where formula_23 is a viscosity tensor that maps the strain rate tensor formula_24 onto the viscous stress tensor formula_25. since the indices in this expression can vary from 1 to 3, there are 81 "viscosity coefficients" formula_23 in total. however, due to spatial symmetries these coefficients are not all independent. for instance, for isotropic newtonian fluids, the 81 coefficients can be reduced to 2 independent parameters. the most usual decomposition yields the standard (scalar) viscosity formula_10 and the bulk viscosity formula_28: