If I want to build a solver for this following problem:

1. There is stagnant water governed by the Navier-Stokes equation in the domain.

2. Air bubbles (µm to mm in size) are produced from an arbitrary point within the domain, and their movement will obey NS-equation as well.

3. Particles (having similar sizes like the air bubbles) in the water interact with the bubbles following a collision-attachment-detachment model, for example like those suggested in this paper.

Yoon, R. H., and G. H. Luttrell. "The effect of bubble size on fine particle flotation." Mineral Procesing and Extractive Metallurgy Review 5.1-4 (1989): 101-122.

What kind of numerical approach should I use? Lagrangian or Eulerian will work better? I guess the former one?

Will Smoothed Particle Hydrodynamics be sufficient and efficient for the problem?

Please also give me some suggestions or share the link if you have seen similar discussions before. Many thanks.


2 Answers 2


What $Re$ and how many particles do you intend to simulate? These put limits on the approaches available to you.

If you have few particles/bubbles, fully-resolved Eulerian methods (diffuse interface, immersed boundary, etc.) might handle this problem, but they will be expensive. Cut cell methods might also be useful, though again, expensive. You might also explore boundary element methods.

I suspect you have many particles and will just need to do some degree of modeling. There is a vast literature in this area. Euler--Euler sub-grid models can handle the bubbles or the particles, though I am unaware of attempts to handle both in the same grid cell simultaneously. However, I suspect it should be possible (and may have already been done). Check the work of R. Fox, O. Desjardins, and the works that cite them. The Euler--Lagrange approach is almost always simpler from a mathematical perspective. This is probably more practical, though the method you choose will depend on the flow physics.

Edit: Some references on the Euler--Lagrange front.

Bubbles [notable keyword is volume averaging]:

Solid particles:

  • $\begingroup$ Thanks a lot for answering and this kind of analysis is what I´ve expected. The Re in the context of my problem should be very low, possibly lower than 1. There will be 100k particles, I guess this is not a small number. Additionally, the particles will have different sizes ranging from several µm to mm. I personally think that the Eulerian methods won´t be the best options, especially when it comes to instantiating various-sized particles within the grid-based domain. Nevertheless, I will still go through the work related to Euler--Euler sub-grid models. $\endgroup$ Commented Nov 12, 2021 at 8:29
  • $\begingroup$ @MichaelGao I think EL could be a reasonable route forward, at least to get started. I will try to put some references for you when I have time! $\endgroup$
    – user20857
    Commented Nov 12, 2021 at 16:40

Welcome to research. It may be that there’s already a good answer to your problem. In which case, you need to spend some more time looking until you are certain the problem has a good solution available. Or, maybe little work has been done on this problem, and you should still spend the time looking until you give up and develop your own. Or, perhaps you are only looking to use a good solver for these kinds of problems, then you should keep digging, including asking everywhere you can (like here).

This sounds like a problem where at least some reasonable work has been done. You have a reference, but have you looked at enough of its citations and it’s author’s relevant recent work to see what’s happened in the field since 1989?

  • $\begingroup$ First of all, thank you very much for this general suggestion. I have already invested a couple of weeks digging into this topic, including the physical models of collision-attachment interactions between particles and bubbles, all the strengths and shortcomings of Finite Methods and Particle-Based Methods, and many more aspects. Of course, there are many relevant works and articles since 1989 regarding the collision-attachment models as well as the SPH, which is a powerful tool for coping with the type of problem like mine. $\endgroup$ Commented Nov 11, 2021 at 21:09
  • $\begingroup$ However, as far as I have searched, I didn´t find a solver which considers both the multiphase (liquid-air) flow and the particle-air phase interactions. This fact is actually logical for me, as in most cases (in which scientists and engineers are interested), the particles within the fluid move with the advective flux, or the solutes are very tiny (therefore they are not considered as particles) so they are relocated with the diffusive flux. $\endgroup$ Commented Nov 11, 2021 at 21:21
  • $\begingroup$ I know my description might not be precise or even not true for experts, I still think that in most cases, either the movement of the fluid or the transport behavior of the solutes is the main research objective. Models up to now have all to some extent sacrificed one unimportant aspect and focused on a more essential one. Well to my interest, both aspects are crucial so I can´t just neglect one. $\endgroup$ Commented Nov 11, 2021 at 21:27
  • $\begingroup$ @MichaelGao,do you have experimental evidence (physical or computational) that one of the two possibilities you describe happens more frequently? $\endgroup$
    – Bill Barth
    Commented Nov 28, 2021 at 15:34
  • $\begingroup$ Are you referring to the advective and diffusive transport of the particles/solutes? When there are only particles or solutes in a liquid phase, then yes. I had both physical and computational evidence before, however that will be another question (which could be solved by SPH (for particles), or by colloid filtration theory (colloids), or by Advection-Dispersion-Reaction Equation for Solute Transport (solutes)). When there are particles and air bubbles both existing in a liquid phase, their interactions and consequences are mostly not explained by any of those models mentioned $\endgroup$ Commented Nov 29, 2021 at 11:48

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.