A Free surface problem in electrohydrodynamics

[hunt_mat]

TL;DR Summary

If I have a constant bulk charge in a fluid, does that correspond to a specific surface charge at the free surface?

I'm thinking about the following problem. I have an electrified fluid with a constant charge density, Q, within the fluid. Will this necessarily yield a surface charge?

Would I have to compute it by looking at the displacement fields on either side of the interface? Would it change if the bulk charge within the fluid remains constant?

 

[pines-demon]

Im reading this on a phone so maybe I am missing something. But the electric field is different inside and outside the medium, the discontinuity is proportional to the surface charge.

 

[hunt_mat]

Yes. The picture I'm thinking about is where there is an electric field within the fluid as well as a constant charge density. There is also an electric field above the fluid. The surface charge is the difference between the electric displacements D. I want to know if given that there is a bulk electric charge, will that give rise to a surface charge that I have to take into account?

 

[pines-demon]

Yes. The picture I'm thinking about is where there is an electric field within the fluid as well as a constant charge density. There is also an electric field above the fluid. The surface charge is the difference between the electric displacements D. I want to know if given that there is a bulk electric charge, will that give rise to a surface charge that I have to take into account?

Can you tell me what is the electric field inside the fluid and outside (at least an example)?

 

[hunt_mat]

That is something to be found as part of the solution. I'm connecting the Euler equations with Maxwell's equations to find the resulting shape of the free surface.

 

[Dale]

The bulk charge does not intrinsically produce a surface charge. Consider the case of a sphere with uniform charge density. In that case the electric field is continuous at the surface so there is no surface charge.

That said, with a charged fluid there will be forces on the bulk fluid due to the charge. So that may lead to motion that does produce a surface charge.

The bulk charge and the surface charge are not directly linked, but they can be indirectly linked through the motion of the fluid.

 

[pines-demon]

That is something to be found as part of the solution. I'm connecting the Euler equations with Maxwell's equations to find the resulting shape of the free surface.

But do you at least have the electric field inside and outside the medium?

 

[renormalize]

But do you at least have the electric field inside and outside the medium?

The OP posits a fluid medium with a constant bulk charge density and so it carries a nonzero total charge. Assuming that the fluid does not extend to infinity in any direction, it can therefore be completely surrounded by a closed surface. The integral of the normal electric field over this surface must then be nonzero by Gauss' Law. Hence, by necessity there exists an electric field outside the medium.

 

[hunt_mat]

The OP posits a fluid medium with a constant bulk charge density and so it carries a nonzero total charge. Assuming that the fluid does not extend to infinity in any direction, it can therefore be completely surrounded by a closed surface. The integral of the normal electric field over this surface must then be nonzero by Gauss' Law. Hence, by necessity there exists an electric field outside the medium.

Unfortunately, the fluid is infinite in extent.

 

[hunt_mat]

But do you at least have the electric field inside and outside the medium?

Yes. I have solved the result when there is no electric charge, I thought that the next logical step(for a mathematician that is) is to look at the case where there is a constant charge density within the fluid.

 

[hunt_mat]

The bulk charge does not intrinsically produce a surface charge. Consider the case of a sphere with uniform charge density. In that case the electric field is continuous at the surface so there is no surface charge.

That said, with a charged fluid there will be forces on the bulk fluid due to the charge. So that may lead to motion that does produce a surface charge.

The bulk charge and the surface charge are not directly linked, but they can be indirectly linked through the motion of the fluid.

Surely only the tangential part of the electric field is continuous at the boundary. You have to make a calculation for the normal component of the electric field.

If the forces aren't too large, then should there be no surface charge? Or, at the very least, the assumption of no surface charge is reasonable.

I'm also assuming that there is no charge below the fluid, so I can assume that the electric field is zero at the bottom?

 

[Dale]

Surely only the tangential part of the electric field is continuous at the boundary.

