In the above equations ρ ρ ( x , t ) is the mass density (current), ρ ρ ˙ ˙ is the material time derivative of ρ ρ , v ( x , t ) is the particle velocity, v ˙ ˙ is the material time derivative of v , σ σ ( x , t ) is the Cauchy stress tensor, b ( x , t ) is the body force density, e ( x , t ) is the internal energy per unit mass, e ˙ ˙ is the material time derivative of e , q ( x , t ) is the heat flux vector, and s ( x , t ) is an energy source per unit mass. The operators used are defined below.