Monaghan

The simplest form of the AV function is that of Monaghan (1989), which combines a linear bulk velocity with a quadratic von Neumann-Richtmeyer viscosity. It takes the form

$$\Pi_{ij}=\frac{-\alpha\mu_{ij}c_{ij}+\beta\mu_{ij}^2}{\rho_{ij}},$$ (17)

where the speed of average sound speed is given by

$$c_{ij}\equiv\frac{c_i+c_j}{2};   c_i\equiv\left(\frac{\parti... ...t)^{\frac{1}{2}}=(\Gamma A_i \rho_i^{\Gamma-1})^{\frac{1}{2}},$$ (18)

the average density is $\rho_{ij}\equiv(\rho_i+\rho_j)/2$, and $\mu_{ij}$ is a function which measures the velocity convergence, given by

$$\mu_{ij}=\cases{ {(\vec{v}_i-\vec{v}_j)\cdot(\vec{r}_i-\vec{... ... if $(\vec{v}_i-\vec{v}_j)\cdot(\vec{r}_i-\vec{r}_j)\ge0$,\cr}$$ (19)

where $h_{ij}\equiv(h_i+h_j)/2$ is the average smoothing length. The parameters $\alpha$, $\beta$, and $\eta^2$ are set in the input files as ALPHA, BETA, and ETA2. We recommend for most purposes $\alpha\approx0.5$, $\beta\approx1$, and $\eta^2\approx 10^{-2}$, but see Lombardi et al. (1999) for more discussion.