Collective molecular dissipation on Navier–Stokes macroscopic scales: Accretion disc viscous modeling in SPH

Abstract

In the nonlinear Navier–Stokes viscous flow dynamics, physical damping is mathematically accomplished by a braking term in the momentum equation, corresponding to a heating term in the energy equation, both responsible of the conversion of mechanical energy into heat. In such two terms, it is essential the role of the viscous stress tensor, relative to contiguous macroscopic moving flow components, depending on the macroscopic viscosity coefficient ν. A working formulation for ν can always be found analytically, tuning some arbitrary parameters in the current known formulations, according to the geometry, morphology and physics of the flow. Instead, in this paper, we write an alternative hybrid formulation for ν, where molecular parameters are also included. Our expression for ν has a more physical interpretation of the internal damping in dilute gases because the macroscopic viscosity is related to the small scale molecular dissipation, not strictly dependent on the flow morphology, as well as it is free of any arbitrary parameter. Results for some basic 2D tests are shown in the smoothed particle hydrodynamics (SPH) framework. An application to the 3D accretion disc modeling for low mass cataclysmic variables is also discussed. Consequences of the macroscopic viscosity coefficient reformulation in a more strictly physical terms on the thermal conductivity coefficient for dilute gases are also discussed.