Theoretical considerations on the constitutive properties of the SPH bulk viscosity

The constitutive properties of the SPH bulk viscosity are discussed in this paper. To this aim, a continuum macroscopic fluid domain is initially modeled as a Hamiltonian system of discrete particles, for which the inter-particle dissipative forces are required to be radial in order to conserve the angular momentum. The resulting system of particles is then reconverted to the continuum domain via the framework of the SPH model. Since a SPH consistent approximation of the Newtonian viscous term in the momentum equation incorporates both inter-particle radial as well as non-radial terms, it is postulated that the latter must be null. In the present work it is shown that this constraint implies that first and second viscosities are equal, resulting in a positive value for the bulk viscosity, in contradiction with the Stokes' hypothesis, i.e., the zeroing of the bulk viscosity. Moreover, it is found that this postulate leads to bulk viscosity coefficients close to values found in the experimental literature for monoatomic gases and common liquids such as water.