Three Dimensional Odd Viscosity in Ferrofluids with Vorticity-Magnetization Coupling

1. A step towards investigation of paryty breaking ferrofluids

1. The Poisson bracket formalism is not the most efficent tool to study hydrodynamics with dissipation.
2. Strong overlap with previous results.

This paper delves into the unique properties of ferrofluids, synthetic magnetic colloids composed of magnetized nanoparticles each surrounded by a repulsive surfactant layer. When exposed to an external magnetic field, ferrofluids acquire a macroscopic magnetization density, resulting in magnetic behavior deeply intertwined with the fluid dynamics of the surrounding medium. The study draws parallels between ferrofluids and chiral active fluids, specifically those containing unidirectionally spinning hematite cubes, which exhibit a two-dimensional non-dissipative odd viscosity term. However, unlike these chiral fluids, standard ferrofluid dynamics typically do not demonstrate 3D parity breaking terms, likely due to the small size of the magnetic particles.
The manuscript aims to explore whether ferrofluids can manifest a unique 3D odd viscosity term through specific mechanisms. The paper posits that coupling the fluid vorticity with the magnetization, especially when magnetization aligns with a uniform and static applied field, introduces parity breaking terms in ferrofluid hydrodynamics. This results in the emergence of a three-dimensional odd viscosity term. The result is interesting; however, I have a couple of critical comments about the construction.
1. The limit I going to zero seems to be a singular limit. At the level of the discussion presented in the paper it is not clear that this limit is always consistent throughout the manuscript. In the Hamiltonian in this limit the terms involving magnetization are negligible.
2. The internal degrees of freedom of particles usually become unimportant in the hydrodynamic regime as they lead to gapped modes that relax quickly. One can imagine that the authors want to include them, but this leads to a framework that goes beyond pure fluid dynamics and this does not seem to be the message of the paper.
3. Finally, I think that the non-Newtonian aspects of the ferrofluids should also be important for the Helle-Shaw solution and may modify the prediction. In the current form the solution seems to be very similar to Ref. [32].
The authors should address the above points.

1- The paper is well written
2- Odd viscous phenomena are explored theoretically in a physical system where it has not been explored previously
3- The experimental proposal is concrete and simple

1- As the authors acknowledge, the overlap with reference [22] is very strong.
2-The fluid mechanical results that lead to the experimental proposal that are presented also hold for general odd viscous systems and these have to a large extent been presented earlier by the same authors in [32]. Mostly what this paper does is implicitly restrict to a certain odd viscosity that happens to coincide with ferrofluids, but also with fluids with an intrinsic angular momentum-vorticity coupling.
3- Apart from the overlap with [22], there is also strong overlap with plasmas that also produce odd viscosity when subjected to a background magnetic field.
4- One of the main strengths is that a new physical system is being explored through the lens of odd viscosity, but little details about these systems are provided.

The authors study a vorticity-magnetization coupling called \gamma which is assumed to be part of the theoretical description of ferrofluids and which they show, with a Poisson bracket formalism, gives rise to a parity breaking term which at least from the title and introduction I assume is considered to be an odd viscous term. They then study the implications of this term for the Hele-Shaw cell.

Throughout the work, the resemblance with reference [22] is stressed, where the main difference is that for [22] there is a coupling between vorticity and intrinsic angular momentum. Below (31) it is stated that the final parity-breaking term that comes from the \gamma term is identical to the momentum balance law with odd viscosity obtained in [22] after taking the incompressible limit, I have some comments about this
1) Despite that the incompressible limit simplifies it to this single term, it would still to understand the extent to which this term can be seen as originating from odd viscosity. This is also important considering that the title of the work involves odd viscosity but the word odd viscosity is then not used at all after the introduction.
2) A previous work by the same authors [32] considered the Hele-Shaw problem for general 3D odd viscosities, so this stresses that more elaboration on (31) would be helpful to understand what is new and what isn’t. I do see that below (43) there is a statement about how there is a relation to the odd parameter of [32] but if (31) would be analyzed more thoroughly from the point of view of odd viscosity this would not come as a surprise since [32] works with general odd viscosities.
3) Is there anything known in the literature about the degree to which ferrofluids are incompressible? It would be nice if there were a discussion of this since it clearly has implications.
Furthermore, as mentioned in 2), it seems that an important part of what is novel about the work is that the authors utilize somewhat known derivations in the context of a different system, namely ferrofluids. Therefore, it would be helpful if more details would be given about ferrofluids. I see some statements below equation (24) about a “heuristic analysis of the structure of the ferrofluid nanoparticles”. This is a great opportunity to provide more details about the nature of ferrofluids, maybe with some numbers that give a sense of what an experimental setup of ferrofluids that could display the phenomena described in this work would look like. The authors also mention that the Hele-Shaw cell is considered by experimentalists in the context of ferrofluids and provide references. Is there anything about what these experimentalists have reported that is worth mentioning because it can perhaps be related to the \gamma term and odd viscosity?
Another key result presented in this work is the relation between the far field rotation and the \gamma term. Also here it would be good to have an understanding of how general the relation between 3d odd viscosity and far field circulation is. Particularly, in [32] a general anisotropic 3d system with odd viscosity is considered and a relation between circulation and the change of a bubble area is presented. Is there anything different here except that a specific odd viscosity is considered?
Lastly, the similarity with [22] is stressed in the sense that the equations would be equivalent if magnetization were replaced with intrinsic angular momentum. However, this also seems very similar to plasmas subject to an ordinary background magnetic field. This connection is made in [22] where it is stated that intrinsic angular momentum which induces odd viscosity could be replaced by a background magnetic field so that it coincides with what was found by Landau-Lifshitz. One could ask whether there is a difference between plasmas subject to a background magnetic field and fluids with magnetization that is turned on by a background magnetic field.
I think that the message that intrinsic angular momentum can be replaced by magnetization and that this allows for a result that follows from a computation identical to [22] is an interesting one but that this alone is not novel enough to warrant publication. With additional fluid mechanical results related to experimental implications the situation could be different, however it seems that the fluid mechanical results currently presented are merely restricted versions of general fluid mechanical results already presented in [32]. I also believe that currently this work does not make a great case for why experimentalists should look for odd viscosity specifically in ferrofluids as opposed to all the other fluids that can potentially display odd viscosity.

1- Make it transparent that the resulting odd term is a term that could be written as an odd viscosity in the stress tensor, and therefore that the fluid mechanical results that were obtained earlier for general 3d odd viscosities can be obtained by restricting to this odd viscosity.
2- Then highlight anything new with respect to these previously obtained fluid mechanical results.
3- Provide more details about what the state of fluid mechanical experiments for ferrofluids is and if there is anything that suggests that odd viscosity could be observed.
4-Comment on magnetized plasmas.

1. The manuscript contains new results. These are the system equations for a ferrofluid with an additional vorticity-magnetization coupling. The Hamiltonian structure of these equations is also presented.

2. The authors propose an experiment which would demonstrate the presence (or absence) of the introduced new term by measuring the circulation of the fluid in a particular Hele-Shaw cell setup.

3. The manuscript is nicely written

4. It contains references to relevant literature

While an additional term proposed by the authors is allowed by symmetry in parity broken fluids, it is not clear what one should expect for the value of the introduced coupling $\gamma$. The authors say that it would depend on the "microscopic properties of the fluid nanoparticles". However, it is not clear whether it can be significant for small size particles. They also say "a heuristic analysis of the structure of the ferrofluid nanoparticles and the atoms within, shows $\gamma$ to be inversely proportional to the molecule Lande g-factor." I think it would be useful for readers if the authors expanded that comment and give that heuristic argument in slightly more detail.

The manuscript introduces an additional symmetry-allowed term into the ferrofluid hydrodynamic equations. They study the effect of the term on the fluid motion and describe a possible test for the presence of the term in the Hele-Shaw cell experiment.

I think that the manuscript is interesting and it contains new results. In my opinion, it should be published in SciPost Physics with a few minor changes described below. (the changes are minor and do not affect any of the conclusions of the authors)

1. Expand the comment on ``heuristic argument'' to make it more clear (see above).

2. Reference in abstract: it is better to have a numbered citation their [32] instead of the full citation which is hard to find in the list of references.

3. The second paragraph in the first column. The comparison to "incompressible fluids" is too abrupt and unnecessary as the active chiral fluids exhibit additional transport coefficients even in comparison with compressible fluids.

4. The second paragraph in the second column on the first page. "A ferrofluid is a type of 3D active matter fluid...". This is not strictly correct. Only driven ferrofluids should be considered as active. The main example of the paper, the fluid with constant magnetization, for example, is not active.

5. After equation (8). I believe that there should be an equilibrium magnetization $M_i^0$ in the sentence "... and the magnetization $M_i$ vanishes..."

6. After eq. (21) "to" is missing in "does lead parity breaking terms"

7. Just above (41). The sentence "... and average over the plate separation..." is not clear. I think the authors meant "... and average over the thickness of the fluid between the plates..." or something similar. Please, clarify.

8. Right before the end of the first column on the page 6. "the two fluid interface." should be "the two-fluid interface."