Communications in Mathematical Sciences

Volume 16 (2018)

Number 5

Non-isothermal electrokinetics: energetic variational approach

Pages: 1451 – 1463

(Fast Communication)



Pei Liu (Department of Mathematics, Pennsylvania State University, University Park, Penn., U.S.A.)

Simo Wu (Department of Mathematics, Pennsylvania State University, University Park, Penn., U.S.A.)

Chun Liu (Department of Applied Mathematics, Illinois Institute of Technology, Chicago, Il., U.S.A.)


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.


electrokinetics, electro-thermal motion, energetic variation approach

2010 Mathematics Subject Classification

35Q35, 35Q79, 76A02, 80A20

The research is partially supported by NSF grants DMS-1714401, DMS-1412005.

Received 24 October 2017

Received revised 28 April 2018

Accepted 28 April 2018

Published 19 December 2018