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The convervation of energy amounts to (Equation (1.551) in [19]):
| \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:
| \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)
| \frac{\partial \rho }{\partial t} + \nabla \cdot (\rho \boldsymbol{v}) = 0, | (694) |
this also amounts to:
| \rho \dot{\varepsilon } = \frac{\partial \rho c_v T}{\partial t} + \nabla \cdot (\rho c_v T \boldsymbol{v}), | (695) |
leading to
| \frac{\partial \rho c_v T}{\partial t} + \nabla \cdot (\rho c_v T\boldsymbol{v}) = \nabla \cdot (\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:
| \frac{\partial \rho c_p T }{\partial t}+ \nabla \cdot (\rho c_p T \boldsymbol{v}) = \nabla \cdot (\lambda \nabla T) + d_{kl}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: