I am working on a problem in Computational Fluid Dynamics, modeling multi-phase fluid flow through porous media. Though there are continuum equations to describe macroscopic flow (darcy's law, buckley leverett equations, etc), these models do not apply for heterogenous media (with transport properties). We could, however, try to use the microscopic model (lattice boltzmann, or pore network models) which would be more faithful to the dynamics of the macroscopic heterogeoneous media. But any computational simulation of this model would run too slowly to be worth it. The principals of conservation laws apply at both scales (conservation of mass, momentum, energy), but the equations that describe these laws differ at each scale. How then, can we upscale microscopic physics in a computationally efficient manner? Are there any techniques for describing microscopic phenomena at the macroscopic level without such a heavy computational cost? Is there any technique to build a continuum description at all scales of the problem?
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$\begingroup$ How is this question related to lattice QFT? Also, I don't understand why you want to use a microscopic model. Probably you just need to generalize a bit the macroscopic equations you started with. I must say I don't quite understand what you mean by heterogenous media. But then I don't know much about fluid dynamics so feel free to ignore my comments (except the first one) $\endgroup$– SquarkCommented Jan 16, 2012 at 20:46
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$\begingroup$ Sorry... I would have tagged it as lattice-boltzmann, if I had the reputation points to do so. $\endgroup$– PaulCommented Jan 16, 2012 at 21:26
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$\begingroup$ This is not my area at all, so this might be a particularly ignorant comment, but can you not simply simulate a small section at the microscopic level for a range of different conditions for the boundaries, and then piece together these into a solution for a larger volume, repeating several times if necessary? The basic idea being that there are many areas for which exactly the same calculation is necessary, and probably relatively few distinct such calculations, so better to build a lookup table than to calculate anew each time. $\endgroup$– Joe FitzsimonsCommented Jan 17, 2012 at 9:17
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$\begingroup$ Lattice Boltzmann methods aren't necessarily microscopic exactly - the particles in the simulation don't have to represent actual molecules. I believe that such methods have been successfully used to model quite large-scale flows, without having to model $\sim 10^{23}$ individual particles. $\endgroup$– N. VirgoCommented Dec 29, 2012 at 11:29
1 Answer
Your problem is highly nontrivial. The theoretical tool to be used is the renormalization group, which extracts the relevant dynamics of the large scales of the system. But if we were able to use it "in a blind way", then we would have a technique to study the macroscopic dynamics of any microscopic system... and this would made a lot of my colleagues unemployed :) The basic idea is to make "blocks" or perform a bit of "coarse-graining" in your original system and see if you can describe the resulting dynamics with the same microscopic laws, but changing a bit the parameters. If you can, then you're lucky. You get a "flow" in your parameter space, and the fixed points give you the macroscopical dynamics: how the system will behave in the thermodynamic limit.
The alternative approach, which is used very often, is to try to write the most general local partial differential equation which is compatible with all your physical requirements and symmetries. These equations will have "open" parameters that you will put later on, in a semi-empirical way. You can see examples in A.L. Barabasi and E.H. Stanley, "Fractal concepts in surface growth", and many other places.
