Non-isothermal electrokinetics: Energetic variational approach

Fluid dynamics accompanies with the entropy production, thus increases the local temperature, which plays an important role in charged systems, such as the ion channel in biological environment and electrodiffusion in capacitors/batteries. In this article, we propose a general framework to derive the transport equations with heat flow through the energetic variational approach. According to the first law of thermodynamics, the total energy is conserved and we can use the least action principle to derive the conservative forces. From the second law of thermodynamics, the entropy increases and the dissipative forces can be computed through the maximum dissipation principle. Combining these two laws, we then conclude with the force balance equations and a temperature equation. To emphasize, our method provides a self-consistent procedure to obtain the dynamical equations satisfying proper energy laws and it not only works for the charge systems but also for general systems.