From a stress tensor, it is possible to determine the stress that is actually applied on a small surface element of matter. Consider a fluid domain and a small surface with normal unit direction given by . The force density applied on this surface can be obtained from

(30–15)

where is the stress tensor. In general, will not be oriented along , but it is possible to select a surface with an orientation such that and have the same direction. In this case,

(30–16)

where is the proportionality factor. This equation can be rewritten as

(30–17)

where is the unit tensor. This is the characteristic equation for , and the solutions for are said to be the eigenvalues.

Ansys Polyflow solves Equation 30–17 by setting the determinant of the first term on the left-hand side to zero:

(30–18)

In general, this leads to a third-order polynomial expression, the roots of which are the eigenvalues. There are three eigenvalues, but for most 2D planar flows, the third one is equal to zero. These quantities are independent with respect to the selected reference frame, and may provide a more convenient description of the stress state in the fluid than the display of individual components of the stress tensor.