Inhomogeneous viscosity

Dear friends,

this question has been posted before, but there was no conclusive answer:
Is it possible - using LBM - to simulate a multi-component fluid with different viscosities, or a single-component fluid with a varying viscosity? The viscosity of my fluid can change by a factor of 5, depending on the circumstances. Could I just introduce a relaxation parameter field tau(location) which passes its value to the collision step for a given lattice node? Has anybody tries this before? Are there any references?


Hi Timm…

        are you using Shan-Chen MultiComponent method or a different method?     I play with the MultiComponent ShanChen method.. and I agree, the viscosity ratio that I can get is never better then 1/10 (obtained playing and trying different combination of relaxation time for the two fluid)... and its very difficult to control the stability of the method when you add some dynamic to the problem.

anyway … two days ago I saw a paper where, instead of using the traditional explicit BGK scheme for MultiComponent fluid flow, the authors use an implicit local scheme for the BGk evolution … and the showy that they can play with a much more with the viscosities. Maybe it can help.

Umm … I’m sorry but I forgot the paper at home and I cannot find it anymore on the web … sorry
Anyway, the second author is X.Shan, the journal is Journal of fluid mechanics and the year is 2001.
If you can wait until tomorrow I can check the reference later.

Can I ask what kind of problem you are trying to deal with?


Hello Andrea,

thank you for the hint.
I am simulating red blood cells. Their interior have a viscosity which is five times larger than that of the ambient fluid. Since both fluid are modeled together (the same set of populations), I am looking for a way to modify the viscosity on the fly. The fluids are spatially separated by a membrane, which is put on top of the fluid grid. In principle I could just define tau as function of position, but I do not know about convergence behavior and so on.
So I just wait for your next post, it is not urgent.


Hi Timm,

  here the reference to the paper I told you about.

Journal of Fluid Mechanics (2002), 452:61-96 Cambridge University Press
Copyright © 2002 Cambridge University Press

Analysis of drag and virtual mass forces in bubbly suspensions using an implicit formulation of the lattice Boltzmann method


I hope it helps


Thank you for the reference, I will read the paper in the next days.


There is another paper talking about the above stated problem:
Ashrafizaadeh, Bakhshaei, A comparison of non-Newtonian models for lattice Boltzmann blood flow simulations, Computers and Mathematics with Applications, in press (doi:10.1016/j.camwa.2009.02.021) or here. There are also some references in the paper.


Hi Timm,

Your problem should be doable using the Lee and Lin formulation. It is slightly different from the usual, and pretty tricky to implement, but it allows for large density differences and also large viscosity differences between two incompressible fluids. The two references below should help:

T. Lee and C.-L. Lin. A stable discretization of the lattice Boltzmann
equation for simulation of incompressible two-phase ?ows at high density
ratio. Journal of Computational Physics, 206:16–47, 2005.

S. Mukherjee and J. Abraham. A pressure-evolution based multi-
relaxation-time high-density-ratio two-phase lattice-boltzmann model.
Computers and Fluids, 36:1149–1158, 2007.

The implementation is tricky. Let me know if you need help with it.

Thank you for those articles!
Right now, this problem is not very pressing, and I do not know whether I even have time to include the viscosity gradient effect into our model. There are more important ingredients which have to be thought of first.
But now I know some references for the case that I need them. :slight_smile:


Dear Timm,

You can try also to use free energy models in latest articles by C. Pooley, H. Kusumatmaja.

Hopefully it helps,

Hi Alex,

free energy models are not what I am looking for. The point is that I do not really have two components which have to be separated by the LBM itself. The seperation is done by implementing a thin membrane using the immersed boundary method. This is actually working quite well. I just wondered how to deal with different viscosities (even the densities of inside/outside are nearly identical). But I think, I have understood the techniques to solve the problem now. If somebody is interested in details, I am glad to explain some basic ideas here and give some references.


Hi Timm,

Yes, that would be always interesting to hear your thoughts.

[quote=I just wondered how to deal with different viscosities (even the densities of inside/outside are nearly identical).][/quote]

It’s exactly what the free energy binary model does. Probably not in your case, but generally.

Thank you,


I’m interested in trying a multi-component model (two immiscible liquids). The first liquid is water, the other is meant to be a theoretical representation of saturated clay. I am not interested in the dynamics within the pore space of the clay, but rather just as a homogeneous liquid with its own density and viscosity. The two liquids will have a density ratio in the neighborhood of 3. I’m not yet sure what the viscosity of the theoretical clay liquid is yet but I imagine it will differ by several orders of magnitude (I am also working under the assumption it will be non-Newtonian). I am ultimately interested in the motion of the theoretical liquid while varying the flow condition of the upper fluid.

I would like to know if anyone has an opinion on the feasibility of such an idea within the LBM framework and if the papers mentioned above are relevant; they seem to focus on liquid-vapor systems.