9.3.6 Summary of the frictional stress tensor

In summary, the viscous stress tensor is given by

$$\tau^{mn} = \rho \, \left( \begin{array}{ccc} e_{11} \, ... ...c^{44} & e_{33} \, (c^{33} - c^{13} ) \end{array} \right).$$ (9.46)

Motivated by Wajsowicz (1993) and Smagorinsky (1993), define the kinematic viscosity coefficients

$$\nu 3 \alpha = (c^{11} + c^{12})/2 - c^{13},$$ = (9.47)

$$\beta = (c^{11} - c^{12})/2$$ A = (9.48)

$$\kappa \gamma = c^{44}/2,$$ = (9.49)

where $\nu, A, \kappa$ is the notation used in Wajsowicz (1993), and $\alpha, \beta, \gamma$ is the notation used in Smagorinsky (1993). The stress tensor components now take the form

$$\tau^{mn} = \rho \, \left( \begin{array}{ccc} (A + \nu) ... ... \kappa \, e_{23} & 2 \, \nu \, e_{33} \end{array} \right)$$ (9.50)

which exposes a total of three viscous degrees of freedom.