# Boundary conditions for 3d current flow through water

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1. Sep 26, 2015

### stevecarson

I've forgotten a lot of field theory so I've been rereading it in a couple of electric field theory textbooks. What seems like a simple problem falls between the cracks. I hope some readers can help - it will be appreciated.

My application seems simple (solution will require numerical FEA but that is fine), but the boundary conditions to the problem are not clear to me. Find current flow through a 3d body of water.

Problem - tank of water of known conductivity (e.g. tap water, σ = 0.03 S/m). Tank dimensions are known (e.g. 2m x 0.8m diameter with insulating boundary and air above the water). Two electrodes of known dimensions and locations (e.g. each is 0.1m high x 0.01m wide x 0.001m thick, placed 0.5m apart across the center of the tank). But I want to play around with these and they won't always be identical and located symmetrically.

Find - potential (voltage) field, current (vector) field, and main objective, the current flow between the electrodes.

Potential field, V(x,y,z) should be easy. But it's not. The potential field, V, is always given as ∑ Qi/ri.4πε. I know the voltage at my two electrodes - because I have applied it from a constant voltage source like a battery. Let's say 5v. How does that relate to the charge distribution needed to calculate the electric field, E and potential E=∇V ??

Current field, J(x,y,z) = σE = σ∇V.
This will be easy once V(x,y,z) is calculated, but is water a conductive or convective medium for current? I think conductive (like current flow in metals rather than electron flow through a vacuum), but it's not clear from any text I have read.

The real equation should probably be the Poisson equation, ∇2V=ρ/ε, which follows from ∇. J=ρ and J = σ∇V, but again the boundary conditions aren't clear for reasons already stated - and do I need to iterate to find the charge distribution from the current flow and insert charge distribution back in the equation?
(That is, first iteration, only the charge at the terminals exists, but second iteration, some charge is in the water flowing between the terminals and affecting the electric field, so keep iterating until result converges).

It should all be clear, but it's not. I would really appreciate insights from any experts.

Thanks
Steve

2. Sep 26, 2015

### Simon Bridge

The potential, electric firld, and charge distribution, are related via Maxwells equations.
Water is both conductive and convective... due to the presence of ions in solution: it is not a simple homogeneous substance.
Your boundary conditions will depend on what the tank is made out of.
Good luck.