Physical viscosity in smoothed particle hydrodynamics simulations of galaxy clusters

Most hydrodynamical simulations of galaxy cluster formation carried out to date have tried to model the cosmic gas as an ideal, inviscid fluid, where only a small amount of (unwanted) numerical viscosity is present, arising from practical limitations of the numerical method employed, and with a strength that depends on numerical resolution. However, the physical viscosity of the gas in hot galaxy clusters may in fact not be negligible, suggesting that a self-consistent treatment that accounts for the internal gas friction would be more appropriate. To allow such simulations using the smoothed particle hydrodynamics (SPH) method, we derive a novel SPH formulation of the Navier-Stokes and general heat transfer equations and implement them in the GADGET-2 code. We include both shear and bulk viscosity stress tensors, as well as saturation criteria that limit viscous stress transport where appropriate. Our scheme integrates consistently into the entropy and energy conserving formulation of SPH employed by the code. Using a number of simple hydrodynamical test problems, e.g. the flow of a viscous fluid through a pipe, we demonstrate the validity of our implementation. Adopting Braginskii parametrization for the shear viscosity of hot gaseous plasmas, we then study the influence of viscosity on the interplay between AGN-inflated bubbles and the surrounding intracluster medium (ICM). We find that certain bubble properties like morphology, maximum clustercentric radius reached or survival time depend quite sensitively on the assumed level of viscosity. Interestingly, the sound waves launched into the ICM by the bubble injection are damped by physical viscosity, establishing a non-local heating process. However, we find that the associated heating is rather weak due to the overall small energy content of the sound waves. Finally, we carry out cosmological simulations of galaxy cluster formation with a viscous ICM. We find that the presence of physical viscosity induces new modes of entropy generation, including a significant production of entropy in filamentary regions perpendicular to the direction of the clusters encounter. Viscosity also modifies the dynamics of mergers and the motion of substructures through the cluster atmosphere. Substructures are generally more efficiently stripped of their gas, leading to prominent long gaseous tails behind infalling massive haloes.