The balance of energy can be expressed as:
where is the mass density, is the internal energy per unit mass, is the Cauchy stress, is the particle velocity, is the heat flux vector, and s is the rate at which energy is generated by sources inside the volume (per unit mass).
Recall the general balance equation
In this case, the physical quantity to be conserved the total energy density which is the sum of the internal energy density and the kinetic energy density, i.e., . The energy source at the surface is a sum of the rate of work done by the applied tractions and the rate of heat leaving the volume (per unit area), i.e, where is the outward unit normal to the surface. The energy source inside the body is the sum of the rate of work done by the body forces and the rate of energy generated by internal sources, i.e., .
Hence we have
Let Ω be a control volume that does not change with time. Then we get
Using the relation , the identity , and invoking the symmetry of the stress tensor, we get
We now apply the divergence theorem to the surface integrals to get
Since Ω is arbitrary, we have
Expanding out the left hand side, we have
For the first term on the right hand side, we use the identity to get
For the second term on the right we use the identity and the symmetry of the Cauchy stress tensor to get
After collecting terms and rearranging, we get
Applying the balance of mass to the first term and the balance of linear momentum to the second term, and using the material time derivative of the internal energy
we get the final form of the balance of energy: