Entropy production and the temperature equation in electrohydrodynamics

The equation for the temperature in moving dielectric fluids is obtained in most texts and papers dealing with electrothermohydrodynamics by simply adding the Joule heating, and more exceptionally by adding also the dielectric heating. Here we derive this equation rigorously from first principles. First, we derive the entropy production, and then we show that additional terms depending on the electric field are present in the equation for the temperature. Among them we have terms of thermoelectric origin, and terms depending on the variation of permittivity with temperature, in addition to Joule and dielectric heating. The different terms in this equation are discussed in detail, and are given a clear physical interpretation.