is there any way how to incorporate anisotropic viscosity into LBM? I would like to simulate liquid with fibers also on macroscopic scale (without modeling the fibers) which makes the fluid anisotropic (different viscosity in different directions).

I just modify relaxation time to obtain a non-Newtonian LBM. But this is not what I need. I would like to have different viscosities in different directions. Probably, I will go with the anisotropic permeability which I hope will give the same results as an anisotropic viscosity.

Maybe it is possible to use different relaxation times, depending on in which direction the populations are propagated? Just an idea…
For example (D2Q9): populations moving along the x-axis have tau_x, populations moving along y-axis tau_y. For populations moving in xy-direction, a suitable combination of tau_x and tau_y should be taken.

I have tried that but I got strange results. The viscosity was dominated by one of the values. I do not remember if it was bigger tau or smaller one. As I am not professional in the LBM I do not dare to do it and there seems to be no papers dealing with it.

Sorry for my ignorance, but what does an anisotropic viscosity mean (except that it it is not varying in the same way depending on the directions)? What would it mean in terms of the Navier-Stokes questions? Could you be a bit more specific?

I think it can be done with the combination of MRT and equilibrium functions. Chapman-Enskog is your tool. If you are interested in anisotropic fluids for advection-diffusion equation you can take a look how Dr.Ginzburg does it.

thanks for the suggestions. What I use are Navier Stokes equations -> simple one relaxation time LBM. By anisotropic viscosity I mean that the viscosity has different values in different orientations so that it reflects presents of oriented fibers in the liquid. Something similar is an anisotropic permeability as described here:

where they introduce the permeability as a permeability force. Since I find the structure of Darcy’s law (velocity of the flow is linearly proportional to the pressure gradient - the proportionality coefficient is permeability) very similar to the structure of laminar shear flow equation (shear rate is proportional to the shear stress - the coefficient is viscosity) I try to find a viscous force in a similar manner. But until now I did not find the correct form.

PS: I do not dare to play with Chapman-Enskog as it would take much more time (to study it, derive it etc.) than I have.