MUSCL scheme

In continuum mechanics, a hydrostatic stress is an isotropic stress that is given by the weight of water above a certain point. It is often used interchangeably with "pressure" and is also known as confining stress, particularly in the field geomechanics. Its magnitude ${\sigma }_h$ can be given by:

${\sigma }_h=\sum _{i=1}^n{\rho }_{i}g{h}_i$

where $i$ is an index denoting each distinct layer of material above the point of interest, ${\rho }_i$ is the density of each layer, $g$ is the gravitational acceleration (assumed constant here; this can be substituted with any acceleration that is important in defining weight), and $h_i$ 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

${\sigma }_{h,sand}={\rho }_{w}g{h}_w=1000{\text{kg/m}}^3\cdot 9.8{\text{m/s}}^2\cdot 10\text{m}=9.8\cdot 10^{4}kg/m{s}^2=9.8\cdot 10^{4}N/m^2$

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

${\sigma }_h\cdot I_3=\left[\begin{array}{ccc}{\sigma }_h& 0& 0\\ 0& {\sigma }_h& 0\\ 0& 0& {\sigma }_h\end{array}\right]$

where $I_3$ is the 3-by-3 identity matrix.