9.3.5 Trace-free frictional stress

The frictional stress tensor under consideration here is a deviatoric stress tensor (e.g., Smagorinsky 1993, Salmon 1998), which is defined to have a zero trace

$$\delta_{ik} \, \tau^{ik} = \tau^{ii} = 0.$$ (9.41)

Consequently,

$$C^{iimn} \, e_{mn} (C^{1111} + C^{1122} + C^{3311}) \, (e_{11} + e_{22}) + C^{3333} \, e_{33}.$$ = (9.42)

Since MOM assumes an incompressible fluid, the trace of the strain or deformation tensor also vanishes

e_mm = u_m,m = 0. (9.43)

As such, a trace-free frictional stress tensor implies the following relation between the viscosity tensor elements

C³³³³ = C¹¹¹¹ + C¹¹²² - C³³¹¹, (9.44)

or

c³³ = c¹¹ + c¹² - c¹³. (9.45)