The temperature equation in electrohydrodynamics

The equation for the temperature in moving dielectric fluids is obtained in most texts and papers dealing with electrothermohydrodynamics simply adding Joule heating, and more exceptionally adding also dielectric heating. Here we derive this equation rigorously from first principles. We show that additional terms depending on the electric field are present. 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 interpreted physically, and discussed in detail.