I’m working on thesis regarding usage of LBM for non-newtonian flow simulations. As a starting point, I investigated the derivations of LBM as a discretisation of Boltzmann equation. I was interested in possibility of transferring some microscopical properties of fluid to the macroscopical level of LBM, possibly via the Chapman-Enskog expansion, somehow altering the collision operator.
But I’m getting more questions than answers. The whole derivation of Boltzmann equation starts with usage of one particle distribution function, that eliminates possible particle corellations, then it proceeds assuming only two-particle collisions, molecular chaos hypothesis assuming non-trivial mean free path between collisions and the equilibrium distribution in BGK operator is the Boltzmann distribution. All these assumptions prohibit any thoughts about simulating something other, than gases! All these assumptions are just plainly violated in dense fluids, like water, as far as I know.
Yet, somehow, via the Chapman-Enskog expansion, there is a limit proven, leading to Navier Stokes equations, that are valid not only for gases, but other fluids of various density, including mentioned water. And many people around the world are happily using the LBM for simulation of such fluids. So why does LBM work for dense fluids, when it “shouldn’t”?
So, please correct me,If I’m wrong. From this a draw a conclusion that:
The derivation of macroscopic properties of fluid (NS equations) from microscopic properties (Boltzmann equation) may be misleading. It may be more true for gases, but the real reason, why LBM dynamics limit to the NS equations is still hidden. Viewed from an abstract viewpoint, for both LBM and LGA, we have a lattice, and some “physically guessed” evolution rules, that rule the game. We’ve found, that these rules are working quite well, but It’s more of a coincidence in case of dense fluids. There possibly might be some derivation, that I don’t know, that starts from microscopic fluids of various densities and derives same or similar lattice game rules, as we have now, but this derivation has to start somwhere else, than in Boltzmann equation. (quick google: maybe Dynamic density functional theory?)
Considering applied assumptions, there is no hope for physically legitimate derivation of some (possibly non-newtonian) property of dense fluid, starting from microscopic model, using the modification of standard Boltzmann -> LBM process (simply because saying “Boltzmann equation” contradicts any dense fluid). The only simple justifiable approach is to connect LBM to NS via C-E expansion (or in general, lattice game rules to partial differential equation)(which leaves us in pure mathematics), and using this connection, transfer various (possibly non-newtonian) properties from classical PDE model, to LBM rules. I believe, this is the case of the so called ‘ad-hoc’ non-newtonian model, that dynamicaly changes relaxation parameter in BGK operator, adjusting the viscosity, according to computed strain rate tensor. Of course, the connection to underlying physics is lost. We just play with the PDE <=> LBM connection.
this is an interesting point, and you should have a look at Succi’s book, especially on p. 42. Succi states that the higher order collision terms are not used to recover hydrodynamics (Navier-Stokes), they are only responsible for making the mean free path smaller. In other words, dense fluids like water are still dominated by two-particle interactions, or - as far as I understand - the higher order collisions do not change the qualitative form of the equations, only the value of the macroscopic viscosity. I am not an expert in kinetic theory…
I think, as you also say, that we are just lucky that the transition LBM => NSE works, especially since the mechanisms for viscosity in gases in liquids are different.
Perhaps you should think about the question in a slightly different way: under what conditions does the Boltzmann equation lead to hydrodynamic-type equations for the conserved (collision-invariant) quantities?
There is a large disparity between the length-scales of the Boltzmann (microscopic) level and macroscopic scale on which hydrodynamic quantities vary. The length-scale of the Boltzmann level is generally taken to be the mean free path. Hilbert and Chapman and Enskog showed that we can exploit this disparity using the ratio of the different scales (the Knudsen number). Since this parameter is small we can expand the distribution function, which leads to an infinite hierarchy of hydrodynamic variables in the limit of the Knudsen number going to zero, ie it is the hydrodynamic limit. Note that this is an infinite hierarchy, which we truncate at a given order. Then the Boltzmann equation asymptotically approximates the Navier-Stokes in the limit of small Knudsen number and the solutions are slowly varying. It is the separation of scales that is key: if the scaling parameter is not small the expansion does not work and we can not call the gas a fluid. This can be made quite rigorous - see, for example, Chapman and Cowling, or the works of Hilbert and Grad. Note also that Maxwell, for example, was able to make accurate predictions of the fluid viscosity from kinetic theory (see Chapman and Cowling again). But this is difficult and tedious, and it’s not so important for lattice Boltzmann theory - we are usually most concerned with the moments of the distribution function, which need to comply with the hydrodynamic tensors.
I probably wouldn’t say that it is a coincidence that the collision rules for lattice Boltzmann work. In kinetic theory the equilibrium function is shown to be the Maxwellian and in lattice Boltzmann we can shown the equilibrium is a discrete analogue of this. For Navier-Stokes, the moments of the equilibrium function and collision term in LBE must give the correct hydrodynamic moments (density, momentum, viscous stress etc). Also, if you are only interested in macroscopic predictions, perhaps you should ask yourself how important the “underlying physics” is. If you want accurate, fast simulations, is it as important as the maths and numerical analysis?
As for non-Newtonian fluids…that’s another matter.
Quite interesting question. For me I am used to consider that LBM is a mathematical solver of certain type of PDEs. It has connection with kinetic theory but a bit weak. While kinetic equation has certain assumptions, especially in terms of collision integral which can be calculated for rigid and non-rigid spheres, the lattice Boltzmann equation has BGK or MRT collision operator which is more general in some sense than the original collision integral. Basically LBE has weak connections with kinetic theory but more like update rule equation with certain collision operators. This update rule allows us to restore the Navier-Stokes equation in certain regime (Knudsen or grid number goes to zero).
Basically I am more in favor for update mathematical framework than for kinetic nature of the LBE.
Dear all,
Landau thought that the equilibrium Maxwell-Boltzmann velocity distribution is valid not only for rarefied gases but also for condensed matter.(L.D. Landu, E.M. Lifshitc. Statistical physics. -Moscow,Nauka,1976.p584[in Russia])
I think we can think it as this: the distribution function for condensed matter is not must Maxwell-Boltzmann velocity distribution, but it will tend to be when at equilibrium. So we can used Maxwell-Boltzmann velocity distribution for condensed matter.
Further discussion will be appreciated
