The analysis of the temperature distribution in a body is based on the first law of thermodynamics (the “energy balance”) and Fourier’s equation of heat conduction (the “rate equation”). The energy balance, applied to an incremental volume of an incompressible medium at rest, may be written

\[\label{eq1} \rho c_p \frac{\partial T}{\partial t}=-(\nabla\cdot\dot{q})\tag{1}\]

Equation 1 states that the internal energy, and so the temperature of the volume, will rise if the divergence of the heat flux is less than zero, that is, if the heat fluxes into the volume exceed the outward-directed ones.

Additional contributions to the internal energy — for example, by conversion of mechanical (“viscous dissipation”), electrical, chemical, or nuclear energy — may be accounted for by adding the appropriate “source” term to the right side of Equation 1.

The heat flux vector can be expressed in terms of the temperature gradient by Fourier’s equation: