content_image/362121.jpg content_image/362123.jpg content_image/362122.jpg content_image/362124.jpg content_image/362125.jpg content_image/362126.jpg content_image/362127.jpg content_image/362128.jpg content_image/362129.jpg
The viscous part of the pressure tensor is Newtonian: 

<img src='content_image/362121.jpg'>

The thermal and viscous parts of the First-Law energy change are then apportioned between the x and y directions so as to be consistent with overall energy conservation: 

<img src='content_image/362123.jpg'>

<img src='content_image/362122.jpg'>

<img src='content_image/362124.jpg'>

The thermal relaxation time τ has been introduced in the evolution equations to guarantee thermal equilibrium far from the shockwave: 

<img src='content_image/362125.jpg'>

In what follows we consider two models for the cold curve and the heat capacity. First, a weak repulsive pair force suggests implementing a “van der Waals model”: 

<img src='content_image/362126.jpg'>

Second, a triangular-lattice-based model, based on Gr¨ uneisen’s ideas, uses the nearest- neighbor static lattice energy and pressure corresponding to the pair potential evaluated at the nearest-neighbor lattice spacing r , φ = (10 /π )(1 − r ) $^{3}$: 

<img src='content_image/362127.jpg'>

<img src='content_image/362128.jpg'>

The corresponding equilibrium equation of state separates the energy and pressure into “cold” and “thermal” parts: 

<img src='content_image/362129.jpg'>

with γ chosen so as to roughly reproduce equation of state data from molecular dynamics. Let us next apply these two simple cold-curve models to the shockwave problem. 

