The system I am considering consists of a conducting liquid sandwiched between two electrodes.
The electrodes supply a constant flux of electrons $J_{ext}$.
In the liquid there are also uncharged particles, call them $A$, that can capture the electrons and gain charge, lets call them $A^-$.
I am looking for the steady state concentration profile of both $A$ and $A^-$.
From the continuity equation for steady state
$$\nabla \cdot j=0$$
where we get the following set of differential equations in 1D
$$\frac{dJ_A}{dx}=-k C_A C_e$$
$$\frac{dJ_{A^-}}{dx}=k C_A C_e$$
$$\frac{dJ_{e}}{dx}=-k C_A C_e$$
where J's are the fluxes and C's the concentrations for the non-charged particles, charged particles and electrons. k is the rate constant for conversion between electrons and the charged particles.
From charge conservation it then follows that $J_{A^-}+J_e=J_{ext}$ so we can remove one of the differential equations by enforcing this constraint.
At the bottom electrode we have no flux of the charged and uncharged species, i.e $J_{A^-}=J_A=0$.
At the top we have the generation of uncharged particles as the charged ones lose their electrons to the electrode. Namely $J_{A^-}=-J_A$.
Thus we have two differential equations of second order (as $j\propto \nabla C$) and 3 boundary conditions, so I am missing one. The current BCs already conserve electric charge and particle number.
What other possible constraints are there on this system that would lead to a well-defined problem?
**Edit: I am asking for the physical concepts, e.g some conserved quantity, that gives me the correct boundary conditions that would allow me to turn this into a well posed differential equation problem. **
