The energy conservation principle can be formulated for a fluid element in the manner similar to the mass and momentum conservation as
where e(x, t ) is the internal energy per unit mass, q(x, t ) is the vector field of the heat flux by thermal conduction, and ˙Q is the rate of internal heat generation by the effects such as, for example, viscous friction or radiation. The conduction heat flux can be described by the Fourier conduction law
where T(x, t ) is the temperature field and κ is the coefficient of thermal conductivity.
The energy conservation equation can also be written in the enthalpy form,
where h = e + p/ρ is the specific enthalpy. Yet another possibility is the equation for the total (internal plus mechanical) energy E
The energy equation has more complex form if extra effects such as exo- and endothermal chemical reactions, radiation heat transfer, or Joule dissipation are explicitly shown in the right-hand side. In some cases, the equation can be brought into much simpler form. This is, in particular, true when the internal heat generation can be neglected and the fluid can be considered incompressible (the Boussinesq approximation). For an incompressible fluid or a solid, the specific internal energy is e = CT, where C = Cp = Cv is the specific heat. The energy equation becomes
substituting second equation of fourier law in it , then it becomes
which, in the case of a quiescent fluid or a solid and constant conduction coefficient κ, reduces to the classical heat conduction equation