## 4. Modified bulk viscosityA smoothed quantity can be calculated at any point in the fluid by Eq. (1). The smoothed velocity at is If this point coincides with a position for a particle, the
smoothed and the particle's individual velocity will be different in
general. This difference is used to construct a modified bulk
viscosity. Benz (1990) suggests that this quantity could be used when
integrating the position to prevent interparticle penetration. It is
true that it prevents interparticle penetration, but it unfortunately
introduces conservation problems when the particles are not moved at
their individal velocity. If the contribution from particle where . The denominator scales the expression
so that , if for all In the standard formulation of the bulk artificial viscosity
particle where the number of neighbours. The constant
around unity, and used in the same way as the
constants and in Eq.
(6). This interaction can be seen as if the particle interacts with a
virtual particle with the smoothed velocity ,
have a mass of and lies at a distance of
Now consider the integration of the velocitites and internal energy
density from time to
with the time step . From Eq. (8) the velocity
for particle which gives the change in kinetic energy for particle If is the change in internal energy for the particle it is possible to conserve the energy, that is require that . The time derivative of the internal energy for the particle is then consequently defined as to conserve the total energy. The artificial also must prevent particle penetration. The modified
artificial viscosity is calculated with respect to an integrated mean
of the neighbours. Therefore a particle will move less than
from the definition of the time step, Eq. (7),
regardless of the neighbours' individual sound speed and individual
velocities. This should be compared with the distance to the closest
which are approximately one If the particles © European Southern Observatory (ESO) 1997 Online publication: April 28, 1998 |