[holgerbre]

Hi,

I try to simulate a particle-laden flow through a duct, in streamwise direction I choose periodic boundary conditions. The particles carry an electric charge (all are positive!), thus, an electric field arises. For the boundary conditions I choose zero electric potential at the pipe wall since I consider the pipe material to be conductive and grounded.

However, I am not sure how to treat the electric potential in streamwise direction. All my particles carry a positive charge, so due to periodicity the potential will blow up.

Any hint is appreciated.

[piu58]

A periodic boundary condition requires, the the conditions at the one and are equal to the ones of the other. You cannot use periodic b.c. for your case.

If you could establish them your case would diverge, because the articles would be accelerated to infinity.

[FMDenaro]

I am not expert in the electric field but I suspect you need to subtract some potential value when the charged particle leaves the outlet and enters the inlet by periodicity.

[holgerbre]

Thank you for both our answers,

Filippo: I am not sure what you mean. The problem would be the same if the particles are immobile, i.e. no particle leaves the domain. If one would compute the potential it becomes infinity in periodic direction because there is no negative charge present that equals it out.

Uwe: I believe there should be a possibility. I quote from the paper "A Fast Algorithm
for Particle Simulations" by L. GREENCARD AND V. ROKHLIN which deals with a very similar problem in section 4.1. Periodic Boundary Conditions:

"Remark. In certain problems (e.g., cosmology), the computational box obviously cannot satisfy the condition of no net charge (mass). This condition is necessary for the potential to be well defined, since the logarithmic term becomes unbounded as z->inf. Force calculations, however, may still be carried out.
(...)
... used to evaluate force fields everywhere, bypassing the difficulty introduced by the
logarithmic term. The only change required is that the initial expansions computed
be the derivatives of the multipole expansions and not the multipole expansions
themselves."

However, they speak about periodic bc in all directions while I have only with one.

[FMDenaro]

Again, I am not an expert of this specific field. I just immagine something like the pressure field in the streamwise periodic direction. Since to be substained the flow a pressure gradient is required, the periodic conditions can be applied for the pressure only for the field after a constant linear field is subtracted.
Something similar happens for the stream-functions in periodic normal to flow condition.

Sorry if I cannot say a specific method directly for your case

[holgerbre]

I see your point, Filippo. I tried already to substract a constant charge field in such a way that the total charge in the domain is zero. But contrary to pressure, I think the boundary for the electric potential should be zero at the pipe walls and not zero gradient.
Or am I maybe wrong with my wall boundary conditions...?

[Jonas Holdeman]

I see some other problems. If the duct walls are conducting, the charged particles would be attracted to the duct wall (unless there is a strong longitudinal magnetic field) and the charge density would be eventually depleted (steady state is no charge). If the duct is non-conducting (and no magnetic field), the mutual repulsion of the particles would drive them to the wall. Then there is the problem of the driving force (unless the fluid is inviscid).

To further analyze the problem, imagine the duct is formed in a circle (torus), large enough that the curvature can be neglected, in which case it would be physically periodic. What is the effect of the induced magnetic field and what force is driving the flow?

[holgerbre]

Hallo Jonas, thanks for your thoughts! You can see it like his: particles gain charge when they collide with the walls due to triboelectric charging. After a number of collisions the particles reach equilibrium charge, i.e. further collisions will not change their charge which is the state I am interested in. The driving force would be a constant pressure force of the gas flow.

[Jonas Holdeman]

Let me see if I understand your problem. You have both production and dissipation of charge at the duct surface, which you might specify as a charge density at the surface. It would seem that the current produces a magnetic field that interacts with that current producing a Lorentz force that balances the mutual repulsion of the charge and attraction to the conducting duct. The viscous force between the fluid and particles must be analogous to an electrical conductivity. I think you should be using the full magneto-hydrodynamic equations rather than treating this as an electrostatic problem. Then the periodicity problem with the electrostatic potential goes away. You have only to deal with the non-periodicity of the pressure, but there are ways around that.... Just some of my thoughts.

[holgerbre]

Since the particles are in equilibrium state no net charge is exchanged with the walls. Since the wall is conductive and grounded it has a zero potential, no current arises at the surface. The only current is associated to the convective velocity of the charged particles which is far less than speed of light, so to my understanding no magnetic fields are produced and I believe the electrostatic assumption is reasonable. Or am I wrong with that?

[Jonas Holdeman]

Some further thoughts ... You are basically balancing the radial charge flow rates due to diffusion of charges produced at the duct surface with that due to the mutual repulsion of charges. With a cylindrical duct and axisymmetric coordinates and pressure-driven flow, the equipotential surfaces should be cylindrical, and invariant (constant) along the axis. Thus you should have no problem with periodic BC except for the pressure jump. Even the pressure jump can be eliminated with the stream-function - velocity method described in the CFD Wiki.