Viscous Heating At Stagnation In Z-Pinches

The viscous heating associated with m = 0 MHD instabilities in the stagnated Z-pinch is developed further. It would appear that the larger numerical (Neumann) viscosity plus De Bar corrections in simulation codes to yield energy conservation might be another way of representing viscous heating, but in this case the viscosity is inserted to smooth shock discontinuities. However the viscous heating per unit volume appears to be independent of the coefficient of viscosity itself because the fastest growing MHD mode is itself determined by the viscous damping. Therefore it could be argued that, though the correct physics is not in the codes, the resulting heating is not sensitive to the fact that numerical viscosity instead is employed. In addition, by chance, the model of magnetic bubbles first introduced by Lovberg et al. and Riley et al., and later by Rudakov et al. to explain phenomenologically extra heating of the ions leads to the same heating rate as in Haines et al. For the stainless steel array in which Ti was predicted and measured to be >200 KeV while Te = 3 KeV the ion viscous heating is dominant. However, for the low current experiment by Kroupp et al. in which the ion kinematic viscosity is much smaller than the resistive diffusivity there is resistive damping of MHD modes, and no ions viscous heating should be expected.