### Lattice-Boltzmann runs for July 3, 2001.

Lattice-Boltzmann simulation starting from a random distribution of velocities at a density of approx 2.2 particles per site. No driving dynamics, the system is at rest.

System in periodic tube boundary conditions, i.e. periodic in x and bounded by a wall in y. The collision rule for the wall is one-step bounce-back condition.

 Size in x: 256 Size in y: 64 Steps: 22224 Particles: 34288.1 Particles per non-obs site: 2.16029 Total P: 1.32339e-7 Px: 1.27847e-7 Py: 3.41846e-8 Tau = 1 Omega = 1 Mu = 0.16667 Mass = 1

#### GRAPHS

The color coding for all the graphs is relative, i.e. everything is normalize between the maximum and minimum of the existing values at that particular moment.

Graph for the total Momentum, sum over all particle momentums per site.

(Clickable pictures)

Flow diagram of a portion from the center of the box.

Momentum in X and Y.

To show that close to the wall (black) the momentum drops to zero.

Graph of the X and Y components of the momentum as a function of time.

Distribution of velocities for the X and Y components of the velocity.

#### DISCUSSION

The first surprise is that the system evolves from a random distribution to patches of collective velocities that indicate a velocity "segregation", much like waves or bodies of masses that move collectively. The second surprise is that the velocity in the y direction shows initial oscillations (where y is bounded by the walls) whereas in the x direction it just decays to a value.

The oscillations in y have to be due to the walls. The time distance between a lower and an upper point in the oscillation (half a period) is approx 105 time steps. Since the sound velocity is 1/sqrt(3) = 0.577 and the size in y is Ly = 64, this corresponds to traversing Ly in 64/.577 = 110.85, which agrees well. An additional question is the decay time of t = 10,000 for the y oscillations to reach the "noise" level. This slow decay may be due to mixing of the x and y momentums (scattering).

The graph for the velocity distributions in x and y shows interesting phenomena. First, the central velocity for both distributions is at 0. This is because the system is on average at rest (initial conditions are of random velocities). The y direction distribution shows a peaked distribution, while in x there are two humps that appear to be formed when the central peak (from the boundary condition of Vx(y=0,L) = 0) reduces the central part from the Maxwell distribution. Presumably for "slip" boundary conditions this distribution would be closer to Maxwell's.

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