Equations of fluid dynamics G Jacobian determinant I Rate of strain tensor, rotation tensor

Appendix H Properties of second order tensors

A second order tensor can be decomposed into a symmetric and an antisymmetric part in the following way²⁹²⁹How does this look like in general curvilinear coordinates?

\displaystyle T_{ij}=\underbrace{\frac{1}{2}{\left(T_{ij}+T_{ji}\right)}}_{% \text{symmetric}}+\underbrace{\frac{1}{2}{\left(T_{ij}-T_{ji}\right)}}_{\text{% antisymmetric}}

(H.1)

It can also be decomposed into an isotropic and deviatoric part by subtracting and adding the trace of the tensor like

\displaystyle T_{ij}=\underbrace{\frac{1}{n}\delta_{ij}T_{kk}}_{\text{% isotropic}}+\underbrace{T_{ij}-\frac{1}{n}\delta_{ij}T_{kk}}_{\text{deviatoric% , tracefree}}

(H.2)

Combining these two relations yields the general decomposition

\displaystyle\begin{split}\displaystyle T_{ij}=&\displaystyle\overbrace{\frac{% 1}{n}\delta_{ij}T_{kk}}^{\text{isotropic}}+\overbrace{\underbrace{\frac{1}{2}{% \left(T_{ij}+T_{ji}-\frac{2}{n}\delta_{ij}T_{kk}\right)}}_{\text{symmetric, % tracefree}}+\frac{1}{2}{\left(T_{ij}-T_{ji}\right)}}^{\text{deviatoric, % tracefree}}\\ &\displaystyle\underbrace{\hphantom{\frac{1}{n}\delta_{ij}T_{kk}+\frac{1}{2}{% \left(T_{ij}+T_{ji}-\frac{2}{n}\delta_{ij}T_{kk}\right)}}}_{\text{symmetric}}% \hphantom{+}\underbrace{\hphantom{\frac{1}{2}{\left(T_{ij}-T_{ji}\right)}}}_{% \text{ antisymmetric}}\end{split}

(H.3)

An interesting relation can be found when computing the contraction of a unsymmetric tensor $U_{ij}\neq U_{ji}$ with a symmetric tensor $V_{ij}=V_{ji}$

\displaystyle U_{ij}V_{ij}=\frac{1}{2}U_{ij}V_{ij}+\frac{1}{2}U_{ji}V_{ji}=% \frac{1}{2}U_{ij}V_{ij}+\frac{1}{2}U_{ji}V_{ij}=\frac{1}{2}{\left(U_{ij}+U_{ji% }\right)}V_{ij}

(H.4)

In analogy one finds for the contraction of an unsymmetric tensor $U_{ij}$ with an antisymmetric tensor $W_{ij}=-W_{ji}$

\displaystyle U_{ij}W_{ij}=\frac{1}{2}{\left(U_{ij}-U_{ji}\right)}W_{ij}

(H.5)