For a uniform ball of charge (eg a solid charged insulator) both the tangential and normal components are continuous at the boundary

 

[hunt_mat]

For a uniform ball of charge (eg a solid charged insulator) both the tangential and normal components are continuous at the boundary

Isn't the normal component of the \mathbf{D}=\epsilon\mathbf{E} continuous across the boundary if there is no surface charge?

 

[Dale]

Isn't the normal component of the \mathbf{D}=\epsilon\mathbf{E} continuous across the boundary if there is no surface charge?

Yes. That is what I am saying. This is an example of a scenario where there is not a surface charge.

 

[hunt_mat]

Yes. That is what I am saying. This is an example of a scenario where there is not a surface charge.

So I can ignore surface charge for my problem, then?

 

[Dale]

So I can ignore surface charge for my problem, then?

Not necessarily. As I said above, surface charge is not intrinsically tied to the bulk charge, but you could get a surface charge through fluid motion. That would depend on the constitutive equations that govern the fluid and the equations of motion

 

[hunt_mat]

If I'm considering only small perturbations from the equilibrium, then it's still a good assumption?

 

[hunt_mat]

I have written down the set up of the problem I'm interested in here.

 

[renormalize]

I have written down the set up of the problem I'm interested in here.

You say in the first line of your abstract:
Research on free-surface flows in electrohydrodynamics has predominantly focused on the limiting cases of perfectly conducting and perfectly insulating fluids.
But magnetohydrodynamics (MHD), which is the shortened name of what Alfvén originally called "electromagnetic-hydrodynamics", regularly considers the flow of finite-conductivity charged fluids with free surfaces. Here's a recent example: https://arxiv.org/abs/2412.13598.
How familiar are you with the literature of MHD?

 

[hunt_mat]

You say in the first line of your abstract:
Research on free-surface flows in electrohydrodynamics has predominantly focused on the limiting cases of perfectly conducting and perfectly insulating fluids.
But magnetohydrodynamics (MHD), which is the shortened name of what Alfvén originally called "electromagnetic-hydrodynamics", regularly considers the flow of finite-conductivity charged fluids with free surfaces. Here's a recent example: https://arxiv.org/abs/2412.13598.
How familiar are you with the literature of MHD?

I am familiar enough to have written a paper on free surface flows in MHD in the Journal of Plasma Physics, so I'm fairly familiar with that particular aspect of the MHD literature. I have enclosed the paper in question.

I have also included a copy of my paper on EHD without any charges. I am looking to generalise this with a constant charge density within the fluid. What I need to know is: Is there any surface charge density that I need to know about? This adds an extra variable into my equations, I therefore need another equation to complete the system.

 

[hunt_mat]

My MHD paper is apparently too large to upload.

 

[Dale]

If I'm considering only small perturbations from the equilibrium, then it's still a good assumption?

It doesn’t matter if it is small or large. What matters is the direction. If the charge is moving normal relative to the surface then you will get a surface charge that will accumulate.

 

[hunt_mat]

It doesn’t matter if it is small or large. What matters is the direction. If the charge is moving normal relative to the surface then you will get a surface charge that will accumulate.

The surface is also moving. So could it theoretically move with the charge, so no surface charge forms?

 

[renormalize]

I am familiar enough to have written a paper on free surface flows in MHD in the Journal of Plasma Physics, so I'm fairly familiar with that particular aspect of the MHD literature. I have enclosed the paper in question.

I have also included a copy of my paper on EHD without any charges. I am looking to generalise this with a constant charge density within the fluid. What I need to know is: Is there any surface charge density that I need to know about? This adds an extra variable into my equations, I therefore need another equation to complete the system.

