Fundamentals of Continuum Mechanics

4.8 Local forms of the conservation laws

We now obtain the local, or pointwise, forms of the Eulerian representations of the thermomechanical integral conservation laws developed in Sections 4.3 and 4.4:

$\frac{\text{d}}{\text{d}t}{\displaystyle \underset{\mathcal{P}}{\int }\rho \text{d}v}=0,$

  (4.35a)

$\frac{\text{d}}{\text{d}t}{\displaystyle \underset{\mathcal{P}}{\int }\mathbf{v}\rho \text{d}v}={\displaystyle \underset{\mathcal{P}}{\int }\mathbf{b}\rho \text{d}v}+{\displaystyle \underset{\partial \mathcal{P}}{\int }\mathbf{Tn}\text{d}a},$

  (4.35b)

$\frac{\text{d}}{\text{d}t}{\displaystyle \underset{P}{\int }\mathbf{x}\times \mathbf{v}\rho \text{d}v}={\displaystyle \underset{P}{\int }\mathbf{x}\times \mathbf{b}\rho \text{d}v}+{\displaystyle \underset{\partial P}{\int }\mathbf{x}\times \mathbf{Tn}\text{d}a},$

  (4.35c)

$\frac{\text{d}}{\text{d}t}{\displaystyle \underset{\mathcal{P}}{\int }\left(\frac{1}{2}\mathbf{v}·\mathbf{v}+\epsilon \right)\rho \text{d}v}={\displaystyle \underset{\mathcal{P}}{\int }\left(\mathbf{b}·\mathbf{v}+r\right)\rho \text{d}v}+{\displaystyle \underset{\partial \mathcal{P}}{\int }\left(\mathbf{Tn}·\mathbf{v}-\mathbf{q}·\mathbf{n}\right)\text{d}a},$

  (4.35d)

where we have used t = Tn and h = q · n (results (4.26) and (4.31)).

We have made suitable ...