Artificial Viscosity

As we mentioned before, so long as the evolution of matter is completely adiabatic, the entropic variable $A_i$, found from the polytropic equation of state $P_i=A_i \rho_i^\Gamma$, remains constant. For most physical systems, however, this is a poor assumption. Several different techniques have been developed to deal with this situation, by adding an ``artificial viscosity'', or ``AV'', which damps the velocity of converging matter flows while heating up the material (thus increasing $A_i$). Our code includes three such prescriptions, all of which are described in great detail in Lombardi et al. (1999). We recommend giving this article a thorough perusal before using the AV schemes.

No matter which AV scheme is used, the effect on various quantities is determined from the laws of thermodynamics. Essentially, for an SPH artificial viscosity term $\Pi_{ij}$, if the viscous acceleration of the matter is given by

$$\vec{a}_{AV}=\sum_j m_j \Pi_{ij}\nabla W_{ij},$$ (15)

then the entropy equation will read

$$\frac{dA_i}{dt}=\frac{\Gamma-1}{2\rho_i^{\Gamma-1}}\sum_j m_j \Pi_{ij} (\vec{v}_i-\vec{v}_j)\cdot \nabla W_{ij}.$$ (16)