Viscosity

The viscous contribution to the total stress is

$$2 \nu D_{ij}$$

where

$$D_{ij} = 0.5 ( \frac{\partial v_i}{\partial x_j} + \frac{\partial v_j}{\partial x_i} )$$

and divergence is neglected since we only model slightly compressible flows.

Viscous heat generation

The viscous energy loss is turned into heat rate per unit volume $q$:

$$q = 2 \nu D_{ij} D_{ij}$$

See group_materi_viscosity_heatgeneration.