Phenomenological approach to viscosity of a two-dimensional square crystal

Constitutive equations for the viscous stress tensor and tensorial molecular field for an incompressible two-dimensional square crystal are proposed, based on general symmetry considerations. The viscous stress derived involves 15 viscosity coefficients and contains terms which are linear as well as quadratic in the velocity gradient. The entropy production is expressed in terms of thermodynamic fluxes and forces. An equilibrium condition for the antisymmetric part of the viscous stress is used to derive a balance of angular momentum. Furthermore, it is found that viscous stresses in a plane Couette geometry are characterized by four effective viscosity coefficients. In the limit of slow flow, the first normal stress difference is predicted to be independent of the shear rate.