Fundamentals of Continuum Mechanics

Appendix E

Energy Formulations and Stability Conditions for Newtonian Fluids

In this section, we review and augment the thermomechanical theory for compressible Newtonian fluids presented in Chapter 7 and discuss the corresponding stability conditions.

E.1 Governing equations

Recall that the fundamental laws for a continuum in the thermomechanical theory are

$\stackrel{.}{\rho }+\rho \text{div}\mathbf{v}=0,\rho \stackrel{.}{\mathbf{v}}=\text{div}\mathbf{T}+\rho \mathbf{b},\rho \stackrel{.}{\epsilon }=\mathbf{T}·\mathbf{D}+\rho r-\text{div}\mathbf{q},$

  (E.1)

which correspond to Eulerian statements of conservation of mass, linear momentum, and energy; refer to Section 5.3. In (E.1), ρ is the density, v is the velocity, T is the Cauchy stress, b is the body force per unit mass, ε is the internal ...