Conservation of Energy (compressible flow)

The convervation of energy amounts to (Equation (1.551) in [19]):

$\displaystyle \rho \dot{\varepsilon} = d_{kl} t_{kl} - p \nabla \cdot \boldsymbol{v} - \nabla \cdot (- \lambda \nabla T) + \rho h,$ (692)

where $ \boldsymbol{d}$ is the deformation rate tensor, $ \lambda$ is the heat conduction coefficient and $ h$ is the heat production per unit of volume. The total time derivative of the energy density $ \varepsilon$ can also be written as:

$\displaystyle \rho \dot{\varepsilon } = \rho \frac{D c_v T}{D t} = \rho \left [ \frac{\partial c_v T}{\partial t} + \boldsymbol{v} \cdot \nabla (c_v T) \right ].$ (693)

$ c_v$ is the specific heat at constant volume. Since (conservation of mass)

$\displaystyle \frac{\partial \rho }{\partial t} + \nabla \cdot (\rho \boldsymbol{v}) = 0,$ (694)

this also amounts to:

$\displaystyle \rho \dot{\varepsilon } = \frac{\partial \rho c_v T}{\partial t} + \nabla \cdot (\rho c_v T \boldsymbol{v}),$ (695)

leading to

$\displaystyle \frac{\partial \rho c_v T}{\partial t} + \nabla \cdot (\rho c_v T...
...dot (\lambda \nabla T) + d_{kl}t_{kl} - p \nabla \cdot \boldsymbol{v} + \rho h.$ (696)

In some other books one may find the completely equivalent expression:

$\displaystyle \frac{\partial \rho c_p T }{\partial t}+ \nabla \cdot (\rho c_p T...
..._{kl}+ \boldsymbol{v} \cdot \nabla p + \frac{\partial p}{\partial t} + \rho h ,$ (697)

where $ c_p$ is the specific heat at constant pressure. Integrating across a volume V one obtains:

$\displaystyle \frac{\partial }{\partial t} \int_{V}^{} \rho c_v T dv + \int_{S}^{} \rho c_v T \boldsymbol{v} \cdot \boldsymbol{n} ds$ $\displaystyle = \int_{A}^{} \lambda \frac{\partial T}{\partial n} da + \int_{V}^{} d_{kl}t_{kl} dv$    
  $\displaystyle - \int_{V}^{} p \nabla \cdot \boldsymbol{v} dv + \int_{V}^{} \rho h dv.$ (698)

The following terms can be distinguished:



Subsections