Cosmic ray propagation with cosmic ray viscosity in hydrodynamic description

Cosmic rays propagate in magnetized plasma through interaction with hydromagnetic waves. A rarely discussed interaction, called cosmic ray viscosity, is involved if the plasma is in shear flow. We derive the hydrodynamic version of cosmic ray propagation equation by suitable momentum space average of the propagation equation with cosmic ray viscosity. We put forward a corresponding self-consistent MHD model for the system. We present possible steady state solutions.