Investigation of steady-state solutions for electrochemical processes

The governing equations of mass transport process in a dilute electrolyte are: (UNFORMATTED TABLE OR EQUATION FOLLOWS)$$\eqalign{{\partial u\sb{i}\over\partial t} = {\partial\over\partial x}(d\sb{i}{\partial u\sb{i}\over\partial x}&+d\sb{i}e\sb{i}{\partial\phi\over\partial x}u\sb{i}),\quad i = 1,\...,m\cr&\sum\limits\sbsp{i=1}{m}e\sb{i}u\sb{i} = 0\cr}$$(TABLE/EQUATION ENDS)where $u\sb{i}$, $d\sb{i}$ and $e\sb{i}$ are the concentration, diffusion coefficient and charge of species i respectively, $\phi$ is the potential in the electrolyte. Both $u\sb{i}$ and $\phi$ are solutions yet to be determined.

We investigate the steady state solution to the above equations subject to some nonlinear boundary conditions for two electrochemical models. In the first model, a species is produced at one electrode and consumed at another. In the second model, species which are supplied from an external source are all consumed at the electrodes. The existence of the steady state solution is proved for each model. Finally, the numerical algorithm has been developed to find the numerical approximation to the steady state solution for each model.