<< mixing
mixing immiscible fluids nature of mathematical modeling  final project. ben fry This applet simulates the action of two fluids of different densities. Ideally, these fluids are immiscible, like oil and water. In the simulation, the denser fluid starts on top, and slowly the two fluids change places. This phenomenon is known as a RayleighTaylor instability. (As an aside, two 'miscible' fluids would simply diffuse into one another, in the manner of milk being poured into coffee.) To do this, I used a lattice gas cellular automata. Neil's book discusses both FHP and HPP lattice gases, so I won't redescribe these here. In this case, I used FHP. In particular, I used FHPIII, which is a variant of FHPI which is described in the book. FHPIII distinguishes itself by having 14 states, and several additional collision rules than those of FHPI described in the book. The states include the original 6 states as used by FHPI, plus a seventh, which is simply a particle at rest (but with stored momentum, to ensure conservation of momentum). There are two sets of these seven, to account for the two fluids. FHPIII is by default a miscible model, so there are a number of methods that exist to simulate immiscibility. I chose a pseudoimmiscible model that 'fakes' it. The details of this method can be found in the third reference listed at the bottom of the page. The basic idea is that any time a single particle of one fluid is at a site with two or more particles of the other fluid, that single particle has a chance of instantaneously turning into the same fluid as the others. The likelihood of this event is set by a constant. This is done the same way for the opposite fluid, so things stay more or less even. Once the pseudoimmiscible part is working, gravity is simulated by flipping the directions of a few particles at each time step to simulate the force of gravity. More specifically, a small portion particles from each fluid are changed from a rest state and put in motion in the direction of (or against) gravity.
I researched many references for this particular
project, I've listed several of the most useful ones.
2. D. d'Humieres, P. Lallemand,
"Numerical Simulations of Hydrodynamics with
Lattice Gas Automata in Two Dimensions"
Complex Systems I, (1987) 599632.
3. H. Cabannes, "Mechanics of Fluids"
C. R. Acad. Sc. Paris, t. 303, Series II, No. 13 (1986) 1169.
4. H. Cabannes, "Hydrodynamics of Lattice Gases"
Computational Fluid Dynamics, (1988) 3.
5. D. Rothman, S. Zaleski, "LatticeGas Cellular Automata"
Cambridge University Press (1997)
6. A. Gunstensen, D. Rothman,
"A GalileanInvariant Immiscible LatticeGas"
Physica D 47 (1991) 5363.
