Hernquist and Katz

Another form for the AV, introduced by Hernquist & Katz (1989) calculates $\Pi_{ij}$ directly from the SPH estimate of the divergence of the velocity field:

$$\Pi_{ij}=\cases{ {q_i\over\rho_{i}^2}+{q_j\over\rho_{j}^2}& ... ...if $(\vec{v}_i-\vec{v}_j)\cdot(\vec{r}_i-\vec{r}_j)\ge0$,\cr}$$ (20)

where

$$q_i=\cases{ \alpha \rho_i c_i h_i \vert\vec{\nabla}\cdot \ve... ...\cr 0& if $\left(\vec{\nabla}\cdot \vec{v}\right)_i\ge0$,\cr}$$ (21)

and

$$(\vec{\nabla}\cdot \vec{v})_i={1 \over \rho_i}\sum_j m_j (\vec{v}_j-\vec{v}_i)\cdot\vec{\nabla} W_{ij}.$$ (22)

We recommend, with the standard caveats, setting $\alpha\approx\beta\approx0.5$.