OK, I can see from your attached paper that you are quite knowledgeable regarding EHD, so providing a detailed answer to your question is clearly above my pay grade!
I will only comment that, just as the shape and evolution of a free surface must be computed from the fluid dynamics, I would guess that the surface-charge at that surface must also be computed, rather than manually imposed to be specific value (like zero). Here is a paper that supports that guess:
Charge distribution in turbulent flow of charged liquid—Modeling and experimental validation
Abstract:
Electric discharges due to the flow of charged organic liquids are a common ignition source for explosions in the chemical and process industry. Prevention of incidents requires knowledge of electric fields above the surface of charged liquids. Quantitative methods often estimate electric fields based on simplifying assumptions like homogeneous volumetric charge distribution and neglect of surface charge. More detailed electrohydrodynamic (EHD) models are only available for laminar flow regimes. This work presents a model for forced turbulent EHD flows of dielectric liquids based on Reynolds-averaged Navier–Stokes equations that predicts the electric field in the gas phase in good agreement with our experiments. We observe diminishing surface charge accumulation at the liquid surface with increasing flow velocities and thereby unify seemingly contradictory previous findings regarding the relevance of surface charge. The model can efficiently be applied to various industrial flow configurations and provide a central tool in preventing electrostatic hazards.
Good luck in your research.

 

[Dale]

The surface is also moving. So could it theoretically move with the charge, so no surface charge forms?

Yes. That is why I said “moving normal relative to the surface”.

 

[hunt_mat]

OK, I can see from your attached paper that you are quite knowledgeable regarding EHD, so providing a detailed answer to your question is clearly above my pay grade!
I will only comment that, just as the shape and evolution of a free surface must be computed from the fluid dynamics, I would guess that the surface-charge at that surface must also be computed, rather than manually imposed to be specific value (like zero). Here is a paper that supports that guess:
Charge distribution in turbulent flow of charged liquid—Modeling and experimental validation
Abstract:
Electric discharges due to the flow of charged organic liquids are a common ignition source for explosions in the chemical and process industry. Prevention of incidents requires knowledge of electric fields above the surface of charged liquids. Quantitative methods often estimate electric fields based on simplifying assumptions like homogeneous volumetric charge distribution and neglect of surface charge. More detailed electrohydrodynamic (EHD) models are only available for laminar flow regimes. This work presents a model for forced turbulent EHD flows of dielectric liquids based on Reynolds-averaged Navier–Stokes equations that predicts the electric field in the gas phase in good agreement with our experiments. We observe diminishing surface charge accumulation at the liquid surface with increasing flow velocities and thereby unify seemingly contradictory previous findings regarding the relevance of surface charge. The model can efficiently be applied to various industrial flow configurations and provide a central tool in preventing electrostatic hazards.
Good luck in your research.

I had a look at their paper, and it looks like nonsense to me. In their equation (2), they seem to mix vectors and scalars in a single equation. However, if one makes the simple assumption of Ohm's law, [\itex]\mathbf{J}=\sigma\mathbf{E}[/itex] then you're led down a rather odd route. I thank you for pointing this out, as I was thinking of looking at a variable charge density in my next paper.

 

[hunt_mat]

Yes. That is why I said “moving normal relative to the surface”.

The question remains, as my model includes electric fields, fluid velocity and a free surface, what equation do I add to close the system for surface charge?

 

[renormalize]

I had a look at their paper, and it looks like nonsense to me. In their equation (2), they seem to mix vectors and scalars in a single equation.

I think that $\nabla^2$ is just a typo; it should instead be the vector-operator $\nabla$ according to Fick's laws of diffusion. The Fick diffusion electric current is distinct from the ohmic drift electric current. See: https://en.wikipedia.org/wiki/Diffusion_current.

 

[hunt_mat]

I think that $\nabla^2$ is just a typo; it should instead be the vector-operator $\nabla$ according to Fick's laws of diffusion. The Fick diffusion electric current is distinct from the ohmic drift electric current. See: https://en.wikipedia.org/wiki/Diffusion_current.

I'm familiar with Fickian diffusion.

 

[renormalize]

I'm familiar with Fickian diffusion.

So do you agree that the authors meant to write $\nabla$ and that their eq.(2) is therefore not nonsense?

 

[hunt_mat]

So do you agree that the authors meant to write $\nabla$ and that their eq.(2) is therefore not nonsense?

I don't know. It's certainly something that I'm not going to be concerned about when I deal with variable charge density, although I am curious where the term $\rho_{v}\mathbf{u}$ comes from. That would couple the variable charge density very nicely.