  In this article we address the study of ion charge transport in the
biological channels separating the intra and extracellular regions of a cell.
The focus of the investigation is devoted to including thermal driving forces
in the well-known velocity-extended Poisson-Nernst-Planck (vPNP)
electrodiffusion model. Two extensions of the vPNP system are proposed: the
velocity-extended Thermo-Hydrodynamic model (vTHD) and the velocity-extended
Electro-Thermal model (vET). Both formulations are based on the principles of
conservation of mass, momentum and energy, and collapse into the vPNP model
under thermodynamical equilibrium conditions. Upon introducing a suitable
one-dimensional geometrical representation of the channel, we discuss
appropriate boundary conditions that depend only on effectively accessible
measurable quantities. Then, we describe the novel models, the solution map
used to iteratively solve them, and the mixed-hybrid flux-conservative
stabilized finite element scheme used to discretize the linearized equations.
Finally, we successfully apply our computational algorithms to the simulation
of two different realistic biological channels: 1) the Gramicidin-A channel
considered in~\cite{JeromeBPJ}; and 2) the bipolar nanofluidic diode considered
in~\cite{Siwy7}.

Numerical Analysis