Artificial Viscosity

As usual in SPH 1 implementations, viscosity is rather an inter-particles pressure than a bulk pressure. It was shown that the use of 式 1 and 式 2 generates a substantial amount of entropy in regions of strong shear even if there is no compression.(1)
πij=qbci+cj2μij+qαμ2ij(ρi+ρj)2πij=qbci+cj2μij+qαμ2ij(ρi+ρj)2
with (2)
μij=dij(vivj)(XiXj)XiXj2+εd2ijμij=dij(vivj)(XiXj)XiXj2+εd2ij
Where, XiXi (resp. XjXj ) indicates the position of particle I (resp. jj ) and cici (resp cjcj ) is the sound speed at location ii (resp. jj ), and qaqa and qbqb are constants. This leads us to introduce 式 3 and 式 4. 2 The artificial viscosity is decreased in regions where vorticity is high with respect to velocity divergence.(3)
πij=qbci+cj2μij+qαμ2ij(ρi+ρj)2πij=qbci+cj2μij+qαμ2ij(ρi+ρj)2
with (4)
μij=dij(vivj)(XiXj)XiXj2+εd2ij(fi+fj)2,fk=v|kv|k+×v|k+εckdk

Default values for qa and qb are respectively set to 2 and 1.

1
Monaghan J.J., 「Smoothed Particle Hydrodynamics」, Annu.Rev.Astron.Astro-phys; Vol. 30; pp. 543-574, 1992.
2
Balsara D.S., 「Von Neumann Stability Analysis of Smoothed Particle Hydrodynamics Suggestions for Optimal Algorithms」, Journal of Computational Physics, Vol. 121, pp. 357-372, 1995.