress is isotropic, it acts equally in all directions. In tensor form, the hydrostatic stress is equal to

${\displaystyle \sigma _{h}\cdot I_{3}=\sigma _{h}\left[{\begin{array}{ccc}1&0&0\\0&1&0\\0&0&1\end{array}}\right]=\left[{\begin{array}{ccc}\sigma _{h}&0&0\\0&\sigma _{h}&0\\0&0&\sigma _{h}\end{array}}\right]}$

where ${\displaystyle I_{3}}$ is the 3-by-3 identity matrix.

Hydrostatic compressive stress is used for the determination of the bulk modulus for materials.